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Dynamics of C^1-perturbed  non-autonomous systems with strong cone conditions and its applications

Dun ZhouPartially supported by NSF of China No.11971232, 12071217, 11601498.

School of Mathematics and Statistics

Nanjing University of Science and Technology

Nanjing, Jiangsu, 210094, P. R. China

Herein, we consider the dynamics of $C^1$-perturbed almost periodic systems with strong cone conditions in the sketch of skew-product semiflows. With some suitable dissipative assumptions, we investigate the asymptotic behaviors of any orbit of such perturbed systems and we obtain that any omega-limit set generated by the skew-product semiflow of the perturbed system contains at most two minimal sets, and any minimal set is an almost automorphic extension of its base flow. These results reveal that the structures of omega-limit sets and minimal sets for unperturbed systems can carry out to the systems with small $C^1$-perturbations. Furthermore, we apply these theoretic results to some typical models, such as scalar parabolic equations with separated boundary conditions and competitive-cooperative
tridiagonal systems.

§ INTRODUCTION

Perturbations exist widely in all kinds of models that built-in physics, chemistry, biology, engineering, etc. In practical problems, the systems we consider are usually very complicated and difficult to study. Using some relative simple systems to approximate those complicated systems is a common and effective research method. To certain extent, those complicated systems can be viewed as small perturbations of certain simple systems. However, the approximation method can be applied effectively only if certain properties of the simple system are maintained under small perturbations. Therefore, it is of great practical significance to study the stability of various systems under small perturbations.

There are many works on the dynamics of perturbed equations in some special regions, e.g,  near a stationary solution, a periodic solution, on an invariant manifold of the unperturbed system. Our goal is to study the global dynamics of perturbed equations such as the structure of any minimal invariant set, or any  $\omega$-limit set of the perturbed system under the assumption that the unperturbed system
is of certain special structure, say, it   possesses  a type of Lyapunov functional  or satisfies certain strong cone conditions.

To provide some insights to systems with strong cone conditions(this structure exists in some typical systems [2, 11, 13, 15, 16, 51, 26, 27, 28, 29, 32, 30] ), consider the following scalar reaction diffusion equation with Neumann boundary condition,

\begin{equation}\label{parabolic-eq-01}
\begin{cases}
u_{t}=u_{xx}+f(t,x,u,u_{x}),\,\,t>0,\, 0<x<1\cr
\end{cases}
\end{equation}

where $f(t,x,u,u_x)$ is a smooth function.
The asymptotic dynamics of (<ref>) has been extensively studied and is quite well understood when $f$ is time independent,
time periodic, or time almost periodic. For example, if $f$ is periodic in $t$ with period $T$, then any bounded solution of (<ref>) converges to a time $T$-periodic solution. In the case that $f$ is almost periodic in $t$,
Shen and Yi [46, 47, 49, 48] systematically studied the asymptotic dynamics of (<ref>) in terms of the skew-product semiflow generated by (<ref>). They proved that the flow on any minimal set $M$ of the skew-product semiflow is an almost automorphic extension of the base flow, and if the induced flow on $M$ is uniquely ergodic, then it can be embedded into  an almost periodically
forced minimal flow on $\RR$ (see [49]).

It is natural to ask whether the above mentioned results for (<ref>) also hold for various perturbed equations of (<ref>). We mention two perturbed equations of (<ref>) in the following.

The first one is  a non-local perturbation of (<ref>) in the following form:

\begin{equation}\label{parabolic-eq-induced}
\begin{cases}
u_{t}=u_{xx}+f(t,x,u,u_{x})+\epsilon c(t,x)\int_0^1\nu(x)u(t,x)dx\,\,\,t>0,\, 0<x<1\\
u_x(t,0)=u_x(t,1)=0\quad \text{or}\ u(t,0)=u(t,1)=0
\end{cases}
\end{equation}

where $\epsilon$ is constant, $c(t,x)$, $\nu(x)$ are continuous functions and $c(t,x)$ is uniformly almost periodic in $t$. There are several reasons for considering (<ref>), for instance, if $\epsilon, c(t,x)>0, \nu(x)\geq 0$, then $u$ in (<ref>) can denote the distribution of temperature along the given wire in a electric ballast resistor(see [4, 12] for the explanation of other terms in (<ref>)); in population biology, $u$ denotes a population density, $f$ denotes the ratio of birth/death, and the integration term represents a coupling, by which the total population affects each individual(see [12]).

The second one is  also a nonlocal perturbation of (<ref>) due to the influence of some chemical substance,

\begin{equation}\label{parabolic-eq-03-1}
\begin{cases}
u_{t}=u_{xx}-\epsilon v_x u_x-\epsilon(v-u)u+f(t,x,u,u_{x})\,\,\,t>0,\, 0<x<1\cr
\end{cases}
\end{equation}

where $v$ is the solution of

\begin{equation}\label{parabolic-eq-03-2}
\begin{cases}
0=v_{xx}-v+u,\quad 0<x<1\cr
\end{cases}
\end{equation}

Observe that (<ref>) and (<ref>) can be combined  into the following chemotaxis system,

\begin{equation}\label{parabolic-eq-03}
\begin{cases}
u_{t}=u_{xx}-\epsilon(u v_x)_x+f(t,x,u,u_{x})\,\,\,t>0,\, 0<x<1\cr
0=v_{xx}-v+u,\quad 0<x<1\cr
\end{cases}
\end{equation}

where $u$ denotes the density of a species and $v$ is the density of some chemical substance, the cross-diffusion
term $-\chi(u v_x)_x$
reflects  the chemotactic movement.
Chemotaxis models which used to describe the aggregation of certain types of bacteria was introduced by Keller and Segel in [24, 25]. Thereafter, a variety of models with chemotaxis terms have been proposed in different disciplines, such as entomology,  material sciences, “gravitational collapse” in astrophysics,etc. The parabolic-elliptic chemotaxis model (<ref>) was first
introduced by Jäger and Luckhaus in [21], and later
studied by many other authors (see [8, 20, 33, 34, 35], etc.) in different cases.

Given a $C^{1}$-smooth function $u:[0,1]\rightarrow \mathbb{R}$, the zero number of $u$ is
defined as

$$Z(u(\cdot))={\rm card}\{x\in [0,1]|u(x)=0\}.$$

Note that the behavior of the difference of any two classical solutions plays an important role in the study of
the asymptotic of (<ref>), and
the proofs of the  results mentioned  in the above for (<ref>) heavily rely on the following zero number or lap number property for solutions of (<ref>):

Suppose that $u_1(t,x)$ and $u_2(t,x)$ are two classical solutions on $(0,\infty)$. Let $v(t,x)$ be a nonzero classical  solution of

\begin{equation*}
\begin{cases}
v_t=v_{xx}+a(t,x)v_x+b(t,x)v,\quad t>0,\,\, 0<x<1 \\
v_x(t,0)=v_x(t,1)=0,\quad t>0,
\end{cases}
\end{equation*}

\begin{equation*}\begin{cases}
a(t,x)=\int_0^ 1 \partial_4 f(t,x, u_1(t,x), u_{2x}(t,x)+s(u_{1x}(t,x)-u_{2x}(t,x)))ds,\cr
b(t,x)=\int_0^ 1 \partial_3
\end{cases}
\end{equation*}

Then the zero number, denoted by $Z(t;v)$, of $v(t,x)$ on $[0,1]$ is  finite and noncreasing in $t\in (0,\infty)$, and if $Z(t;v)$
drops at $t_0\in (0,\infty)$,  then $v(t_0,x)$ has a multiple zero in $[0,1]$ and

Z(t_0-\epsilon;v)\ge Z(t_0+\epsilon;v)+1\quad \forall\, 0<\epsilon\ll 1.

(Note that if $v(0,x)=u_1(0,x)-u_2(0,x)$, then $v(t,x)=u_1(t,x)-u_2(t,x)$ for all $t>0$).

However, $Z$ highly depends on the form of the equation (<ref>), to be more specific, the properties of $Z$ for (<ref>) can not be inherited by of (<ref>) and (<ref>). As a consequence, it is unclear
whether the dynamics for minimal sets and $\omega$-limit sets of (<ref>) can be carried over to (<ref>) and (<ref>) when $0<\epsilon\ll 1$.
To compensate for this, we describe the above mentioned zero number property for solutions of (<ref>) in terms of geometrical method, which is called strong cone conditions, see [54].

In the following, we assume that  all classical solutions of (<ref>) exist globally and are bounded(it is correct when (<ref>) satisfies some dissipative assumptions).
Let $X^\alpha$ be a fractional power space associated with the operator $u\rightarrow
-u_{xx}:H^{2}(0,1)\rightarrow L^{2}(0,1)$ with domain $\{u\in H^2(0,1)|u_x(0)=u_x(1)=0\}$
satisfies $X^\alpha \hookrightarrow C^{1}[0,1]$. Let $H(f)$ be the hull of $f$. Then (<ref>) generates a skew-product semiflow $\Lambda_t$ or $X^\alpha \times H(f)$,

\Lambda_t (u_0,g)=(\phi(t,x;u_0,g),g\cdot t),

where $\phi(t,x;u_0,g)$ is the solution of

\begin{equation}\label{parabolic-eq-01-g}
\begin{cases}
u_{t}=u_{xx}+g(t,x,u,u_{x}),\,\,t>0,\, 0<x<1\cr
\end{cases}
\end{equation}

with  $\phi(0,\cdot;u_0,g)\in X^\alpha$ and $g\in H(f)$,  $g\cdot t(\cdot,x,u,u_x)=g(\cdot+t,x,u,u_x)$.

A closed subset $C$ of  $X^\alpha $ is a $k$-cone  if  $\lambda C=C$ for all $\lambda\in \mathbb{R}\setminus\{0\}$ and there are $k$-dimensional space $V_0$ and $k$-codimensional space $X_0$ such that $V_0\subset C$ and $X_0\cap C=\{0\}$.
For any $i\in \{1,\cdot,n,\cdots\}$, define the following set

\begin{equation}
\label{parabolic-cone-eq}
C_i:=cl\{u\in X^\alpha :u(\cdot)\ \text{has only simple zero and }Z(u(\cdot))\leq i-1\}
\end{equation}

and put $C_0=\{0\}$. Then $C_i$ is a $i$-cone  in $X^\alpha$ (see Lemma <ref>).
Define the map $\pi$ by

\[
u(\cdot)\longmapsto u(0),\quad X^\alpha \to \mathbb{R},
\]

and  let

\begin{equation}
\label{parabolic-hyper-plane-eq}
\mathcal{H}=\pi^{-1}(0).
\end{equation}

Then, $\mathcal{H}$ consists of functions that equal $0$ at $x=0$. Since $\pi\in X^{\alpha *}\setminus \{0\}$ ($X^{\alpha *}$ is the dual space of $X^\alpha$), $\mathcal{H}$ is a $1$-codimensional subspace of $X^\alpha$.

For any $z=(u_1,u_2, g)\in X\times X\times H(f)$, consider the following linear parabolic equation:

\begin{equation}\label{linear-para-eq-00}
\begin{cases}
v_t=v_{xx}+a^z(t,x)v_x+b^z(t,x)v,\quad t>0,\,\, 0<x<1 \\
v_x(t,0)=v_x(t,1)=0,\quad t>0,
\end{cases}
\end{equation}

\begin{equation*}
\begin{cases}
a^z(t,x)=\int_0^ 1 \partial_4 g(t,x,\phi(t,x;u_1,g), \phi_x(t,x;u_2,g)+s(\phi_x(t,x;u_1,g)-\phi_x(t,x;u_2,g)))ds\cr
b^z(t,x)=\int_0^ 1 \partial_3
\end{cases}
\end{equation*}

Let $T(t,u_1,u_2,g)$ be the solution operator of (<ref>).  Observe that

T(t,u_1,u_2,g)=\int_0^1 \phi(t,\cdot;u_2+s(u_1-u_2),g)ds

Moreover,  $T(t,u_1,u_2,g)$ satisfies  the following properties:

(i) $T(t_1+t_2,u_1,u_2,g)=T(t_2,\phi(t_1,\cdot;u_1,g),\phi(t_1,\cdot;u_2,g),g\cdot t_1)\circ T(t_1,u_1,u_2,g)$ for any
$t_1,t_2\ge 0$, $\lim_{t\to 0}T(t,u_1,u_2,g)v=v$ for any $v\in X^\alpha$, and  the function $(0,\infty)\times X^\alpha\times X^\alpha\times H(g)\ni (t, u_1,u_2,g)\mapsto T(t,u_1,u_2,g)\in L(X^\alpha,X^\alpha)$ is continuous, where $ L(X^\alpha,X^\alpha)$ is the space of bounded linear maps from $X^\alpha$ to $X^\alpha$.

(ii) $T(t,u_1,u_2,g)\in L(X^\alpha, X^\alpha)$ is compact for any $t>0$.

(iii) For each $t>0$ and $v\in X^\alpha\setminus \{0\}$, $T(t,u_1,u_2,g)v\in C_i\setminus\{0\}$ for some $i\in\NN$.

(iv) For exach $t>0$ and $v\in X^\alpha \setminus\{0\}$, there is an open neighborhood $O$ of $v$ such that
$T(t,u_1,u_2,g) O\subset C_i\setminus \{0\}$.

(v) For each $t>0$, if $ T(t, u_1,u_2,g)v \in \left( \left( C _ { i } \backslash C _ { i - 1 } \right) \cap \mathcal{H} \right)\setminus\{0\}$  for an $i\ge 1$,  then $v \notin C _ { i }$.

Observe that properties (i)-(ii) follow from the regularity and a priori estimates for parabolic equations, and properties (iii)-(v) follow from the property (a) described in the above. It is clear that properties (i)-(ii) can be carried over to (<ref>) and (<ref>) when $0<\epsilon\ll 1$.  It is then natural to ask whether properties (iii)-(v) are preserved under small perturbations in the sense that there are  sequence of cones $\tilde C_i\subset X^\alpha$ and  one-codimensional
subspace $\mathcal{\tilde H}$ such that (iii)-(v) with $C_i$ and $\mathcal{H}$ being repalced  by $\tilde C_i$ and $\mathcal{\tilde H}$ hold for (<ref>) and (<ref>)
when $0<\epsilon\ll 1$, and if so, whether (<ref>) and (<ref>) have similar asymptotic
dynamics as (<ref>).

The answers to the above questions are yes by Tereščák's work [54] when $f$ is time independent or periodic and $c$ is time independent or the $f$ and $c$ share the same period. However, it can't say anything, when $f$ in $t$ is non-periodic, or even if $f$, $c$ of $t$ in (<ref>) are periodic but their periods are irrationally dependent. As a matter of fact, the non-periodic cases are widespread in the systems we've encountered.

Motivated by the above questions, we investigate the dynamics of $C^1$-perturbed almost periodic systems with strong cone conditions. We first establish some general theory for $C^1$-perturbed almost periodic skew-product semiflows with strong condition conditions(see Theorems <ref>,<ref>,<ref>) and then apply the theory to some special almost periodic differential equations, including (<ref>) and almost periodic competitive-cooperative tridiagonal systems(see Theorems <ref>,<ref>, <ref>). In short, for those almost periodic systems we considered, the structure of $\omega$-limit set and minimal set for unperturbed systems can carry over to the systems with small $C^1$-perturbations.

The rest of the paper is organized as follows.  In section 2, we introduce some conceptions and properties of skew product semiflows, almost-periodic (almost-automorphic) functions (flows), cone and exponential separations; and we then present our main results related to skew-product semiflows of non-autonomous systems with strong cone conditions, and characterize the structure of minimal sets as well as limit sets for those perturbed systems with $1$-codimension subspaces. In section 3,  proofs of our main results are given based on Proposition <ref>. In section 4, under some suitable dissipative assumptions, we apply our main results to study the dynamics of (<ref>), (<ref>) and competitive-cooperative
tridiagonal systems. In section 5, we give some remarks on our setting as well as on the main results . Finally, we give a detail proof for Proposition <ref>.

§ DEFINITIONS,  ASSUMPTIONS,  AND STATEMENTS OF  THE MAIN RESULTS

Through out this section, we
assume that $(X,\|\cdot\|)$ is a Banach space, $Y$ is a metric space, $\Theta$ and $Z$ are  compact metric spaces, and  $\mathcal{ E } = X \times \Theta$.

§.§ Definitions: skew-product semiflows, exponential separations, almost periodic (almost automorphic) functions and flows

In this subsection, we introduce the definitions of cones,  skew-product semiflows, exponential separations, almost periodic (almost automorphic) functions and flows, and recall some basic properties.

First,  we recall that  a closed subset $C$ of $X$ is a $k$-cone  if and only if $\lambda C=C$ for all $\lambda\in \mathbb{R}\setminus\{0\}$ and there are $k$-dimensional space $V_0$ and $k$-codimensional space $X_0$ such that $V_0\subset C$ and $X_0\cap C=\{0\}$. A cone is called solid if it has non-empty interior.

Next, we introduce the definitions of skew-product semiflows and exponential separations.

A continuous function
$\sigma:Y\times \RR\to Y, (y,t)\mapsto y\cdot t$, denoted by $(Y,\sigma)$ or $(Y,\RR)$ is called  a continuous flow on $Y$ if it satisfies that  $\sigma(y,0)=y$ and  $\sigma(\sigma(y,s),t)=\sigma(y,s+t)$ for all $y\in Y$, $s,t\in\mathbb{R}$. A subset
$\tilde Y\subset Y$ is invariant if $\sigma_t(\tilde Y)=\tilde Y$ for every $t\in
\RR$. A subset $\tilde Y\subset Y$ is called minimal if it is
compact, invariant and the only non-empty compact invariant subset
of it is itself.  Every compact and $\sigma$-invariant set contains a minimal subset and a subset $\tilde Y$ is minimal if and only if every trajectory is
dense. The continuous flow $(Y,\sigma)$ is called
recurrent or minimal if $Y$ is minimal.

The following lemma  follows  from <cit.>.

Let $p:(Z,\mathbb{R})\rightarrow(\Theta,\mathbb{R})$ be an epimorphism of flows. Then the set

$\Theta'=\{\theta_{0}\in \Theta: for \ any\ z_{0}\in p^{-1}(\theta_{0}),\ \theta\in \Theta \ and\ any\ sequence\
\{t_{i}\}\subset \RR \ with\ \theta\cdot t_{i}\rightarrow \theta_{0},\ there\ is\ a\ sequence\
\{z_{i}\}\in p^{-1}(\theta)\ such\ that\ z_{i}\cdot t_{i}\rightarrow z_{0} \}$

is residual and invariant. In particular, if $(Z,\mathbb{R})$ is minimal and distal, then $\Theta'=\Theta$.

Let $p:(Z,\mathbb{R})\rightarrow(\Theta,\mathbb{R})$ be a homomorphism of minimal flows. $(Z,\mathbb{R})$ is said to be an almost $1$-cover or almost automorphic extension of $(\Theta,\mathbb{R})$ if there is $\theta_0\in \Theta$ such that card $p^{-1}(\theta_0)=1$.

Suppose that $\sigma ( \theta , t ) = \theta \cdot t$ is a compact flow on $\Theta$.  A
skew-product semiflow $\Lambda_t:\mathcal{ E } \rightarrow \mathcal{ E } $ is a semiflow
of the following form

\begin{equation}\label{skew-product-semiflow}
\Lambda_{t}(x,\theta)=(\phi(t,x,\theta),\theta\cdot t),\quad t\geq 0,\, (x,\theta)\in \mathcal{ E }.
\end{equation}

We always write the above defined skew-product semiflow as $\Lambda=(\phi,\sigma)$.

Let  $L(X,X)$ be the set of all bounded linear operators on $X$. Suppose $(Z,\sigma)$ is a compact flow and
let $\left\{ T(t,z) : t > \right.$
$0 , z \in Z \}$ be a family of bounded linear maps on a Banach space $X$ with the
following properties:

$\bullet$  $\lim_{t\to 0+}T(t,z)x=x$ for all $z\in Z$ and $x\in X$.

$\bullet$  the function ${ (t,z)} \mapsto T(t,z) : (0, + \infty )\times Z   \rightarrow L ( X , X )$ is continuous

$\bullet$  $T(t_1+t_2,z) = T(t_1,z\cdot t_2) \circ T(t_2,z)$ for all $t _ { 1 } , t _ { 2 } > 0$ and $z \in Z$.

$\{T(t,z)\}$  a linear skew product semiflow on $X\times Z$. We denote it by $\bar\Lambda=(T,\sigma)$.

Let $K=X\times Z$, a subset $\tilde  {K}=\bigcup_{z\in Z}\tilde K(z)=\bigcup_{z\in Z}X_z\times \{z\}$ of $K$ is called a $k$-dimensional vector subbundle of $K$ on $Z$, if for each $z\in Z$, $X_z\subset\ X$ is a $k$-dimensional subspace of $X$.

A linear skew-product semiflow $\bar\Lambda=(T, \sigma)$ on $X\times Z$ is called admits a $k$-dimensional
continuous separation along $Z$ associated with a $k$-cone $C$ if there are $k$-dimensional subbundles $ \cup_{z\in Z}  V _z \times \{z\}$ ($V_z\subset C$) and $\cup_{z\in Z}L _ { z }  \times \{z\}$ of  $X\times Z$ and
$X^{*}\times Z$ ($X^*$ is the dual space of $X$), respectively,  such that the following properties are satisfied:

(1) (Continuity) $ \cup_{x\in Z}V _ { z }\times \{z\}$ and $\cup_{z\in Z}L _ { z }\times
\{z\}$ are continuous bundles, i.e., for any convergent sequence $z_n\in Z$, $x_n\in V_{z_n}$ (resp. $x^*_n\in L_{z_n}$) satisfying $z=\lim_{n\to\infty} z_n$ and $x=\lim_{n\to\infty}x_n$ (resp. $x^*=\lim_{n\to\infty}x^*_n$), then $x\in V_z$ (resp. $x^*\in L_z$).

(2) (Invariance) $T{ (t,z)}V _ { z} = V _ {z\cdot t }$ and $T^ { * }{(t,z)} L _ { z\cdot t} = L _ { z}$ for all $t > 0 , z \in Z$, where $T^ { * }{(t,z)}$ is the adjoint operator of $T{(t,z)}$.

(3)(Exponential separation) There are constants $M,\gamma > 0$ such that

\[
\|T{ (t,z)}w\|\leq M e^{-\gamma t}\|T{(t,z)}v\|
\]

for all $z \in Z$, $w \in \operatorname{Anih} \left( L _ { z } \right) \cap S , v \in V _ { z } \cap S$ and $t \geq 1$, where $\operatorname{Anih}(L_z)=\{v\in X:l(v)=0 \ \text{for all}\ l\in L_z\}$ ($ \operatorname{Anih} \left( L _ { z } \right)\cap C=\{0\}$) and $S=\{u: \|u\|=1, u\in X\}$.

The following exponential separation lemma is from <cit.>.

Let $(X,\|\cdot\|)$ be a Banach space, $C$ be a $k$-cone in $X$, $\bar\Lambda=(T,\sigma)$ be a linear skew-product semiflow on $X\times Z$. Assume that  $T(t,z)$ is compact for each ${ (t,z)}\in (0, + \infty )\times Z$, and for any given $v \in C \backslash \{ 0 \}$ and  ${ (t,z)}\in (0, + \infty )\times Z$, there is an open neighborhood of $\mathcal{N}(t,z,v)$ of $v$ in $X$ such that
$T(t,z) \mathcal{N}\subset C \backslash \{ 0 \}$. Then the skew-product semiflow $\bar\Lambda$ admits a $k$-dimensional continuous separation along $Z$ associated with the $k$-cone $C$.

We now introduce the definition of almost periodic (almost automorphic) functions and flows.

Assume that $D$ is a nonempty subset of $\RR^m$. We list some definitions and properties related to almost periodic (automorphic) functions (flows) in the following.

A function $f\in C(\RR\times \mathbb{R}^n,D)$ is said to be admissible if for any compact subset $K\subset \mathbb{R}^n$, $f$ is bounded and uniformly continuous on
$\RR\times K$. $f$ is $C^r$ ($r\ge 1$)  admissible if $f(t,w)$ is $C^r$ in $w\in \mathbb{R}^n$ and Lipschitz in $t$, and $f$ as well as its partial derivatives to order $r$ are admissible.

Let $f\in C(\RR\times \mathbb{R}^n,D)$  be an admissible function. Then
$H(f)={\rm cl}\{f\cdot\tau:\tau\in \RR\}$ (called the hull of
$f$) is compact and metrizable under the compact open topology (see [39, 50]), where $f\cdot\tau(t,\cdot)=f(t+\tau,\cdot)$. Moreover, the time translation $g\cdot t$ of $g\in H(f)$ induces a natural
flow on $H(f)$ (cf. [39]).

(1) A function $f\in C(\RR, D)$ is almost periodic if for every $\{t'_k\}\subset\mathbb{R}$ there is a subsequence $\{t_k\}\subset \{t'_k\}$
such that $\{f(t+t_k)\}$ converges uniformly in $t\in\RR$.

(2) $f\in C(\RR,D)$ is almost automorphic if for every $\{t'_k\}\subset\mathbb{R}$ there is a subsequence $\{t_k\}\subset \{t'_k\}$
and a function $g:\mathbb{R}\to \mathbb{R}$ such that $f(t+t_k)\to g(t)$ and $g(t-t_k)\to f(t)$ for each $t\in\RR$.

(3)  A function $f\in C(\RR\times \mathbb{R}^n,D)$ is uniformly almost periodic in $t$ (resp. uniformly almost automorphic in $t$), if $f$ is both admissible and, for each fixed $d\in \mathbb{R}^n$, $f(t,d)$ is almost periodic (resp. almost automorphic)  with respect to $t\in \RR$.

Assume that $(X,\mathbb{R})$ be a compact flow. For $x\in X$ and a net $\alpha=\{t_n\}$ in $\mathbb{R}$, define $T_{\alpha}x=\lim_{n\to\infty}x\cdot t_n$, if the limit exists.

A compact flow $(X,\mathbb{R})$ is called almost periodic, if any nets $\alpha'$, $\beta'$ in $\mathbb{R}$ have subnets $\alpha$, $\beta$ such that

for all $x\in X$, where $\alpha+\beta=\{t_n+s_n\}$, $\alpha=\{t_n\}$, $\beta=\{s_n\}$.

Let $(X,\mathbb{R})$ be a compact flow. A point $x\in X$ is an almost automorphic point if any net $\alpha'$ in $\mathbb{R}$ has a subnet $\alpha=\{t_n\}$ such that $T_{\alpha}x$, $T_{-\alpha}T_{\alpha}x$ exist and $T_{-\alpha}T_{\alpha}x=x$. $(X,\mathbb{R})$ is called almost automorphic, if there is an almost automorphic point $x_0\in X$ with dense orbit.

Assume that $f$ is a uniformly almost periodic (automorphic) function in $t$. Then the following hold.

(1) $H(f)$ is always minimal, and $(H(f),\mathbb{R})$ is an almost periodic (automorphic) minimal flow.

(2)  $g$ is a uniformly almost periodic (automorphic) function for all (residually many) $g\in H(f)$.

(3) There is a unique $F\in C(H(f)\times \mathbb{R}^n, V)$ such that $F(g\cdot t,z)\equiv g(t,z)$; moreover, if $f$ is $C^r$ admissible, then $F$ is $C^r$ in $z\in \mathbb{R}^n$ and Lipschitz in $g\in H(f)$(see, e.g. [50]).

Before ending this subsection, we introduce the so called $C^1$-functions on a subset of $X$.  Let $B\subset X$ be a bounded open set, define

C^1(B,X):=\{F:B\to X:F \text{ is continuously  Fr\'{e}chet differetiable with bounded derivative on } B\}

with the norm

\|F\|_{C^1(B,X)}=\sup_{x\in B}\|F(x)\|_X+\sup_{x\in B}\|DF(x)\|_{L(X,X)}.

§.§ Assumptions

In this subsection, we  introduce  the standing assumptions.

Let $X$ be a Banach space and  $L(X,X)$ be the space of bounded linear operators on $X$.
Assume that there exists a sequence of solid $k _ { i }$-cones  satisfying $C _ { 0 } = \{ 0 \} , C _ { N } = X$ and $C _ { i } \supset C _ { i - 1 } , k _ { i } > k _ { i - 1 }$ for all $1 \leq i < N$, where $N \in \mathbb { N }\cup \{+ \infty \}$. Let $(\Theta,\sigma)$ be an almost periodic minimal flow  and $P : X \times \Theta \rightarrow \Theta$ be the natural projection.
$\mathcal{U}_0\subset X\times \Theta$ be an open set with $P(\mathcal{U}_0)=\Theta$. Let  $\Lambda_t=(\phi(t,\cdot,\theta),\theta\cdot t)$($t>0$) be a skew-product semiflow on $\mathcal { U } _ { 0 }$,
which is $C^1$ in $x$. Our  objective is to study the asymptotic  dynamics of $C^1$-perturbed skew-product semiflows
of $\Lambda_t$ on $\mathcal{U}_0$.

Define $\mathcal{U}_0(\theta)=\{x|(x,\theta)\in\mathcal{U}_0\} $ for each $\theta\in \Theta$.
Denote $\mathcal{U}_0^e=\bigcup_{\theta\in\Theta}\{(x,y,\theta)|x,y\in\mathcal { U}_0(\theta)\}$.
For given $t\ge 0$,  $(x,y,\theta)\in \mathcal{U}_0^e$, let  $T(t,x,y,\theta)\in L(X,X)$ be defined by

\begin{equation}
\label{operator-T-eq0}
T(t,x,y,\theta)=\int_0^1 \p_2 \phi(t,y+s(x-y),\theta)ds.
\end{equation}

Observe that

\begin{equation}
\label{operator-T-eq1}
T(t,x,y,\theta)( x - y ) = \phi(t,x,\theta) - \phi(t,y,\theta)\quad \forall\, t\ge 0,\,\, (x,y,\theta)\in \mathcal{U}_0^e.
\end{equation}

We point out that the behavior of the difference of two orbits of $\Lambda_t$ plays
an important role in the study of the asymptotic dynamics of $\Lambda_t$.

The following are the standing assumptions on $T(t,x,y,\theta)$.

(H1) $T(t_1+t_2,x,y,\theta)=T(t_2, \phi(t_1,x,\theta),\phi(t_1,x_2,\theta),\theta\cdot t)\circ T(t_1,x,y,\theta))$ for all $t_1,t_2\ge 0$, $x,y\in X$, and $\theta\in\Theta$,  $\lim_{t\to 0}T(t,x,y,\theta)v=v$ for each $v\in X$, and the function $(t, x, y, \theta) \mapsto T(t,x,y,\theta) : (0, 1] \times\mathcal { U }_{ 0 }^e \rightarrow L ( X , X )$ is continuous

(H2) $T(t,x,y,\theta)$ is compact operator for any $t\in [\frac{1}{2}, 1]$.

(H3) For each $t>0$, $v \in X \backslash \{ 0 \}$ implies $T(t,x,y,\theta)v \in C _ {i} \backslash \{ 0 \}$ for some $i \in \mathbb { N }$.

(H4) For each $t>0$, if $ T(t,x,y,\theta)( C _ { i } \backslash \{ 0 \})\subset \operatorname{int}( C _ { i })$.

(H5) There exists a $d$-codimension linear subspace $\mathcal{H}$ of $X$ such that, if $ T(t,x,y,\theta)v \in \left( \left( C _ { i } \backslash C _ { i - 1 } \right) \cap \mathcal{H} \right)\setminus\{0\}$ for $t>0$ and  $1 \leq i \leq N$,  then $v \notin C _ { i }$ (if $N=\infty$, we require $1 \leq i < N$).

(1)   (H1) and (H2) are two basic assumptions on
$\Lambda_t$.   Assumptions (H3)-(H5) are referred to as strong cone conditions for $\Lambda_t$. By (H1) and (H3), if $x\not =y$, then for each  $t>0$, $\phi(t,x,\theta)-\phi(t,x,\theta)\in C_i\setminus\{0\}$ for some $i\in\NN$.
By (H4), $C_i$ is invariant under $T(t,x,y,\theta)$. (H5) implies that  if $x-y\in C_i\setminus\{0\}$, then for each $t>0,$ either $\phi(t,x,\theta)-\phi(t,y,\theta)\in \mathcal{H}\cap C_{i-1}$ or $\phi(t,x,\theta)-\phi(t,y,\theta)\in C_i\setminus{(\mathcal{H}\cup C_{i-1})}$; furthermore, combine with (H4) and by repeating the process, $\phi(t,x,\theta)-\phi(t,y,\theta)$ can stay in $\mathcal{H}$ for finite many times, that is, there exist $t_0>0$ and $i_0\in {1,\cdots,i}$ such that $\phi(t,x,\theta)-\phi(t,y,\theta)\in \operatorname{int}( C_{i_0})\setminus{(\mathcal{H}\cup C_{i_0-1})}$ for all $t>t_0$, and it helps us to reduce the system into a $d$-dimensional system.

(2) Let $X^\alpha$ be a fractional power space associated with the operator $u\rightarrow
-u_{xx}:H^{2}(0,1)\rightarrow L^{2}(0,1)$ with domain $\{u\in H^2(0,1)|u_x(0)=u_x(1)=0\}$
satisfies $X^\alpha \hookrightarrow C^{1}[0,1]$. Let $H(f)$ be the hull of $f$. Let $\{T(t,u_1,u_2,g): t\ge 0,u_1,u_2\in X^\alpha, g\in H(f)\}$ be the family of bounded linear operator in $X^\alpha$ generated by  (<ref>).
Let $C_i$  $(i=1,2,\cdots$) and $\mathcal{H}$ be as in (<ref>) and (<ref>), respectively. Then $T(t,u_1,u_2,g)$ satisfies  (H1)-(H5)  with $X=X^\alpha$, $\Theta=H(f)$,  $\mathcal{U}_0=X^\alpha\times H(f)$.

(3) When $(\Theta,\sigma)$ is a $T$-periodic flow.
Under the  assumptions (H1)-(H5),  Tereščák in [54] proved that  any $\omega$-limit set of the associated Poincaré map  can be embedded into a $d$-dimension plane  (<cit.>). Particularly, if $d=1$,  any  $\omega$-limit of the associated Poincaré map is a fixed point(<cit.>).

§.§ Statements of main results

In this subsection, we state the main general results of the paper.

Suppose that (H1)-(H4)  hold. Let $K _ { 0 } \subset \mathcal { U }_ { 0 }$ be a compact, invariant set under $\Lambda_{ t }$. For any given $\theta\in \Theta$, let $K_0(\theta)=\{x|(x,\theta)\in K_0\}$. Then there are numbers $N_{0} \in \mathbb{N}$,  $\epsilon_{0}>0,$ open bounded neighborhood $\mathcal{V}_{0}(\theta)$ of $K_{0}(\theta)$ for each $\theta\in \Theta$ and a sequence of  $k_{i}$-cone $\mathcal{C}_{i}$ ($i=1,\cdots,N_0$) satisfying

$$\mathcal{C}_{0}=\{0\}\subset\mathcal{C}_{1} \subset \operatorname{int} \mathcal{C}_{2}\cup\{0\} \subset
\mathcal{C}_{2} \subset \ldots \mathcal{C}_{N_{0}-1} \subset \operatorname{int}\left(\mathcal{C}_{N_{0}}\right) \cup\{0\} \subset \mathcal{C}_{N_{0}}

such that for any skew-product semiflow $\tilde\Lambda_t=(\tilde \phi,\theta)$ on $\mathcal{U}_0$ with $\tilde \phi(t,\cdot,\theta) \in C^{ 1 } ( \mathcal { V }_0(\theta), X ) $ for $ t \in \left[\frac{1}{2}, 1 \right]$ and

\|\phi(t,\cdot,\theta) - \tilde \phi(t,\cdot,\theta) \| _ {C ^ { 1 } ( \mathcal { V }_0(\theta), X )} < \epsilon_0, \quad \frac{1}{2} \leq t \leq 1, \ \theta\in \Theta,

if the given two distinct points $x, y \in \mathcal { V } _ { 0 }(\theta)$ satisfy  $\tilde \phi(t,x,\theta), \tilde \phi(t,y,\theta)\in \mathcal{V}_0(\theta\cdot t)$  for all $t>0$, then one has the following:

(1) There is $T^*>0$ such that

\begin{equation}\label{exponential-decay}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \notin \mathcal { C } _ { N_0 },\quad t \geq T^*
\end{equation}
and $\|\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\|$ exponentially decays to zero;
or there exists some $i \in \{ 1,2 , \cdots, N_0\}$ such that

\begin{equation}\label{stayin-cone}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i }) \backslash \mathcal { C } _ { i - 1 },\quad  t \geq T^*.
\end{equation}

(2) For any two distinct points $x,y\in \mathcal { V } _ { 0 }(\theta)$, if $\tilde \phi(t,x,\theta),\tilde\phi(t,y,\theta)$ are defined for all $t\leq 0$, $\|\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\|\nrightarrow 0$ as $t\to -\infty$, $\mathrm{cl}\{(\tilde \phi(t,x,\theta),\theta\cdot t)\,:\, t\leq 0\}$ and  $\mathrm{cl}\{(\tilde \phi(t,y,\theta),\theta\cdot t)\,:\, t\leq 0\}$ ($\mathcal{V}_0=\bigcup_{\theta\in\Theta}\{(u,\theta)|u\in \mathcal{V}_0(\theta)\}$) are compact sets contained in $\mathcal{V}_0$, then, only (<ref>) will happen; moreover, there exist $T^{**}>0$ and an integer $1\leq i\leq i'\leq N_0$ such that

\begin{equation}\label{stayin-cone-negative}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 },\quad t \leq -T^{**}.
\end{equation}

(3) Assume moreover that (H5) is satisfied, if $\tilde \phi(t_0,x,\theta)-\tilde \phi(t_0,y,\theta)\in \mathcal{C}_{N_0}$ for some $t_0>0$, then $\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\notin \mathcal{H}$ for $t\gg 1$.

Suppose that (H1)-(H4) are hold. Let $E\subset \mathcal { V } _ { 0 }$ be a compact invariant set of the skew-product semiflow $\tilde \Lambda_t$, then, for any two distinct points $(x,\theta), (y,\theta) \in E$, there exist $T^*>0$ and some $i\in \{ 1,2 , \cdots, N_0 \}$ such that

\begin{equation*}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i }) \backslash \mathcal { C } _ { i - 1 },\quad  t \geq T^*.
\end{equation*}

If moreover $\|\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\|\nrightarrow 0$ as $t\to -\infty$, then there also exist $T^{**}>0$ and an integer  $i'$ with $ i\leq i'\leq N_0$ such that

\begin{equation*}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 },\quad t \leq -T^{**}.
\end{equation*}

Suppose that (H1)-(H5) are hold. Let $E\subset \mathcal { V } _ { 0 }$ be a compact invariant set of the skew-product semiflow $\tilde \Lambda_t$. If for any two distinct points $(x,\theta),(y,\theta)\in E$, there exists some $i\in \{1,\cdots,N_0\}$ such that $\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\in \operatorname{int} (\mathcal{C}_i)\setminus\mathcal{C}_{i-1}$ for all $t\in\mathbb{R}$. Then, the skew-product flow on $E$ is topologically conjugate to a skew-product flow on a compact invariant set $\hat E$ of $\mathbb{R}^d\times \Theta$.

In this case, the dynamics on $E$ can be reduced to a skew-product semiflow on an compact invariant set of  $d$-dimensional space.

(Almost $1$-covering property)

Assume moreover that $\mathcal{H}$ in assumption (H5) is of 1-codimension subspace of $X$.  Then for any  compact invariant minimal set
$\mathcal{\tilde M}\subset \mathcal{V}_0$ of $\tilde \Lambda_t$,  there is a residual subset $\Theta_0$ of $\Theta$, such that for any $\theta\in \Theta_0$, one has $P^{-1}(\theta)\cap \mathcal{\tilde M}$ is a single point.

Assume moreover that $\mathcal{H}$ in assumption (H5) is of 1-codimension subspace of $X$.
Suppose that for the given $(x,\theta)\in \mathcal{V}_0$, $\omega$-limit set $\tilde \Omega=\tilde\omega(x_0,\theta_0)$ of $\tilde \Lambda_t$ is compact and contained in $\mathcal { V }_0$, then $\tilde \Omega $ contains at most two minimal sets. Moreover, one of the following is true:

(1) $\tilde\Omega$ is a minimal invariant set.

(2) $\tilde\Omega=\mathcal{ \tilde M}_1\cup \mathcal{\tilde M}_{11}$, where $\mathcal{\tilde M}_1$ is minimal, $\mathcal{\tilde M}_{11}\neq \emptyset$, and $\mathcal{\tilde M}_{11}$ connects $\mathcal{\tilde M}_1$ in the sense that if $(x_{11},\theta)\in \mathcal{\tilde M}_{11}$, then $\mathcal{\tilde M}_1\subset\tilde\omega(x_{11},\theta)$ and $\mathcal{\tilde M}_1\subset\tilde\alpha (x_{11},\theta)$.

(3)  $\tilde \Omega=\mathcal{\tilde M}_1\cup \mathcal{\tilde M}_2\cup \mathcal{\tilde M}_{12}$, where $\mathcal{\tilde M}_1$, $\mathcal{\tilde M}_2$ are minimal sets, $\mathcal{\tilde M}_{12}\not =\emptyset$, and for any $u_{12}\in \mathcal{\tilde M}_{12}$, either  $\mathcal{\tilde M}_1\subset \tilde\omega(x_{12},\theta)$ and $\mathcal{\tilde M}_2\cap \tilde \omega(x_{12},\theta)=\emptyset$, or $\mathcal{\tilde M}_2\subset \tilde\omega(x_{12},\theta)$ and $\mathcal{\tilde M}_1\cap \tilde \omega(x_{12},\theta)=\emptyset$, or  $\mathcal{\tilde M}_1\cup \mathcal{\tilde M}_2\subset \tilde \omega(x_{12},\theta)$ (and analogous for $\tilde \alpha(x_{12},\theta)$).

Theorem <ref> and Theorem <ref> tell us that the structures of minimal sets and $\omega$-limit sets can keep for the skew-product semiflows with $C^1$-small perturbations. Note that if $\tilde \Lambda_t=\Lambda_t$, $\mathcal{V}_0=\mathcal{U}_0$ (usually $\mathcal{U}_0=X\times \Theta$), it is the un-perturbed case.

§ PROOF OF THE MAIN RESULTS

In this section, we give detailed proofs for our main results.

§.§ Proof of Theorem <ref> and its corollaries

Before proving Theorem <ref>, we list an important proposition which can be viewed as a generalization of <cit.> for discrete mapping to continuous skew-product semiflow.

Suppose that (H1)-(H4) are satisfied and $K_0(\theta)$ is as in Theorem <ref>. Then there are numbers $N_{0} \in \mathbb{N}$, $0<\lambda_0<1, T_{0},\epsilon_{0}>0,$ open bounded neighborhood $\mathcal{V}_{0}(\theta)$ of $K_{0}(\theta)$ for each $\theta\in \Theta$ and a sequence of  $k_{i}$-cone $\mathcal{C}_{i}$ ($i=1,\cdots,N_0$) satisfying

$$\mathcal{C}_{1} \subset \operatorname{int} \mathcal{C}_{2}\cup\{0\} \subset
\mathcal{C}_{2} \subset \ldots \mathcal{C}_{N_{0}-1} \subset \operatorname{int}\left(\mathcal{C}_{N_{0}}\right) \cup\{0\} \subset \mathcal{C}_{N_{0}}

such that for any skew-product semiflow $\tilde\Lambda_t=(\tilde \phi,\theta)$ on $\mathcal{U}_0$ satisfying that  $\tilde \phi(t,\cdot,\theta) \in C^{ 1 } ( \mathcal { V }_0(\theta), X )$ for  $t \in \left[\frac{1}{2}, 1 \right]$ and

\|\phi(t,\cdot,\theta) - \tilde \phi(t,\cdot,\theta)\| _ {C ^ { 1 } ( \mathcal { V }_0(\theta), X )} < \epsilon_0 \quad \text { for all }\,\,\, \frac{1}{2} \leq t \leq 1, \ \theta\in \Theta,

the following hold  for any $x, y \in \mathcal{V}_{0}(\theta)$.

(1) If $x-y \in \mathcal{C}_{i} \backslash\{0\}$ for some $1 \leq i \leq N_{0}$, and $\tilde \phi(s,x,\theta), \tilde \phi(s,y,\theta)\in \mathcal{V}_0(\theta\cdot s)$, $s\in[0,t]$ for some given $t \geq T_{0}$, then
$\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta) \in \operatorname{int}\mathcal{C}_{i}$.

(2) If $\tilde \phi(s,x,\theta), \tilde \phi(s,y,\theta)\in \mathcal{V}_0(\theta\cdot s)$, $s\in[0,t]$, and $\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \notin \mathcal{C}_{N_{0}}$ for some given  $t \geq T_{0}$,  then

\begin{equation}
\|\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\| \leq \lambda_0^t\|x-y\|.
\end{equation}

(3) Assume moreover (H5) is satisfied. If $\tilde \phi(s,x,\theta), \tilde \phi(s,y,\theta)\in\mathcal{V}_0(\theta \cdot s)$ for all $0\leq s\leq 2T_0$, and $x-y,\tilde \phi(2T_0,x,\theta)- \tilde \phi(2T_0,y,\theta)\in \mathcal{C}_{i} \backslash \mathcal{C}_{i-1}$
for some $i \in\left\{1,2, \ldots, N_{0}\right\}$, then $\tilde \phi(T_0,x,\theta)-\tilde \phi(T_0,y,\theta)\notin \mathcal{H} $.

We now use Proposition <ref> to prove Theorem <ref>, while the proof of this proposition will be given in Section <ref>.

If (<ref>) is established, then by Proposition <ref>(2), $\|\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\|$ exponentially decays to zero.

Suppose that (<ref>) is not established, then there exists a sequence $t_n\to\infty$ such that

\tilde \phi(t_n,x,\theta)-\tilde \phi(t_n,y,\theta) \in \mathcal {C}_ {N_0 }.

Let $k_1\in \mathbb{N}$ be such that $t_{k_1}\geq T_0+t_1$, then by Proposition <ref> (1),

\[
\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta) \in \operatorname{int}\mathcal{C}_{N_0},\quad t\geq t_{k_1}.
\]

\[
\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta) \in \operatorname{int}(\mathcal{C}_{N_0})\setminus \mathcal { C } _ { N_0 - 1 },\quad t\geq t_{k_1}
\]

then we obtain (<ref>), otherwise there exists $T'\geq t_{k_1}$ such that

\[
\tilde \phi(T',x,\theta) - \tilde \phi(T',y,\theta) \in \mathcal { C } _ { N_0 - 1 }.
\]

And there is $t_{k_2}>T'+T_0$ such that

\[
\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta) \in \operatorname{int}\mathcal { C } _ { N_0 - 1 },\quad  t\geq t_{k_2}.
\]

Since $N_0$ is finite, by repeating the above process finite many times, there are $i\in \{1,\cdots,N_0\}$ and $T^*>0$ such that (<ref>) is established, (1) is proved.

To prove (2), suppose (<ref>) is established. Then, by the assumptions, (<ref>) can be improved to

\begin{equation}\label{exponential-decay2}
\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta)\notin \mathcal { C } _ { N_0 } \text { for all } t \in\mathbb{R}.
\end{equation}

Indeed, if there exists $T_1^{*}<T^*$ such that

\[
\tilde \phi(T_1^{*},x,\theta) - \tilde \phi(T_1^{*},y,\theta)\in \mathcal { C } _ { N_0 }.
\]

Then by Proposition <ref> (1), one has

\[
\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta)\in \mathcal { C } _ { N_0 }\quad t\geq T_1^*+T_0,
\]

a contradiction to (<ref>).

Now by Proposition <ref> (2), equation (<ref>) implies that

\[
\|x-y\|\leq \lambda_0^t\|\tilde \phi(-t,x,\theta) - \tilde \phi(-t,y,\theta)\|\leq 2c_0\lambda_0^t, \forall t>0
\]

where $0<\lambda_0<1$ and $c_0$ is the upper bound of $\|\tilde \phi(-t,x,\theta) - \tilde \phi(-t,y,\theta)\|$, $t>0$. Therefore, $x=y$, a contradiction. Thus, only (<ref>) will happen.

Without loss of generality, we may assume that $\tilde \Lambda_{t_n}(x,\theta)\to (x^*,\theta^*)$ and $\tilde \Lambda_{t_n}(y,\theta)\to (y^*,\theta^*)$ as $t_n\to-\infty$. By using (<ref>), there exist $i'\in \{1,\cdots,N_0\}$ and $T^*_2>0$ such that

\begin{equation*}
\tilde \phi(t,x^*,\theta^*)-\tilde \phi(t,y^*,\theta^*) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 },\quad t \geq T_2^*.
\end{equation*}

\begin{equation*}
\tilde \phi(T^*_2,x^*,\theta^*)-\tilde \phi(T^*_2,y^*,\theta^*) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 }.
\end{equation*}

Since $\operatorname{int}\mathcal { C } _ { i' } \backslash \mathcal { C } _ { i' - 1 }$ is an open subset in $X$, there is $N_1\in \mathbb{N}$ such that

\begin{equation}\label{negative-sequence}
\tilde \phi(T^*_2+t_n,x,\theta)-\tilde \phi(T^*_2+t_n,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 },\quad n\geq N_1.
\end{equation}

For any given $t\in \mathbb{R}$, there is $n\geq N_1$ satisfies $t_n+T_2^*+T_0<t$, and therefore by using Proposition <ref>(1)

\begin{equation}\label{stayin-i'}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\mathcal { C } _ { i' }.
\end{equation}

Suppose there is some $t_0\leq T^*_2+t_{N_1}-T_0$, such that

\begin{equation*}
\tilde \phi(t_0,x,\theta)-\tilde \phi(t_0,y,\theta) \in \mathcal { C } _ { i' - 1 }.
\end{equation*}

Then, again by using Proposition <ref>(1), one has

\begin{equation*}
\tilde \phi(T^*_2+t_{N_1},x,\theta)-\tilde \phi(T^*_2+t_{N_1},y,\theta) \in \operatorname{int}\mathcal { C } _ { i' - 1 },
\end{equation*}

a contradiction to that (<ref>). Therefore,

\begin{equation*}
\tilde \phi(t_0,x,\theta)-\tilde \phi(t_0,y,\theta) \notin\mathcal { C } _ { i' - 1 }.
\end{equation*}

This combines with (<ref>) imply that

\begin{equation*}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 }, \quad \text{for all }t\leq T^*_2+t_{N_1}-T_0.
\end{equation*}

(2) is then proved.

Finally, we prove (3). By (1), we know that there exist some $i_0\in \{1,\cdot,N_0\}$ and $T^*>0$ such that

\begin{equation*}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i_0 }) \backslash \mathcal { C } _ { i_0 - 1 },\quad t \geq T^*.
\end{equation*}

By assumption (H5), we know that $	\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \notin \mathcal{H}$ for all $t>T^*$.

The proof of this theorem is completed.

Corollary <ref> is a direct result of Theorem <ref> (1) and (2). We give a short proof for Corollary <ref>.

By Theorem <ref>(3), we know that

\begin{equation}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\notin \mathcal{H}, \quad t\in \mathbb{R}
\end{equation}

for any distinct points $(x,\theta), (y,\theta)\in E$. Let $\mathcal{H}'$ be a $d$-dimensional subspace of $X$ such that $\mathcal{H}'\oplus \mathcal{H}=X$. Denote by $p:X\mapsto \mathcal{H}'$ the projection of $X$ on $\mathcal{H}'$ along $\mathcal{H}$ and $h:X\times \Theta\mapsto \mathcal{H}'\times \Theta$ is the following:

\begin{equation}
h:(x,\theta)\to (p(x),\theta)
\end{equation}

Noticing that $E$ is compact and $h$ is continuous, $E\to \hat E=h(E)$ is a homeomorphic. Thus, we denote $(\hat x,\theta)$ be the point in $\hat E$, where $\hat x=p(x)$.
For any $(\hat x,\theta)\in \hat E$, define $\hat \phi(t,\hat x,\theta)=p(\tilde\phi(t,x,\theta))$ for all $t\in \mathbb{R}$. Then,

\begin{equation}\label{induced-skew-product}
\hat \Lambda_t: \hat E\to\hat E,\quad \hat \Lambda_t (\hat x,\theta)=(\hat \phi(t,\hat x,\theta),\theta\cdot t).
\end{equation}

is a flow on $\hat E$, and

h \tilde \Lambda_t (x,\theta)=\hat\Lambda_t h(x,\theta)\quad \forall \, t\in \mathbb{R},\, \, (x,\theta)\in E.

That is, the following diagram is commutative:

\begin{CD} E @>h>> \hat E\\ @VV\tilde \Lambda_tV @VV\hat\Lambda_tV\\ E @>h>> \hat E.
\end{CD}$$

Indeed, for any $(\hat x,\theta)\in \hat E$ and $t_1,t_2\in \mathbb{R}$, we have

\[
\begin{split}
\hat\phi(t_1+t_2,\hat x,\theta)&=p(\tilde\phi(t_1+t_2,x,\theta))\\
&=p(\tilde\phi(t_2,\tilde\phi(t_1,x,\theta),\theta\cdot t_1))\\
&=\hat\phi(t_2,p(\tilde\phi(t_1,x,\theta)),\theta\cdot t_1)\\
&=\hat\phi(t_2,\hat\phi(t_1,\hat x,\theta),\theta\cdot t_1)
\end{split}
\]

which implies that $\hat \Lambda_t$ is a skew-product flow on $\hat E$. The commutative diagram is then done.

§.§ Proof of Theorems <ref> and  <ref>

We always assume $E,\mathcal{\tilde M}\subset \mathcal { V } _ { 0 }$ be compact invariant set and compact minimal invariant set of the skew-product semiflow $\tilde \Lambda_t$ respectively.

§.§.§ Proof of Theorem <ref>

To prove Theorem <ref>, we need several lemmas for the preparation.

Fix $\theta,\ \theta^*\in \Theta$. Let $(x_{i},\theta), (x^*_{i},\theta^*)\in \mathcal{V}_0 (i=1,\ 2,\ x_{1}\neq x_{2},\ x^*_{1}\neq x^*_{2})$ be such that $\tilde \Lambda_t(x_{i},\theta)\in\mathcal{V}_0$ is defined on $\mathbb{R}^{+}$ (resp. $\mathbb{R}^{-}$) and $\tilde \Lambda_t(x^*_{i},\theta^*)\in\mathcal{V}_0$ is defined on $\mathbb{R}$. If there exists a sequence $t_{n}\rightarrow +\infty$ (resp. $s_{n}\rightarrow -\infty$) as $n\rightarrow \infty$, such that $\tilde \Lambda_{t_{n}}(x_{i},\theta)\rightarrow (x^*_{i},\theta^*)$ (resp. $\tilde \Lambda_{s_{n}}(x_{i},\theta)\rightarrow (x^*_{i},\theta^*)$) as $n\rightarrow \infty (i=1,2)$, then there exists $i\in \left\{ 1,2, \cdots, N_0 \right\}$ such that

\tilde \phi(t,x^*_1,\theta^*)-\tilde \phi(t,x^*_2,\theta^*) \in \operatorname{int}\ (\mathcal { C } _ { i }) \backslash \mathcal { C } _ { i - 1 }

for all $t\in \mathbb{R}$.

We only prove the case  $s_n\to -\infty$, while the case $t_n\to \infty$ is similar. First of all, one can use similar deductions as those in Theorem <ref>(2) to obtain that (<ref>) will not happen for $\tilde \phi(t,x^*_1,\theta^*)-\tilde \phi(t,x^*_2,\theta^*) $, and hence, there exist $T^{*}>0$ and an integer $1\leq i'\leq N_0$ such that

\begin{equation*}
\tilde \phi(t,x^*_1,\theta^*)-\tilde \phi(t,x^*_2,\theta^*) \in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 }\text { for all } t \leq -T^{*}.
\end{equation*}

We claim that there exists $T^{**}>0$ such that

\begin{equation}\label{negative-constant1}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i'}) \backslash \mathcal { C } _ { i' - 1 }\text { for all } t \leq -T^{**}.
\end{equation}

Indeed, since $\tilde\Lambda_{s_{n}}(x_{i},\theta)\rightarrow (x^*_{i},\theta^*)$, there exists $N_1>0$ such that

\begin{equation*}
\tilde \phi(s_n-T^*,x_1,\theta)-\tilde \phi(s_n-T^*,x_2,\theta)\in \operatorname{int}\ (\mathcal { C } _ { i' }) \backslash \mathcal { C } _ { i' - 1 },\quad n\geq N_1.
\end{equation*}

Let $T^{**}=-(s_{N_1}-T_0-T^*)$, on the one hand, by Proposition <ref>(1)

\begin{equation*}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\notin  \mathcal { C } _ { i' - 1 },\quad t\leq -T^{**}
\end{equation*}

one the other hand, fix $t\leq -T^{**}$ and choose $n_0\gg 1$, such that $s_{n_0}-T^*+T_0\leq t$, then again by Proposition <ref>(1)

\[
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} \mathcal { C } _ { i' }
\]

and (<ref>) is proved.

By equation (<ref>) and Theorem <ref> (1), there exist $1\leq i\leq N_0$ and some $T^+>0$ such that

\[
\tilde \phi(t,x^*_1,\theta^*)-\tilde \phi(t,x^*_2,\theta^*) \in \operatorname{int}\ (\mathcal { C } _ { i}) \backslash \mathcal { C } _ { i - 1 }\text { for all } t \geq T^+.
\]

Also there exists $N^+>0$ such that

\[
\tilde \phi(s_n+T^+,x_1,\theta)-\tilde \phi(s_n+T^+,x_2,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i}) \backslash \mathcal { C } _ { i - 1 }
\]

for all $n\geq N^+$. Let $n> N^+$ be such that $s_n+T^+\leq -T^{**}$, then

\[
\tilde \phi(s_n+T^+,x_1,\theta)-\tilde \phi(s_n+T^+,x_2,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i'}) \backslash \mathcal { C } _ { i' - 1 }.
\]

Note that $(\operatorname{int}\ (\mathcal { C } _ { i}) \backslash \mathcal { C } _ { i - 1 })\cap(\operatorname{int}\ (\mathcal { C } _ { i'}) \backslash \mathcal { C } _ { i' - 1 })=\emptyset$ if $i\neq i'$, hence, $i=i'$, we have proved
\tilde \phi(t,x^*_1,\theta^*)-\tilde \phi(t,x^*_2,\theta^*) \in \operatorname{int}\ (\mathcal { C } _ { i }) \backslash \mathcal { C } _ { i - 1 }$, for all $ t\in\mathbb{R}.

Since $\mathcal{H}$ is a 1-codimension subspace of $X$, there exists 1-dimension subspace $\mathcal{H}'=\mathrm{span}\{u^+\}$ ($u^+\in X\setminus \mathcal{H}$ with $\|u^+\|=1$) such that $\mathcal{H}'\bigoplus\mathcal{H}=X$. Define $X^+=\{s u^++u|s\in\mathbb{R}^+, u\in \mathcal{H}\}$ and $X^-=\{s u^++u|s \in\mathbb{R}^-, u\in \mathcal{H}\}$.

Let $(x,\theta), (y,\theta) \in \mathcal { V } _ { 0 }$ be two distinct points such that $\tilde \phi(t,x,\theta), \tilde \phi(t,y,\theta)\in \mathcal { V } _ { 0 }$ for all $t>0$. Suppose (<ref>) not happen, then there exists $T>0$ such that $\tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta)$ always lie in $X^+$ or $X^-$ for all $t\geq T$.

Since (<ref>) is not happen, by Theorem <ref>(1), there exist $i\in\{1,\cdots,N_0\}$ and $T^*>0$ such that

\begin{equation*}
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in \operatorname{int}\ (\mathcal { C } _ { i }) \backslash \mathcal { C } _ { i - 1 } \text { for all } t \geq T^*.
\end{equation*}

Without loss of generality, assume that $T^*=0$ and let $u=x-y$, $\tilde z= (x,y,\theta)$ and $\tilde T(t,\tilde z)u= \tilde \phi(t,x,\theta) - \tilde \phi(t,y,\theta)$, then

\begin{equation}\label{stay-in-cone}
\tilde T(t,\tilde z)u \in \operatorname{int}\ (\mathcal { C } _ { i })  \backslash \mathcal { C } _ { i - 1 },\quad t\geq 0.
\end{equation}

Noticing that $\mathcal{H}$ is a $1$-codimension subspace of $X$,  Proposition <ref>(3) can be strengthened to:  if $ u \in \mathcal { C } _ { i } \backslash \mathcal { C } _ { i - 1 }$ and $ \tilde T(2T_0,\tilde z) u \in \mathcal { C } _ { i } \backslash \mathcal { C } _ { i - 1 }$ for the given $i$, then

\begin{equation}\label{strength-eq}
\tilde T(T_0,\tilde z)u \in X^{\pm}.
\end{equation}

It then follows from (<ref>), that

\tilde T(t,\tilde z) u\notin \mathcal{H}\text { for all }  t\geq T_0.

Since $\tilde T(t,\tilde z)$ is continuously depend on $t$, the above equation equivalents to that $\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta) \in X ^ { + }$ or $X ^ { - }$ for all $t\geq T_0$.

For any $\theta \in \Theta ,$ define the relation $``\geq" \left( ``> " \right)$ on $P ^ { - 1 } ( \theta)$ ($P:X\times \Theta\mapsto \Theta$ is the nature projection) in the following:
for any $\left(x ,\theta \right) , \left( y, \theta \right) \in P ^ { - 1 } ( \theta ) , \left( x, \theta \right) \geq ( > ) \left(y, \theta \right)$ if there is $T > 0$ such that

\[
\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\in X^+\bigcup \{0\} (\tilde \phi(t,x,\theta)-\tilde \phi(t,y,\theta)\in X^+),\quad \text{ for all }t\geq T.
\]

Assume $E\subset \mathcal{V}_0$ be a compact invariant set of $\tilde \Lambda_t$. For any $\theta \in \Theta$, $``\geq"$ defines a total ordering on $P ^ { - 1 } (\theta)\cap E,$ i.e., $``\geq"$ is a
partial ordering of $P ^ { - 1 } (\theta)\cap E$.

By Lemma <ref>, it is not hard to see that if one fix $\theta\in \Theta$ and choose any two points $(x,\theta),(y,\theta)\in P^{-1}(\theta)\cap E$, then either $(x_1,\theta)\geq (x_2,\theta)$ or $(x_2,\theta)\geq (x_1,\theta)$ and the “$=$” holds if and only if $x_1=x_2$.

All that remains is to show that “$\geq$” satisfies transitivity, that is for any three points $(x_1,\theta),(x_2,\theta),(x_3,\theta)\in P^{-1}(\theta)\cap E$ with $(x_1,\theta)\geq (x_2,\theta)$, $(x_2,\theta)\geq(x_3,\theta)$, then $(x_1,\theta)\geq (x_3,\theta)$. It is sufficient to prove that “$>$” satisfies transitivity.

First of all, there exists $T>0$ such that

\[
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in X^+ \quad \text{and } \tilde \phi(t,x_2,\theta)-\tilde \phi(t,x_3,\theta)\in X^+,\quad \forall t\geq T.
\]

In other words, fix some $t\geq T$, then

\[
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)=s_{12}u^++u_{12} \quad \text{and } \tilde \phi(t,x_2,\theta)-\tilde \phi(t,x_3,\theta)=s_{23}u^++u_{23}
\]

where $s_{12},s_{23}\in \mathbb{R}^+$ and $u_{12},u_{23}\in \mathcal{H}$. Therefore,

\[
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_3,\theta)=(s_{12}+s_{23})u^++u_{12}+u_{23}\in X^+.
\]

Since the the choosing of $t\geq T$ is arbitrary,

\[
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_3,\theta)\in X^+,\quad \forall t\geq T.
\]

That is $(x_1,\theta)>(x_3,\theta)$.

Suppose $\mathcal{\tilde M}\subset \mathcal{V}_0$ be a minimal set of $\tilde \Lambda_t$. For any $\theta \in \Theta$, let $A(\theta )=P ^ { - 1 } (\theta)\cap \mathcal{\tilde M}$. Assume
$A(\theta)$ is not a single point set,  by Corollary <ref>, for any two distinct points $\left(x_1 , \theta \right) , \left( x_2, \theta\right) \in A ( \theta)$, we can define

\[

\[\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta) \in\operatorname{int}\ (\mathcal { C } _ { i }) \backslash \mathcal { C } _ { i - 1 },\ t\gg 1.
\]

I_m( A ( \theta ) ) = \min \left\{ I \left( x_ { 1 } , x_ { 2 } , \theta \right) | \left( x_ { 1 } , \theta \right) , \left( x_ { 2 } , \theta \right) \in A (\theta) , x_ { 1 } \neq x _ { 2 } \right\}.

For any $\left( x_ { 0 } , \theta \right) \in A (\theta) ,$ define

\begin{equation}\label{neighborhood}
A \left( x _ { 0 } , \theta \right) = \left\{ ( x, \theta) \in A ( \theta ) | x \neq x _ { 0 } , I \left( x , x_ { 0 } , \theta \right) = I_m( A ( \theta ) ) \right\}.
\end{equation}

Immediately we have the following two lemmas concerning $A(\theta)$ and $A(x_0,\theta)$.

Let $ \Theta' \subset \Theta$ be the residual subset in Lemma <ref>. Assume $\theta \in \Theta'$ be such that $A(\theta)$ contains at least two points, then $A( x_ { 0 } , \theta)\neq \emptyset$ for any $( x_ { 0 } , \theta) \in A (\theta)$.

Let $( x _ { 1 } ^ { * } , \theta), ( x _ { 2 } ^ { * } , \theta) \in A ( \theta )$
satisfy $I(x_ { 1 } ^ { * } , x_ { 2 } ^ { * } , \theta) =I_m( A ( \theta) )$. Assume $T > 0$ be such that

\tilde \phi( t , \cdot x_ { 1 } ^ { * } , \theta) - \tilde \phi( t , \cdot , x _ 2 ^ { * } , \theta)\in \operatorname{int} (\mathcal{C}_{I(x_1^* , x_{ 2 }^{ * }, \theta)})\setminus \mathcal{C}_{I(x_ { 1 } ^ { * } , x_ { 2 } ^ { * } , \theta)-1}, \quad t \geq T.$$

By view of Lemma <ref> and the minimality of $\mathcal{\tilde M}$, for any $(x_0 , \theta) \in A ( \theta )$, there are $t_{ n } \rightarrow \infty$ and
$\left\{ \left( x_ { n } , \theta \right) \right\} \subset A ( \theta )$ such that $\theta \cdot t _ { n } \rightarrow \theta , \tilde\Lambda_ { t _ { n } } \left( x_ { 0 } , \theta \right) \rightarrow \left( x_ { 1 } ^ { * } , \theta\right) ,$ and $\tilde\Lambda _ { t _ { n } } \left( x _ { n } , \theta \right)$
$\rightarrow \left( x_ { 2 } ^ { * } , \theta \right)$ as $n \rightarrow \infty .$ It follows that there exists $n _ { 0 } > 0$ such that
$\tilde \phi \left( t _ { n } + T , \cdot, x _ { 0 } , \theta\right) - \tilde \phi \left( t _ { n } + T , \cdot , x _ { n } , \theta \right)\in \operatorname{int} (\mathcal{C}_{I \left(x_ { 1 } ^ { * } , x_ { 2 } ^ { * } , \theta \right)})\setminus \mathcal{C}_{I \left(x_{ 1 }^{ * }, x_ { 2 }^{ * } , \theta \right)-1}$ for $n \geq n _ { 0 } .$ Since $I_m( A  ( \theta ) )$ is the minimum index on $A ( \theta)$, one has that $I\left( x _ { 0 } , x_ { n } , \theta \right) =$
$I_m( A  ( \theta ) )$ for $n \geq n _ { 0 }$. Thus, $(x _ { n },\theta)\in A(x_0,\theta)\setminus\{(x_0,\theta)\}$ for $n \geq n _ { 0 }$. We have proved this lemma.

For any $\theta \in \Theta'$ with
$A(\theta)$ contains at least two points, there is a $\left( x_ { 0 } , \theta \right) \in A ( \theta )$ such that $\left( x _ { 0 } , \theta \right) \geq ( x , \theta)$ for any
$( x, \theta) \in A \left( x_ { 0 } , \theta\right) .$

Suppose the lemma is not true. Let $\theta^ { * }\in \Theta'$ be the point such that for any $(x^ { * } , \theta^ { * }) \in A( \theta ^ { * })$, there is a $( x_ { 0 }^{ * }, \theta^ { * }) \in A(x^ { * }, \theta^*)$
satisfies $(x_ { 0 }^{ * } ,\theta^{*})>(x^{ * },\theta^ { * })$.

Noticing that $I(x_ { 0 } ^ { * }, x^ { * } , \theta^ { * }) = I_m( A( \theta^ { * }))$, by Corollary <ref>, there is a $T > 0$ such that

\[\tilde \phi(t,x_ { 0 } ^ { * },\theta^{*})-\tilde \phi(t,x^ { * },\theta^{*})\in (\operatorname{int} (\mathcal{C}_{I_m(A( \theta^ { * }))})\setminus \mathcal{C}_{I_m( A ( \theta^ { * }))-1})\cap X^+,\quad t \geq T.\]

The continuity of $\tilde \phi(T,\cdot,\cdot)$ implies that there is an open neighborhood $B( x^ { * } , \theta^ { * })$ of
$( x^ { * } , \theta^ { * })$ in $A(\theta^ { * })$ such that

\[\tilde \phi(T,  x_ { 0 } ^ { * } , \theta^ { * }) - \tilde \phi(T, x , \theta^ { * })\in (\operatorname{int} (\mathcal{C}_{I_m(A( \theta^ { * }))})\setminus \mathcal{C}_{I_m(A( \theta^ { * }))-1})\cap X^+,\quad (x, \theta^ { * }) \in B( x^ { * } , \theta^ { * }).\]

By the definition of $I_m(A( \theta^ { * }))$, one has

\[
I(x,x^*,\theta^*)=I_m( A  (\theta^ { * })),\quad (x,\theta^*)\in B( x^ { * } , \theta^ { * }).
\]

Therefore, $(x_0^*,\theta^*)>(x,\theta^*)$ for any $(x,\theta^*)\in B(x^*,\theta^*)$, in other words, $(x^*_0,\theta^*)$ can be viewed as an upper bound of $B(x^*,\theta^*)$ in $A(\theta^*)$. Since $\{B( x ^ { * } , \theta ^ { * }) :( x^ { * } , \theta ^ { * }) \in A( \theta^{ * })\}$ is an open covering of the compact set $A( \theta^ { * })$, there are $( x _ { i } ^ { * } , \theta ^ { * }) \in A( \theta^{ * }) , i = 1,2 , \ldots , n _ { 0 } ,$ for some $n _ { 0 } \geq 1 ,$ such that

\bigcup _ { i = 1 } ^ { n _ { 0 } } B( x _ { i } ^ { * } ,\theta^ { * }) = A(\theta^ { * }).

Let $( x _ { 0 } , \theta ^ { * }) \in A( \theta ^ { * })$ be the maximum element of $\{(x_ { 0j} ^ { * } , \theta^ { * })\}_{j=1}^{n_0}$
(i.e., $( x_ { 0 } , \theta ^ { * }) \geq( x _ { 0 i } ^ { * } , \theta^ { * })$, $i = 1,2 , \ldots , n _ { 0 })$, where $( x _ { 0 i } ^ { * } , \theta ^ { * }) \in A(\theta^ { * }) , i =$
$1,2 , \ldots , n _ { 0 } ,$ are upper bounds of $B( x _ { i } ^ { * } , y ^ { * })$. Then $(x _ { 0 } , \theta^ { * })$ is a maximum element of $A( \theta^ { * })$, and is also an upper bound of $A \left( x _ { 0 } , \theta^ { * } \right) $. By Lemma <ref>, $A( x_ { 0 } , \theta^ { * })$ is nonempty, for any $\left( x, \theta^ { * } \right) \in$
$A \left( x _ { 0 } , \theta^ { * } \right) , \left(x_ { 0 } , \theta^ { * } \right) > \left( x ,\theta^ { * } \right)$. A contradiction to our assumption.

Suppose on the contrary that there $A(\theta)$ is not a single point set for some $\theta \in \Theta'$. It then follows from Lemma <ref> that there exists $(x_0, \theta) \in A ( \theta )$ such that $(x_{ 0 } , \theta)\geq (x, \theta)$ for all $(x, \theta) \in A(x _0, \theta)$. Choose $(x _1^*, \theta ), (x _2^*, \theta) \in A (\theta)$ with $x_ { 1 } ^ { * } \neq x_ { 2 } ^ { * }$ be such that $I( x_ { 1 } ^ { * } , x _ { 2 } ^ { * } , \theta) =I_m( A (\theta))$, and assume that

(x_1^*, \theta)>(x_2^*, \theta),\quad \tilde \phi(t, x_1^*, \theta) - \tilde \phi(t,  x_2^ { * } , \theta)\in \operatorname{int} (\mathcal{C}_{I_m(A(\theta))})\setminus \mathcal{C}_{I_m(A(\theta))-1}\cap X^+,\quad t \geq 0.

It then follows from the minimality of $\mathcal{\tilde M}$ and Lemma <ref> that there are sequences $t _ { n } \rightarrow \infty$ and $(x_ { n }, \theta) \in A(\theta)$, such that $\tilde \Lambda _ { t _ { n } }(x_0, \theta) \rightarrow(x_2^*, \theta)$,
$\tilde\Lambda _ { t _ { n } }( x_ { n } , \theta) \rightarrow (x_1^*, \theta)$ as $n \rightarrow \infty$. Using similar arguments as those in Lemma <ref> and Lemma <ref>, there is a $n_{0}> 0$ such that $\tilde \phi( t_n,  x_n, \theta) - \tilde \phi(t_ n,x_0, \theta)\in \operatorname{int} (\mathcal{C}_{I_m( A  ( \theta ) )})\setminus \mathcal{C}_{I_m( A  ( \theta ) )-1}\cap X^+$ and $I( x_ { n } , x _ { 0 } , \theta) =I_m( A ( \theta ) )$ for $n \geq n_0$. Therefore,
$(x _ { n }, \theta)\in A( x _ { 0 } , \theta)$ and $(x_n,\theta)>(x_0,\theta)$ for $n\geq n_0$, a contradiction to the choosing of $x_0$.

§.§.§ Proof of Theorem <ref>

Let $\mathcal{\tilde M}_1,\mathcal{\tilde M}_2\subset \mathcal{V}_0$ be two minimal sets of $\tilde \Lambda_t$. Then there is an integer $N'\in \{1,\cdots,N_0\}$, such that for any $\theta\in \Theta$, $(x_i,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)$($i=1,2$), one has

\begin{equation}\label{constant-on-twosets1}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1},\quad t\in \mathbb{R}.
\end{equation}

Fix any $\theta\in \Theta$, $(x_i,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)$($i=1,2$), we claim that there is an integer $N'\in \mathbb {N}$ such that

\begin{equation}\label{constant-onfibre}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1},\quad t\in \mathbb{R}.
\end{equation}

We first note that there are $T>0$ and two integers $N_1\leq N_2$ such that

\begin{equation}\label{positive-constant}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N_1})\setminus \mathcal{C}_{N_1-1}, \quad  t\ge T
\end{equation}

\begin{equation}\label{negative-constant}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N_2})\setminus \mathcal{C}_{N_2-1}, \quad  t\le -T.
\end{equation}

In fact, (<ref>),(<ref>) follow directly from Corollary <ref>.

To prove the claim, we only need to prove that $N_1=N_2$. Choose a sequence $t_n\to \infty$ such that $\tilde\Lambda_{t_n}(x_2,\theta)\to (x_2,\theta)$ as $n\to \infty$. Without loss of generality, we assume that $\tilde \Lambda_{t_n}(x_1,\theta)\to (\bar x_1,\theta)$. By Lemma <ref>, there is an integer $N'\in \{1,\cdots,N_0\}$ satisfying that

\begin{equation}\label{constant-1}
\tilde \phi(t,\bar x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1}, \quad t\in\mathbb{R}.
\end{equation}

Since $(x_1,\theta)$, $(\bar x_1,\theta)\in \mathcal{\tilde M}_1\cap P^{-1}(\theta)$, it then follows from Theorem <ref> and the minimality of $ \mathcal{\tilde M}_i$, $i=1,2$ that there are $\{(x^*_i,\theta^*)\}= \mathcal{\tilde M}_i\cap P^{-1}(\theta^*)$, $i=1,2$ and sequence $t^*_n\to\infty$ ($s^*_n\to \infty$) as $n\to \infty$ such that $\tilde \Lambda_{t^*_n}(x_2,\theta)\to (x^*_2,\theta^*)$, $\tilde \Lambda_{t^*_n}(x_1,\theta)\to (x^*_1,\theta^*)$ and $\tilde \Lambda_{t^*_n}(\bar x_1,\theta)\to (x^*_1,\theta^*)$ ($\tilde \Lambda_{-s^*_n}(x_2,\theta)\to (x^*_2,\theta^*)$, $\tilde \Lambda_{-s^*_n}(x_1,\theta)\to (x^*_1,\theta^*)$ and $\tilde \Lambda_{-s^*_n}(\bar x_1,\theta)\to (x^*_1,\theta^*)$) as $n\to \infty$. Then,

\begin{equation*}
\tilde \phi(t,x^*_1,\theta^*)-\tilde \phi(t,x^*_2,\theta^*)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1}, \quad t\in\mathbb{R}.
\end{equation*}

By (<ref>)-(<ref>), we get $N_1=N'=N_2$. That is,

\begin{equation}\label{constant-2}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1},\quad |t|\ge T.
\end{equation}

Thus, we have proved our claim.

Finally, we show that $N'$ is actually independent of $\theta\in \Theta$ and $(x_i,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)$ ($i=1,2$). Indeed, for any $\theta\in \Theta$ and any $(x_i,\theta)$, $(\hat x_i,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)$($i=1,2$). By the claim above, there are $N_1,N_2\in\{1,\cdots,N_0\}$ such that

\begin{equation*}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N_1})\setminus \mathcal{C}_{N_1-1},\quad t\in \mathbb{R},
\end{equation*}

\begin{equation*}
\tilde \phi(t,\hat x_1,\theta)-\tilde \phi(t,\hat x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N_2})\setminus \mathcal{C}_{N_2-1},\quad t\in \mathbb{R}.
\end{equation*}

It again follows from Theorem <ref> that there exist $\theta'\in\Theta'$, $\{(x_i',\theta')\}= \mathcal{\tilde M}_i\cap P^{-1}(\theta')$, and sequences $t_n\to\infty$, $t'_n\to\infty$ such that
$\tilde\Lambda_{t_n}(x_i,\theta)\to(x'_i,\theta')$, $\tilde\Lambda_{t'_n}(\hat x_i,\theta)\to(x'_i,\theta')$ as $n\to\infty$. By (<ref>), we may assume that

\begin{equation*}
\tilde \phi(t,x'_1,\theta')-\tilde \phi(t,x'_2,\theta')\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1}.
\end{equation*}

Note that $\operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1}$ is an open subset in $X$.

\begin{eqnarray*}
\begin{split}
&\tilde \phi(t_n,x_1,\theta)-\phi(t_n, x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1},\quad \text{for }\ n\gg 1\\
&\tilde \phi(t'_n,\hat x_1,\theta)-\phi(t'_n,\hat x_2,\theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1},\quad \text{for }\ n\gg 1.
\end{split}
\end{eqnarray*}

That is $N_1=N_2$. Moreover, for any $\theta,\hat \theta\in \Theta$ and $(x_i,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)$, $(\hat x_i,\hat \theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\hat \theta)$($i=1,2$). Again, one can choose a sequence  $t_n\to -\infty$ and $(\bar x_ 2, \hat \theta)\in \mathcal{\tilde M}_2\cap P^{-1}(\hat \theta)$ such that $\tilde\Lambda_{t_n}(x_1,\theta)\to (\hat x_1,\hat \theta)$ and $\tilde\Lambda_{t_n}(x_2,\theta)\to (\bar x_2,\hat \theta)$ as $n\to \infty$. Therefore, we have

\begin{eqnarray*}
\begin{split}
\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta), \tilde \phi(t,\hat x_1,\hat \theta)-\tilde \phi(t,\hat x_2,\hat \theta)\in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1}
\end{split}
\end{eqnarray*}

for all $t\in \mathbb{R}$. Thus, we have proved that $N'$ is independent of $\theta\in \Theta$ and $(x_i,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)$($i=1,2$), which completes the proof of the lemma.

Recall that $X=\mathcal{H}\bigoplus \mathrm{span}\{u^+\}$, we then have the following

Any two minimal invariant sets $\mathcal{\tilde M}_ { 1 } , \mathcal{\tilde M} _ { 2 }\subset \mathcal{V}_0$ of $\tilde \Lambda_t$ are separated in
the following sense:

(i) $\left[m_ { 1 } (\theta ) , M_ { 1 } ( \theta) \right] \cap \left[ m_ { 2 } (\theta) , M_ { 2 } (\theta) \right] = \varnothing$ for all $\theta \in \Theta ,$ where

\begin{equation*}
\begin{split}
& m _ { i } (\theta) = \min \{s_x|x=x_u+s_xu^+,\ x_u\in \mathcal{H}, (x,\theta)\in  P ^ { - 1 } (\theta )\cap \mathcal{\tilde M}_i\} \\
& M _ { i } (\theta) = \max \{s_x|x=x_u+s_xu^+,\ x_u\in \mathcal{H}, (x,\theta)\in  P ^ { - 1 } (\theta )\cap \mathcal{\tilde M}_i\}\\
&( i = 1,2 )
\end{split}
\end{equation*}

If $m_ { 2 } \left( \theta ^ { * } \right) - M _ { 1 } \left(\theta^ { * } \right) > 0$ for some $\theta^ { * } \in \Theta,$ then there is a $\delta > 0$ such that
$m_ { 2 } ( \theta) - M_ { 1 } (\theta ) \geq \delta$ for all $\theta \in \Theta$.

We first claim that (i) holds for some $\tilde \theta\in\Theta$. Suppose the claim is not correct, then one has $m_2(\theta)\le M_1(\theta)$ and $m_1(\theta)\le M_2(\theta)$ for all $\theta\in \Theta$. Given $\theta,\theta^*\in \Theta$, let $(x_1,\theta)\in \mathcal{\tilde M}_1\cap P^{-1}(\theta)$ be such that $s_{x_1}=m_1(\theta)$ and $(x_2,g)\in \mathcal{\tilde M}_2\cap P^{-1}(\theta)$ be such that $s_{x_2}=M_2(\theta)$. By Lemma <ref> that

$$\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta) \in \operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1},\quad \forall t\in\mathbb{R}\ \text{and some } N'\in\{1,\cdots,N_0\}.

Therefore, by Proposition <ref>(3), $\tilde \phi(t,x_1,\theta)-\tilde \phi(t,x_2,\theta)\notin \mathcal{H}$ for all $t\in\mathbb{R}$, that is, $s_{\tilde \phi(t,x_1,\theta)}-s_{\tilde \phi(t,x_2,\theta)}\neq 0$ for all $t\in \mathbb{R}$. Note that $s_{x_1}=m_1(\theta)\le M_2(\theta)=s_{x_2}$. Then, one has

\begin{equation}\label{smaller}
s_{\tilde \phi(t,x_1,\theta)}<s_{\tilde \phi(t,x_2,\theta)},\quad \text{for all}\ t\in\mathbb{R}.
\end{equation}

By the minimality of $\mathcal{\tilde M}_1$, there is a sequence $t_n\to \infty$, such that $\tilde\Lambda_{t_n}(x_1,\theta)\to (x^*_1,\theta^*)$ as $n\to \infty$, where $(x^*_1,\theta^*)\in \mathcal{\tilde M}_1$ with $s_{x^*_1}=m_1(\theta^*)$. Without loss of generality, we can also assume that $\tilde\Lambda_{t_n}(x_2,\theta)\to (x^*_2,\theta^*)$ as $n\to \infty$. By (<ref>), $m_1(\theta^*)=s_{x^*_1}\leq s_{x^*_2}\le M_2(\theta^*)$. Moreover, it follows from Lemma <ref> and Lemma <ref> that $m_1(\theta^*)\neq M_2(\theta^*)$, which implies that $m_1(\theta^*)<M_2(\theta^*)$ (Otherwise, $x_1^*-x_2^* \in \mathcal{H}$, a contradiction to that $\tilde \phi(t,x_1^*,\theta)-\tilde \phi(t,x_2^*,\theta) \in (\operatorname{int} (\mathcal{C}_{N'})\setminus \mathcal{C}_{N'-1})\cap (X^+\cup X^-)$ for all $t\in \mathbb{R}$). However, similarly as above, one can also obtain that $M_2(\theta^*)<m_1(\theta^*)$, a contradiction.

Without loss of generality, we assume that $M_1(\tilde \theta)<m_2(\tilde \theta)$ for some $\tilde \theta\in \Theta$. Then we will show that $M_1(\theta)<m_2(\theta)$ for all $\theta\in \Theta$. Suppose that there is a $\theta^*\in \Theta$ such that $M_1(\theta^*)\ge m_2(\theta^*)$. Let $(x^*_2,\theta^*)\in \mathcal{\tilde M}_2\cap P^{-1}(\theta^*)$ with $s_{x^*_2}=m_2(\theta^*)$ and choose $(x^*_1,\theta^*)\in \mathcal{\tilde M}_1\cap P^{-1}(\theta^*)$ with $s_{x^*_1}=M_1(\theta^*)$. By the minimality of $\mathcal{\tilde M}_2$, we can find a sequence $t_n\to \infty$, such that $\tilde\Lambda_{t_n}(x^*_2,\theta^*)\to (x^{**}_2,\tilde \theta)$ as $n\to \infty$ and $s_{x^{**}_2}=m_2(\tilde \theta)$. Without loss of generality, one may also assume that $\tilde\Lambda_{t_n}(x^*_1,\theta^*)\to (x^{**}_1,\tilde \theta)$ as $n\to \infty$. By the same arguments as in the previous paragraph, one has

\begin{equation*}
m_2(\tilde \theta)=s_{x^{**}_2}\le s_{x^{**}_1}\le M_1(\tilde \theta),
\end{equation*}

contradicting to our assumption. Therefore, $m_2(\theta)>M_1(\theta)$ for all $\theta\in \Theta$. We have proved (i).

(ii) By (i), it is clear that $m_2(\theta)>M_1(\theta)$ for all $\theta\in \Theta$. Suppose that there is sequence  $m_2(\theta_n)>M_1(\theta_n)$ satisfying that $m_2(\theta_n)-M_1(\theta_n)\to 0$ as $n\to \infty$. Then, we may assume that $\theta_n\to \theta^*\in \Theta$, $m_2(\theta_n)\to c$ and $M_1(\theta_n)\to c$ as $n\to\infty$, for some $c\in\mathbb{R}$. Since $\mathcal{\tilde M}_i$ ($i=1,2$) are compact, $c\in[m_1(\theta^*),M_1(\theta^*)]\cap[m_2(\theta^*),M_2(\theta^*)]$, a contradiction.

Suppose that $\tilde \Omega=\tilde \omega(x_0,\theta_0)$ contains three minimal sets $\mathcal{\tilde M}_i$ ($i=1,2,3$). Define

\begin{equation*}
A_i(\theta)=\{s_x|x=s_xu^++x_u,\,x_u\in\mathcal{H},\,(x,\theta)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta)\}
\end{equation*}

\begin{equation*}
m_i(\theta)=\min A_i(\theta),\quad M_i(\theta)=\max A_i(\theta),
\end{equation*}

for all $\theta\in \Theta$ and $i=1,2,3$. By virtue of Lemma <ref>(ii), we may assume without loss of generality that there is a $\delta>0$ such that

\begin{equation}\label{inequality}
M_1(\theta)+\delta\leq m_2(\theta)\le M_2(\theta)<M_2(\theta)+\delta\le m_3(\theta),
\end{equation}

for all $\theta\in \Theta$.

Now choose $(x_i,\theta_0)\in \mathcal{\tilde M}_i\cap P^{-1}(\theta_0)$, $i=1,2,3$, and consider $(x_0,\theta_0)$ and $(x_2,\theta_0)$. It follows from Theorem <ref>, Lemma <ref> and Proposition <ref>(2) either

\begin{equation}\label{exponential-decay3}
\|\tilde \phi(t,x_2,\theta_0)-\tilde \phi(t,x_0,\theta_0)\|\to 0,\ \text { as }t\to\infty
\end{equation}

that there are $N'\in \{1,\cdots,N_0\}$, $T>0$ such that

\begin{equation}\label{stayin-cone2}
\tilde \phi(t,x_2,\theta_0)-\tilde \phi(t,x_0,\theta_0)\in (\operatorname{int}(\mathcal{C}_{N'})\setminus\mathcal{C}_{N'-1})\cap X^+(\text{ or }X^-),\quad t\ge T.
\end{equation}

Since $\tilde \Omega$ is not minimal, only (<ref>) will happen. Without loss of generality we assume that  $\tilde \phi(t,x_2,\theta_0)-\tilde \phi(t,x_0,\theta_0)\in (\operatorname{int}(\mathcal{C}_{N'})\setminus\mathcal{C}_{N'-1})\cap X^+$ for all $t\ge T$, that is,

s_{\tilde \phi(t,x_2,\theta_0)}>s_{\tilde \phi(t,x_0,\theta_0)},\quad t\geq T.

Since $\mathcal{\tilde M}_3\subset \tilde \omega(x_0,\theta_0)$, there is a sequence $\{t_n\}$, $t_n\to \infty$, such that $s_{\tilde \phi(t_n,x_0,\theta_0)}\to m_3(\theta^*)$ as $n\to \infty$. Let $s_{\tilde \phi(t_n,x_2,\theta_0)}\to \beta(\theta^*)$ with $\beta(\theta^*)\in [m_2(\theta^*),M_2(\theta^*)]$. Consequently,

\[
m_3(\theta^*)\leq \beta(\theta^*)\le M_2(\theta^*),
\]

a contradiction to (<ref>). Thus, $\tilde\omega(x_0,\theta_0)$ contains at most two minimal sets. Let $\tilde \omega(x_0,\theta_0)=\mathcal{\tilde M}_1\cup \mathcal{\tilde M}_2\cup \mathcal{\tilde M}_{12}$, where $\mathcal{\tilde M}_1$, $\mathcal{\tilde M}_2$ are minimal sets. If $\mathcal{\tilde M}_1\neq \mathcal{\tilde M}_2$, since $\tilde\omega(x_0,\theta_0)$ is connected, $\mathcal{\tilde M}_{12}\neq \emptyset$. Choose $(x_{12},\theta)\in \mathcal{\tilde M}_{12}$, it is clear that $\tilde \omega(x_{12},\theta)\cap (\mathcal{\tilde M}_1\cup \mathcal{\tilde M}_2)$ and $\tilde \alpha(x_{12},\theta)\cap (\mathcal{\tilde M}_1\cup \mathcal{\tilde M}_2)$ are not empty, for otherwise, either $\tilde \omega(x_{12},\theta)$ or $\tilde \alpha(x_{12},\theta)$ would contain a minimal set and therefore $\tilde \omega(x_0,\theta_0)$ would have three minimal sets. For the case $\tilde \omega(x_0,\theta_0)$ contains only one minimal set, that is, $\mathcal{\tilde M}_1=\mathcal{\tilde M}_2$. If $\mathcal{\tilde M}_{12}\neq \emptyset$, then a similar argument shows that $\tilde \omega(x_{12},\theta)\cap \mathcal{\tilde M}_1\neq \emptyset$, $\tilde \alpha(x_{12},\theta)\cap \mathcal{\tilde M}_1\neq \emptyset$ for any $(x_{12},\theta)\in \mathcal{\tilde M}_{12}$. Thus, we have completed our proof.

§ APPLICATIONS

In this section, we use Theorems <ref>, <ref>, <ref> to investigate the dynamics of some typical systems.

§.§ Perturbations of  parabolic equations

In this subsection, we focus on $C^1$-small perturbations of

Assume that $f(t,x,u,p)\in C^2(\mathbb{R}\times [0,1] \times \mathbb{R}\times \mathbb{R},\mathbb{R})$ with $f$ and  all its partial derivatives (up to order $2$) are uniformly almost periodic in $t$. Let $\Theta=H(f)$, then by Remark <ref>(iii), there is a unique function $F\in C^2(\Theta\times [0,1] \times \mathbb{R}\times \mathbb{R},\mathbb{R})$, such that $F(\theta\cdot t,x,u,p)\equiv \theta(t,x,u,p)$ (cf. <cit.>). For simplicity, we write $F(\theta\cdot t,x,u,p)$ as $f(\theta\cdot t,x,u,p)$. Equation (<ref>) naturally
induces a family of equations associated to each $\theta\in \Theta$,

\begin{equation}\label{induced-para-eq}
\begin{cases}
u_t=u_{xx}+f(\theta\cdot t,x,u,u_x),\quad t>0,\,\, 0<x<1 \\
u_x(t,0)=u_x(t,1)=0,\quad t>0,
\end{cases}
\end{equation}

Let $X=X^{\alpha}$ be a fractional power space associated with the operator $u\rightarrow
-u_{xx}:H^{2}(0,1)\rightarrow L^{2}(0,1)$ satisfies $X\hookrightarrow C^{1}[0,1]$ (that is, $X$ is compactly embedded into $C^{1}[0,1]$). For any $u\in X$, (<ref>) defines (locally) a unique solution $\phi(t,\cdot;u,\theta)$ in $X$ with $\phi(0,\cdot;u,\theta)=u(\cdot)$ and it continuously depends on $\theta\in \Theta$ and $u\in
X$. Consequently, (<ref>) admits a (local) skew-product semiflow $\Lambda_{t}$ on $X\times
\Theta$:

\begin{equation}\label{para-skew-produ}
\Lambda_{t}(u,\theta)=(\phi(t,\cdot;u,\theta),\theta\cdot t),\quad t\ge 0.
\end{equation}

We impose the following
dissipative condition(D) (see [5, 6, 23, 38] for similar assumptions): there exist constants $\delta, \zeta, \eta>0$ and a continuous function $C(\cdot)$ maps $[0,\infty)$ to $[0,\infty)$, such that $f$ satisfies the following:

\begin{equation}\label{dissipative-condition}
\begin{split}
&\forall \ l>0, \ p \in \mathbb{R}, \quad \sup_{(t,x,u)\in\mathbb{R}\times[0,1]\times[-l,l]} \quad|f(t,x,u, p)| \leq C(l)\left(1+|p|^{2-\eta}\right),  \\
& \forall\ |u| \geq \delta, \ t \in \mathbb{R}, \quad u f(t,x, u, 0)\leq -\zeta.
\end{split}
\end{equation}

Then for any $\theta\in \Theta$ we have

\begin{equation}\label{dissipative-condition-2}
\begin{split}
&\forall \ l>0, \ p \in \mathbb{R}, \quad \sup_{(t,x,u)\in\mathbb{R}\times[0,1]\times[-l,l]} \quad|f(\theta\cdot t,x,u, p)| \leq C(l)\left(1+|p|^{2-\eta}\right),  \\
& \forall\ |u| \geq \delta, \ t \in \mathbb{R}, \quad u f(\theta\cdot t,x, u, 0) \leq -\zeta.
\end{split}
\end{equation}

By comparison principle  for parabolic equations, for any $u_0\in X$, the solution $\phi(t,\cdot;u_0,\theta)$ of (<ref>) is globally defined. Fix $\delta_0\geq\delta$ and let $\mathcal{B}=\{\|u\|_{\infty}\leq \delta_0, u\in X\}$, then by strong comparison principle we have the following statements (cf. <cit.>):

(D1) for any given $(u_0,\theta)\in X\times \Theta$ there exists $t_{(u_0,\theta)}>0$ such that $\phi(t,\cdot;u_0,\theta)\in \mathcal{B}$ for all $t>t_{(u_0,\theta)}$;

(D2) $\mathcal{B}$ is positive invariant: $\phi(t,\cdot;u_0,\theta)\in \mathcal{B}$ for all $u_0\in\mathcal {B}$ and $t\geq 0$, particularly, $\phi(t,\cdot;u_0,\theta)\in \mathcal{\mathring{B}}=\{u|\|u\|_{\infty}< \delta_0, u\in X\}$ for all $u_0\in\mathcal{B}$ and $t>0$;

(D3) (<ref>) or (<ref>) admits a compact global attractor $\mathcal{A}$, that is, we have

\[
\mathcal{A}=\bigcap_{T>0}cl(\cup_{\{t>T,\theta\in \Theta\}}\Lambda_t(\mathcal{B},\theta)).
\]

Let $C_i$ be as in (<ref>) and $\mathcal{H}$ be as in (<ref>).

$C_i$ is an $i$-cone in $X$.

It is not hard to see that $w_n(t,x)=e^{-n^2\pi^2t}\cos n\pi x$ solves the following equation

\begin{equation}
\begin{cases}
w_t=w_{xx},\quad x\in (0,1),\ t\in \mathbb{R}\\
\end{cases}
\end{equation}

Define $W_0=\operatorname{span}\{1\}$ and for any $n\in\mathbb{N}$, let $W_n=\operatorname{span}\{\cos n\pi x\}$; $V_i=\oplus_{j=0}^{i-1} W_j$ and $L_i:=cl(\operatorname{span}\{\cos n\pi x,\ n\geq i\})$. Then, $V_i$ and $L_i$ are $i$-dimensional and $i$-codimensional subspaces of $X$ respectively. Moreover, $V_i\subset C_i$ and $L_i\cap C_i=\{0\}$. Indeed,  any $w\in X$ can be written as the following:

\begin{equation*}
w(x)=\sum_{n=0}^{\infty}a_n\cos n\pi x.
\end{equation*}

If $w(0,\cdot)=w(\cdot)\in V_i\setminus\{0\}$, then there is a index $m\leq i-1$ such that $a_m\neq 0$ and $a_n=0$ for all $n>m$. Then,

w(t,x)=\sum_{k=0}^{m}a_ke^{-k^2\pi^2t}\cos k\pi x.

\lim_{t\to-\infty}e^{m^2\pi^2t}w(t,\cdot)=a_m\cos m\pi x.

Note that $\cos m\pi x$ has exactly $m$ simple zeros in $(0,1)$. Hence, $Z(w(t,\cdot))=m$ for $t\ll -1$. By the non-increasing property of $Z(w(t,\cdot))$ with respect to $t$, $Z(w(t,\cdot))\leq m$ for all $t\in\mathbb{R}$, that is, $w(t,\cdot)\in C_i$.

If $w(\cdot)\in L_i\setminus\{0\}$, let $m>i$ be the minimal index such that $a_m\neq 0$. Suppose on the contrary that $w(\cdot)\in C_i$. Then, $Z(w(t,\cdot))\leq i-1$ for all $t\geq 0$. On the other hand,

\[
\lim_{t\to\infty}e^{m^2\pi^2t}w(t,\cdot)=a_m\cos m\pi x
\]

implies that $Z(w(t,\cdot))=m>i-1$ for $t\gg 1$, a contradiction. Thus, the maximal dimension of subspaces of $X$ which contained in $C_i$ is $i$. Note that $C_i$ is closed and $\lambda C_i=C_i$ for any $\lambda\neq 0$, that is, $C_i$ is a $i$-cone.

Let $\mathcal{U}_0=\mathcal{\mathring{B}}\times \Theta$, $K_0=\mathcal{A}$ and $K=\{(u_1,u_2,\theta)|(u_1,\theta), (u_2,\theta)\in K_0\}$. Then, it is not hard to see that $\Lambda_t$ satisfies (H1)-(H5), particularly, $d=1$ in (H5), by Theorems <ref>, <ref>, <ref>, we have the following

There exists an open neighborhood $\mathcal{V}_0$ of $K_0$ (i.e., there exists a family of open sets $\mathcal{V}_0(\theta)$ of $K_0(\theta)$ such that $\mathcal{V}_0=\{(u,\theta)|u\in K_0(\theta),\,\theta\in\Theta\}$) and $\epsilon_0>0$ such that for any
continuous skew-product semiflow $\tilde\Lambda=(\tilde \phi,\theta)$ on $\mathcal { V }_0$ with $\tilde \phi(t,\cdot,\theta) \in C ^ { 1 } ( \mathcal {U}_0(\theta), X ) , t \in \left[\frac{1}{2}, 1 \right]$ and

\begin{equation}\label{parabolic-c1}
\left\|\phi(t,\cdot,\theta) - \tilde \phi(t,\cdot,\theta) \right\| _ {C ^ { 1 } ( \mathcal {U}_0(\theta), X )} < \epsilon_0 \quad \text { for all } \frac{1}{2} \leq t \leq 1, \ \theta\in \Theta
\end{equation}

one has any minimal set of $\tilde \Lambda_t$ contained in $\mathcal{V}_0$ is an almost $1$-cover of the base flow $\Theta$; moreover, any $\omega$-limit set $\tilde \omega(u_0,\theta)\subset \mathcal{V}_0$ of $\tilde \Lambda_t$ contains at most two minimal sets.

We now consider the perturbed system (<ref>). Since both $f$ and $c$ are uniform almost periodic functions, then $F=(f,c)$ is a uniform almost periodic vector function. Let $\tilde \Theta=H(F)$, then by Remark <ref>(iii), there is a unique function $\tilde F=(\tilde f, \tilde c)\in C^2(\Theta\times [0,1] \times \mathbb{R}\times \mathbb{R},\mathbb{R}^2)$, such that $\tilde F(\tilde\theta\cdot t,x,u,p)=(\tilde f(\tilde \theta\cdot t,x,u,p),\tilde c(\tilde\theta\cdot t,x))\equiv \tilde\theta(t,x,u,p)$. For simplicity, we also write $\tilde f $, $\tilde c$, $\tilde\Theta$ and $\tilde \theta$ as $f$, $c$, $\theta$ and $\Theta$ respectively. And consider the following family of equations:

\begin{equation}\label{parabolic-eq-induced1}
\begin{cases}
u_{t}=u_{xx}+f(\theta\cdot t,x,u,u_{x})+\epsilon c(\theta\cdot t,x)\int_0^1\nu(x)u(t,x)dx\,\,\,t>0,\, 0<x<1, \quad  \theta\in \Theta \\
\end{cases}
\end{equation}

For any $u\in X$, (<ref>) defines (locally) a unique solution $\phi^N(t,\cdot;u,\theta)$ in $X$ with $\phi^N(0,\cdot;u,\theta)=u(\cdot)$ and it continuously depends on $\theta\in \Theta$ and $u\in
X$. Let $\Lambda^N_t:X\times
\Theta\to X\times
\Theta$ be the (local) skew-product semiflow generated by (<ref>) as follows:

\begin{equation}\label{perturbed-para-skew-produ}
\Lambda^N_{t}(u,\theta)=(\phi^N(t,\cdot;u,\theta),\theta\cdot t),\quad t\ge 0.
\end{equation}

Let $\|c\|_{\infty}=\sup_{(t,x)\in \mathbb{R}\times [0,1]}|c(t,x)|$ and $\|\nu\|_{\infty}=\max_{x\in [0,1]}|\nu(x)|$.
Here, we impose a little stronger condition on (<ref>) than (D) by assuming $f$ satisfies (N):
there exists $\xi>0$ such that

\begin{equation}
\label{condition-eq1}
\begin{cases}
f(t,x,u,p)\le -\xi u\quad {\rm for}\quad u\gg 1, \,\, |p|\ll 1, \, (t,x)\in\mathbb{R}\times [0,1]\cr
f(t,x,u,p)\ge -\xi u\quad {\rm for}\quad  u\ll -1,\,\, |p|\ll 1, \, (t,x)\in\mathbb{R}\times [0,1].
\end{cases}
\end{equation}

Immediately, we have

For any given $(u_0,\theta)\in X\times \Theta$, the positive orbit $\{\Lambda^N_t(u_0,\theta),t>0\}$ is precompact in $X\times \Theta$. Let $\omega^N(u_0,\theta)$ be the omega limit set generated by $\Lambda^N_t(u_0,\theta)$, then there exists $\epsilon_1>0$ (independent the choosing of $(u_0,\theta)\in X\times \Theta$) such that for any $0<\epsilon <\epsilon_1$ in (<ref>), $\omega^N(u_0,\theta)$ contains at most two minimal sets and every minimal set is an almost $1$-cover of the base flow $\Theta$.

Suppose $c(t,x)$ and $\nu(x)$ are non-negative functions, then Theorem <ref> is still correct when $\overline\lim_{|u|\to\infty}\frac{f(t,x,u,0)}{u}\le -\xi$ (a weaker condition than (<ref>)), the deductions are similar to that for Theorem <ref>.

To prove Theorem <ref>, we first prove that (<ref>) admits a global attractor.

By (<ref>), there are $M_0>0$ and $\eta_0>0$  such that

\begin{equation}
\label{condition-eq2}
\begin{cases}
f(\theta\cdot t,x,u,p)\le -\xi u +M_0\quad {\rm for}\quad u\ge 0,\,\, |p|\le \eta_0, \,(t,x,\theta)\in \mathbb{R}\times[0,1]\times\Theta\cr
f(\theta\cdot t,x,u,p)\ge -\xi u-M_0\quad {\rm for}\quad  u\le 0,\,\, |p|\le \eta_0, \, (t,x,\theta)\in\mathbb{R}\times [0,1]\times\Theta.
\end{cases}
\end{equation}

\begin{equation}
\label{condition-eq0}
0\leq \epsilon <\frac{\xi}{\|c\|_\infty \|\nu\|_\infty}
\end{equation}

and let

\begin{equation}
\label{m-star-eq}
M^*= \frac{M_0}{\xi-\epsilon\|c\|_\infty \|\nu\|_\infty}.
\end{equation}

Then, we have the following

Given $u_0\in C([0,1])$, let $\phi^N (t,x;u_0,\theta)$ be the solution of  (<ref>)
with $\phi^N(0,x;u_0,\theta)=u_0(x)$ on the existence interval $[0, T_{\max}(u_0,\theta))$. Then $T_{\max}(u_0,\theta)=\infty$

$$\limsup_{t\to\infty}\|\phi^N(t,\cdot;u_0,\theta)\|_\infty\le M^*.

It suffices to prove that $\|\phi^N(t,\cdot;u_0,\theta)\|_\infty$ decreases at $t$ when $\|\phi^N(t,\cdot;u_0,\theta)\|_\infty\ge M^*$.
Let $\phi^N(t,x)=\phi^N (t,x;u_0,\theta)$, then

\begin{equation}\label{parabolic-eq-induced-eq1}
\begin{cases}
\phi^N_{t}\le \phi^N_{xx}+f(\theta\cdot t,x,\phi^N,\phi^N_{x})+\epsilon\|c\|_\infty \|\nu\|_\infty \|\phi^N(t,\cdot)\|_\infty ,\,\,\, 0<x<1\cr
\phi^N_x(t,0)=\phi^N_x(t,1)=0,
\end{cases}
\end{equation}

\begin{equation}\label{parabolic-eq-induced-eq2}
\begin{cases}
\phi^N_{t}\ge \phi^N_{xx}+f(\theta\cdot t,x,\phi^N,\phi^N_{x})-\epsilon\|c\|_\infty \|\nu\|_\infty \|\phi^N(t,\cdot)\|_\infty ,\,\,\, 0<x<1\cr
\phi^N_x(t,0)=\phi^N_x(t,1)=0
\end{cases}
\end{equation}

for $t\in (0,T_{\max}(u_0,\theta))$.

Assume $t_0\ge 0$ be such that $\|\phi^N(t_0,\cdot;u_0,\theta)\|_\infty\ge M^*$.
Let $\phi^+(t,x;u_0^+,\theta\cdot t_0)$ and $\phi^-(t,x;u_0^-,\theta\cdot t_0)$ be the solutions of

\begin{equation}\label{parabolic-eq-induced-eq1}
\begin{cases}
u_{t}= u_{xx}+f((\theta \cdot t_0)\cdot t,x,u,u_{x})+\epsilon\|c\|_\infty \|\nu\|_\infty \|\phi^N(t_0+t,\cdot;u_0,\theta)\|_\infty ,\,\,\, 0<x<1\cr
\end{cases}
\end{equation}

\begin{equation}\label{parabolic-eq-induced-eq2}
\begin{cases}
u_{t}= u_{xx}+f((\theta \cdot t_0)\cdot t,x,u,u_{x})-\epsilon \|c\|_\infty \|\nu\|_\infty \|\phi^N(t_0+t,\cdot;u_0,\theta)\|_\infty ,\,\,\, 0<x<1\cr
\end{cases}
\end{equation}

for $t\in (0,T_{\max}(u_0,\theta)-t_0)$, respectively. Then by the comparison principle for parabolic equations,

\phi^-(t,x;u_0^-,\theta\cdot t_0)\le \phi^N (t_0+t,x;u_0,\theta)\le \phi^+(t,x;u_0^+,\theta\cdot t_0)\quad \forall\, x\in [0,1],\,\, t\in [0,T_{\max}(u_0,\theta)-t_0).

Let $v^+(t,v_0^+)$ and $v^-(t,v_0^-)$ be the solutions of

\begin{equation}
\label{ode-eq1}
\begin{cases}
v_t=-\xi v+M_0 +\epsilon \|c\|_\infty \|\nu\|_\infty \|\phi^N(t_0+t,\cdot;u_0,\theta)\|_\infty , \,\, 0<t<T_{\max}(u_0)-t_0\cr
\end{cases}
\end{equation}

\begin{equation}
\label{ode-eq2}
\begin{cases}
v_t=-\xi v-M_0 -\epsilon \|c\|_\infty \|\nu\|_\infty \|\phi^N(t_0+t,\cdot;u_0,\theta)\|_\infty , \,\, 0<t<T_{\max}(u_0)-t_0\cr
\end{cases}
\end{equation}

By (<ref>),

f((\theta \cdot t_0)\cdot t,x;\phi^+(t,x;u_0^+,\theta\cdot t_0),\phi^+_x(t,x;u_0^+,\theta\cdot t_0))\le -\xi \phi^+(t,x;u_0^+,\theta\cdot t_0)+M_0\quad \forall\, 0\le t\ll 1,\,\, x\in [0,1]

f((\theta \cdot t_0)\cdot t,x;\phi^-(t,x;u_0^-,\theta\cdot t_0),\phi^-_x(t,x;u_0^-,\theta\cdot t_0))\ge -\xi \phi^-(t,x;u_0^-,\theta\cdot t_0)-M_0\quad \forall\, 0\le t\ll 1,\,\, x\in [0,1]

Then by the comparison principle for parabolic equations again,

v^-(t,v_0^-)\le \phi^-(t,x;u_0^-,\theta\cdot t_0)\le \phi^N (t_0+t,x;u_0,\theta)\le \phi^+(t,x;u_0^+,\theta\cdot t_0)\le v^+(t,v_0^+)\quad \forall\, 0\le t\ll 1.

By the assumption that $\|u_0^\pm\|_\infty=|v_0^\pm|=\|\phi^N(t_0,\cdot;u_0,\theta)\|_\infty>M^*$, we have

v_t^+(0)<0\quad {\rm and}\quad v_t^-(0)>0.

Hence, $\|\phi^N(t_0,\cdot;u_0,\theta))\|_\infty$ decreases at $t_0$. The lemma then follows.

Therefore, by a priori estimates for parabolic equations, for any $u_0\in X$, the solution $\phi^N(t,\cdot;u_0,\theta)$ is globally defined. Assume $\epsilon$ and $M^*$ satisfy (<ref>) and (<ref>) respectively, let $\mathcal{B}^N=\{\|u\|_{\infty}\leq M^*, u\in X\}$, then we have the following statements:

(N1) for any given $(u_0,\theta)\in X\times \Theta$ there exists $t_{(u_0,\theta)}>0$ such that $\phi^N(t,\cdot;u_0,\theta)\in \mathcal{B}^N$ for all $t>t_{(u_0,\theta)}$;

(N2) $\mathcal{B}^N$ is positive invariant: $\phi^N(t,\cdot;u_0,\theta)\in \mathcal{B}^N$ for all $u_0\in\mathcal {B}^N$ and $t\geq 0$, particularly, $\phi(t,\cdot;u_0,\theta)\in \mathcal{\mathring{B}}^N=\{u|\|u\|_{\infty}< M^*, u\in X\}$ for all $u_0\in\mathcal{B}^N$ and $t>0$;

(N3) (<ref>) admits a compact global attractor $\mathcal{A}^N$, that is, we have

\[
\mathcal{A}^N=\bigcap_{T>0}cl(\cup_{\{t>T,\theta\in \Theta\}}\Lambda_t^N(\mathcal{B}^N,\theta)).
\]

Assume (<ref>) and (<ref>) are satisfied. Choose $\delta_0\geq M^*$ in (D1)-(D2), then $\mathcal{B}^N\subset \mathcal{B}$. By the regularity of parabolic equations, the positive orbits of $\phi^N(t,\cdot;u_0,\theta)$ and $\phi(t,\cdot;u_0,\theta)$ are bounded in $X$. Moreover, by continuous dependence of the solution with parameters (see <cit.>), one can choose $0<\epsilon\ll 1$ such that

\|\phi(t,\cdot;u_0,\theta)-\phi^N(t,\cdot;u_0,\theta)\|+\|\phi_u(t,\cdot;u_0,\theta)-\phi^N_u(t,\cdot;u_0,\theta)\|<\epsilon_0,\quad  \frac{1}{2}\leq t\leq 1

for all $(u_0,\theta)\in\mathcal{U}_0$, where $\epsilon_0$ is given in (<ref>).

Note that $K_0$ is a global attractor of $\Lambda_t$, there is $0<\delta'\ll 1$ such that $K_{\delta'}=\{(u',\theta)|\|u'-u\|<\delta', (u,\theta)\in K_0, u'\in X\}\subset \mathcal{V}_0$; moreover, there exists $T_1>0$ such that $\cup_{t>T_1,\theta\in\Theta}\Lambda_t(\mathcal{B},\theta)\subset K_{\frac{\delta'}{2}}\subset \mathcal{V}_0$.

For the given $T_1$, $\epsilon$ can be chosen sufficiently small such that

\begin{equation*}
\|\phi^N(t,\cdot;u_0,\theta)-\phi(t,\cdot;u_0,\theta)\|\leq \frac{\delta'}{4},\quad \text{for all } (u_0,\theta)\in\mathcal{B}\times\Theta, t\in (0,2T_1].
\end{equation*}

\[
\Lambda^N_t(u_0,\theta)\in K_{\delta'}\subset\mathcal{V}_0,\quad \forall t\in[T_1,2T_1].
\]

Repeat this process for for $[2T_1,3T_1]$,$[3T_1,4T_1]$,$\cdots$, then

\begin{equation}\label{asym-attractor2}
cl(\cup_{t\geq T_1,\theta\in\Theta}\Lambda^N_t(\mathcal{B},\theta))\subset K_{\delta'}\subset \mathcal{V}_0\subset \mathcal{U}_0.
\end{equation}

Recall that for any $(u_0,\theta)\in X\times \Theta$, there exists $T_{(u_0,\theta)}>0$ such that $(\phi(t,\cdot;u_0,\theta),\theta\cdot t)\in\mathcal{U}_0$ for $t>T_{(u_0,\theta)}$. Therefore, $\epsilon_0$ can be chosen small enough such that $\Lambda^N_t(u_0,\theta)\in K_{\delta'}\subset\mathcal{V}_0$ for all $t\geq T_1$ and $(u_0,\theta)\in \mathcal{U}_0$. The proof of Theorem <ref> is completed by using Proposition <ref>.

Next, we consider the equation (<ref>). Similarly as in (<ref>), (<ref>)
induces the following family of equations:

\begin{equation}\label{para-ellip-induced}
\begin{cases}
u_{t}=u_{xx}-\epsilon(u v_x)_x+f(\theta\cdot t,x,u,u_{x})\,\,\,t>0,\, 0<x<1,\ \theta\in \Theta\cr
0=v_{xx}-v+u,\quad 0<x<1\cr
\end{cases}
\end{equation}

where $\Theta$ is an almost periodic minimal set.

Note that (<ref>) can be written as

\begin{equation}
\label{para-ellip-induced-1}
\begin{cases}
u_t=u_{xx}-\epsilon u_x v_x -\epsilon u v +
\epsilon u^2 +f(\theta\cdot t,x,u,u_x),\quad t>0,\,\, 0<x<1\cr
\end{cases}
\end{equation}

where $v(t,x)$ is the solution of

\begin{equation}
\begin{cases}
v_{xx}-v+u=0,\quad 0<x<1\cr
\end{cases}
\end{equation}

Hence, let $u(t,x)$ be the solution of (<ref>) with $u_0\in C[0,1]$, then by Liouville formula

\begin{equation}\label{solution-expresion}
\begin{cases}
v(t,x)=c(t)(e^x+e^{-x})+\int_0^x\frac{e^{x-y}-e^{y-x}}{2}u(t,y)dy, \quad t\geq 0,x\in[0,1]\cr
\end{cases}
\end{equation}

Let $C=\frac{3e^2-e}{2(e-1)}$, then

\begin{equation}\label{estimate-v}
\|v(t,\cdot)\|_{\infty}\leq C \|u(t,\cdot)\|_{\infty}.
\end{equation}

In order to display that the solution $u(t,x)$ depends on $\theta\in \Theta$ and the initial value $u_0\in C[0,1]$, and $v(t,x)$ depends on $u$, we write $u(t,x)$, $v(t,x)$ as $\phi^u(t,x;u_0,\theta)$
and $\phi^v(t,x;u_0,\theta)$ respectively. It is not hard to see from (<ref>) that the convergence property of $\phi^v(t,x;u_0,\theta)$ is determined by $\phi^u(t,x;u_0,\theta)$. Thus, to investigate the longtime behavior of bounded solutions of (<ref>), it is sufficient to study the convergence property of $\phi^u(t,x;u_0,\theta)$.

Here, we impose a little stronger condition on (<ref>) than (D) by assuming $f$ satisfies (P):
there exists $\xi>0$ such that

\begin{equation}
\label{condition-eq1-1}
\begin{cases}
f(t,x,u,p)\le -\xi u^2\quad {\rm for}\quad u\gg 1, \,\, |p|\ll 1\cr
f(t,x,u,p)\ge -\xi u^2\quad {\rm for}\quad  u\ll -1,\,\, |p|\ll 1.
\end{cases}
\end{equation}

Let $\Lambda_t^u:X\times \Theta\to X\times \Theta$ be the skew-product semiflow defined as the following:

\Lambda^u_t(u_0,\theta)=(\phi^u(t,\cdot,u_0,\theta),\theta\cdot t),\quad (t,u_0,\theta)\in \mathbb{R}^+\times X\times \Theta.

We have

For any given $(u_0,\theta)\in X\times \Theta$, the positive orbit $\{\Lambda^c_t(u_0,\theta),t>0\}$ is precompact in $X\times \Theta$. Moreover, $\omega^c(u_0,\theta)$ be the omega limit set generated by $\Lambda^c_t(u_0,\theta)$, then there exists $\epsilon_1>0$ (independent the choosing of $(u_0,\theta)\in X\times \Theta$) such that for any $0<|\epsilon| <\epsilon_1$, $\omega^c(u_0,\theta)$ contains at most two minimal sets and every minimal set is an almost $1$-cover of the base flow.

We also make some preparations before proving Theorem <ref>.

First, by assumption (P), there are $M_1>0$ and $\epsilon_1>0$ such that

\begin{equation}
\label{condition-eq1-2}
\begin{cases}
f(\theta\cdot t,x,u,p)\le -\xi u^2+M_1\quad {\rm for}\quad u\ge 0,\,\, |p|\le \epsilon_1\cr
f(\theta\cdot t,x,u,p)\ge \xi u^2-M_1\quad {\rm for}\quad  u\le 0,\,\, |p|\le \epsilon_1.
\end{cases}
\end{equation}

Assume moreover

\begin{equation} \label{condition-eq0-0}
\end{equation}

and let

M_1^*=\sqrt {\frac{M_1}{\xi-(C+1)|\epsilon|}}.

Then, we have the following

For given $u_0\in C([0,1])$, let $(\phi^u(t,x;u_0,\theta),\phi^v(t,x;u_0,\theta))$ be the solution of  (<ref>)
with $\phi^u(0,x;u_0)=u_0(x)$ on the existence interval $[0, T_{\max}(u_0,\theta))$. Then $T_{\max}(u_0,\theta)=\infty$

$$\limsup_{t\to\infty}\|\phi^u(t,\cdot;u_0,\theta)\|_\infty\le M_1^*.

The prove is analogous to that for Lemma <ref>, we only need to prove $\|\phi^u(t,\cdot;u_0,\theta)\|_\infty$ decreases at $t$ with  $\|\phi^u(t,\cdot;u_0,\theta)\|_\infty\ge M_1^*$.

By (<ref>), one has

\begin{equation}
\begin{split}
-(C+1)|\epsilon| \|\phi^u(t,\cdot;u_0,\theta)\|_{\infty}^2 &\le -\epsilon \phi^u(t,x;u_0,\theta) \phi^v(t,x;u_0,\theta)+\epsilon (\phi^u(t,x;u_0,\theta))^2 \\
& \le (C+1)|\epsilon|\|\phi^u(t,\cdot;u_0,\theta)\|^2_\infty\quad \forall\, 0<t<T_{\max}(u_0,\theta).
\end{split}
\end{equation}

Assume $t_0\in [0,T_{\max}(u_0,\theta))$ be such that $\|\phi^u(t,\cdot;u_0,\theta)\|_\infty\ge M_1^*$.
Let $\phi^+(t,x;u^+_0,\theta\cdot t_0)$ and $\phi^-(t,x;u^-_0,\theta\cdot t_0)$ be the solutions of

\begin{cases}u_t=u_{xx}-\epsilon u_x v_x +(C+1)|\epsilon| \|\phi^u(t_0+t,\cdot;u_0,\theta)\|^2_\infty +\epsilon u^2 +f((\theta\cdot t_0 )\cdot t,x,u,u_x),\,\, 0<x<1\cr
0=v_{xx}-v+u,\quad 0<x<1\cr
\end{cases}

\begin{cases}u_t=u_{xx}-\epsilon u_x v_x -(C+1)|\epsilon| \|\phi^u(t,\cdot;u_0,\theta)\|^2_\infty +\epsilon u^2 +f(\theta\cdot t,x,u,u_x),\,\, 0<x<1\cr
0=v_{xx}-v+u,\quad 0<x<1\cr
\end{cases}

By the comparison principle for parabolic equations,

\phi^-(t,x;u^-_0,\theta\cdot t_0)\le \phi^u(t_0+t,x;u_0,\theta)\le \phi^+(t,x;u^+_0,\theta\cdot t_0)\quad \forall\, t\in [0,T_{\max}(u_0,\theta)-t_0),\,\, 0\le x\le 1.

Let $w^+(t)$ and $w^-(t)$ be the solutions of

\begin{cases}
w_t=(C+1)|\epsilon|\|\phi^u(t_0+t,\cdot;u_0,\theta)\|_\infty -\xi w^2+M_1\cr
\end{cases}

\begin{cases}
w_t=-(C+1)|\epsilon|\|\phi^u(t_0+t,\cdot;u_0,\theta)\|_\infty +\xi w^2-M_1\cr
\end{cases}

Note  that

f((\theta \cdot t_0)\cdot t,x;\phi^+(t,x;u_0^+,\theta\cdot t_0),\phi^+_x(t,x;u_0^+,\theta\cdot t_0))\le -\xi (\phi^+(t,x;u_0^+,\theta\cdot t_0))^2+M_1\quad \forall\, 0\le t\ll 1,\,\, 0<x<1

f((\theta \cdot t_0)\cdot t,x;\phi^-(t,x;u_0^-,\theta\cdot t_0),\phi^-_x(t,x;u_0^-,\theta\cdot t_0))\ge \xi (\phi^-(t,x;u_0^-,\theta\cdot t_0))^2-M_1\quad \forall\, 0\le t\ll 1,\,\, 0<x<1.

Then  by the comparison principle for parabolic equations,

w^-(t,v_0^-)\le \phi^-(t,x;u_0^-,\theta\cdot t_0)\le \phi^u (t_0+t,x;u_0,\theta)\le \phi^+(t,x;u_0^+,\theta\cdot t_0)\le w^+(t,v_0^+)\quad \forall\, 0\le t\ll 1.

Note that $w_t^+(0)<0$ and $w^-_t(0)>0$, and $w^+(0)=\|\phi^u(t_0,\cdot;u_0,\theta)\|_\infty=-w^-(0)$. Hence $\|\phi^u(t_0,\cdot;u_0,\theta)\|_\infty$ decreases at $t_0$. The proof is completed.

We now prove Theorem <ref>.

The sketch of the proof is very similar to that for Theorem <ref>, we omit the derivation here.

§.§ Competitive-cooperative tridiagonal systems

In this subsection, we consider the perturbations of $n$-dimension system in the following form:

\begin{equation}\label{tri-equation-g}
\dot{x}=f(t,x)+\epsilon g(t,x)
\end{equation}

where $g(t,x)=(g_1(t,x),\cdots,g_n(t,x))$ is a bounded and $C^1$ uniformly almost periodic function with $t$ on $x\in \mathbb{R}^n$, and $f=(f_1,f_2,\cdots,f_n)$ is a $C^{1}$-uniformly almost periodic function defined on $\mathbb{R}\times \mathbb{R}^n$ satisfies the tridiagonal structure:

\begin{equation}\label{tri-equation-f1}
\begin{split}
\dot{x}_1 &=f_1(t,x_1,x_2),\\
\dot{x}_i &=f_i(t,x_{i-1},x_i,x_{i+1}),\quad 2\leq i\leq n-1,\\
\dot{x}_n &=f_n(t,x_{n-1},x_n),\\
\end{split}
\end{equation}

The standard assumption on the tridiagonal system (<ref>) is that the variable $x_{i+1}$ affects
$\dot{x}_i$ and  $x_{i}$ affects
$\dot{x}_{i+1}$ monotonically in the same fashion. More precisely, there are $\epsilon_0>0$ and $\delta_i\in \{-1,+1\}$, $i=1,\cdots,n-1$, such that

\begin{equation*}\label{assumption}
\delta_i\dfrac{\partial f_i}{\partial x_{i+1}}(t,x)\geq \epsilon_0,\quad \delta_i\dfrac{\partial f_{i+1}}{\partial x_i}(t,x)\geq \epsilon_0,\quad 1\leq i\leq n-1,
\end{equation*}

for all $(t,x)\in \RR\times \RR^n$. This is one of the most commonly studied tridiagonal systems, called competitive-cooperative systems, in which individuals either compete or cooperate with their neighboring species. In particular, if $\delta_i=-1$ for
all $i$, then (<ref>) is called competitive.
If $\delta_i=1$ for all $i$, then (<ref>) is called
cooperative. Of course, when $n=2$, system (<ref>) naturally reduces to a two-dimensional competitive or cooperative system.

The equation (<ref>) is often used in modelling ecosystems of $n$
species $x_1,x_2,\cdots,x_n$ with a certain hierarchical structure. In such
hierarchy, $x_1$  interacts only with $x_2$, $x_n$ only with $x_{n-1}$, and for $i = 2,\cdots,n-1$, species $x_i$ interacts with $x_{i-1}$ and $x_{i+1}$.

The asymptotic behavior of bounded solutions of (<ref>) are thoroughly investigated by many others.
When $f$ is independent of $t$, then all bounded trajectories of system  converge to equilibria (see Smillie [51]). In the case that $f$ of $t$ is $T$-periodic, then every bounded solution is asymptotic to a $T$-periodic solution (Smith [52]). For $f$ of $t$ is almost-periodic,  the $\omega$-limit set of any bounded orbit contains at most two minimal sets, and each minimal set is an almost automorphic extension of the base flow; particularly, any hyperbolic $\omega$-limit set is a 1-1 extension of the base flow(see Wang [56], Fang, Gyllenberg and Wang [9]).

Similar as in [52], we now let $\hat{x}_i=\mu_ix_i, \mu_i\in\{+1,-1\}$, $1\le
i\le n,$ with $\mu_1=1, \mu_i=\delta_{i-1}\mu_{i-1}$. Then (<ref>) transforms into a new system of the same type with $\hat{\delta}_i=\mu_i\mu_{i+1}\delta_i=\mu_i^2\delta_i^2=1$.
Therefore, we can always assume, without loss of generality, that the tridiagonal
system (<ref>) is cooperative, which means that

\begin{equation}\label{positive-jacobi-condition}
\dfrac{\partial f_i}{\partial x_{i+1}}(t,x)\geq \epsilon_0,\quad  \dfrac{\partial
f_{i+1}}{\partial x_i}(t,x)\geq \epsilon_0, \qquad 1\le i\le n-1, \,(t,x)\in \RR\times \RR^n.
\end{equation}

Similar as the discussions for parabolic equations, we also consider the following induced equations

\begin{equation}\label{tri-equation-induced}
\begin{split}
\dot{x}_1 &=f_1(\theta\cdot t,x_1,x_2),\\
\dot{x}_i &=f_i(\theta\cdot t,x_{i-1},x_i,x_{i+1}),\quad 2\leq i\leq n-1,\\
\dot{x}_n &=f_n(\theta\cdot t,x_{n-1},x_n),\\
\end{split}
\end{equation}

\begin{equation}\label{tri-equation-g1}
\begin{split}
\dot{x}_1 &=f_1(\theta\cdot t,x_1,x_2)+\epsilon g_1(\theta\cdot t, x_1,\cdots,x_n),\\
\dot{x}_i &=f_i(\theta\cdot t,x_{i-1},x_i,x_{i+1})+\epsilon g_i(\theta\cdot t, x_1,\cdots,x_n),\quad 2\leq i\leq n-1,\\
\dot{x}_n &=f_n(\theta\cdot t,x_{n-1},x_n)+\epsilon g_n(\theta\cdot t, x_1,\cdots,x_n),\\
\end{split}
\end{equation}

where $\Theta=H(f,g)$ and $f$ satisfies

\begin{equation}\label{positive-jacobi-condition2}
\dfrac{\partial f_i}{\partial x_{i+1}}(\theta \cdot t,x)\geq \epsilon_0,\quad  \dfrac{\partial
f_{i+1}}{\partial x_i}(\theta \cdot t,x)\geq \epsilon_0, \qquad 1\le i\le n-1, \,(t,x)\in \RR\times \RR^n
\end{equation}

Then, the above two families of equations generate two skew-product flows $\Lambda_t,\Lambda_t^g: \mathbb{R}^n\times \Theta\to \mathbb{R}^n\times \Theta$,

\[
\begin{split}
\Lambda_t(x^0,\theta)=(\phi(t,x^0,\theta),\theta\cdot t)\\
\Lambda^g_t(x^g,\theta)=(\phi^g(t,x^g,\theta),\theta\cdot t)
\end{split}
\]

where $\phi(t,x^0,\theta)$, $\phi^g(t,x^g,\theta)$ are solutions of (<ref>), (<ref>) with $\phi(0,x^0,\theta)=x^0$, $\phi^g(0,x^g,\theta)=x^g$ respectively.

We impose dissipative assumption (D-T) on (<ref>) (see similar assumptions in [10, 58]) in the following:

\[
f_i(t,x_{i-1},x_i,x_{i+1})x_i\leq -\delta,\quad \text{for any}\ |x_i|\geq  C,|x_{i\pm 1}|\leq |x_i|\ \text{and all}\ t\in\mathbb{R}
\]

for some $\delta,C>0$. Then, the above assumption (D-T) forces any solution $\phi(t,x)$ of (<ref>) to enter the box $\mathcal{B}=\{x\in\mathbb{R}^n:|x_i|\leq C\ \text{for all }i\}$ at some finite time and then remains there (see Lemma 4.3 in [58]). Therefore, (D-T) also imply that equation (<ref>) admits a global attractor $\mathcal{A}$.

To investigate the global dynamics of (<ref>)(or (<ref>)) we assume that $\mathbb{R}^n$ equipped with the maximum norm, that is, $\|x\|=\max\{|x_1|,\cdots,|x_n|\}$ for any $x=(x_1,\cdots,x_n)$; and impose a stronger condition (D-T-1) than (D-T) in the following: there are $\delta,C>0$ such that

\[
\begin{split}
& f_i(t,x_{i-1},x_i,x_{i+1})\leq -\delta ,\quad \text{for any}\ x_i\geq  C,|x_{i\pm 1}|\leq x_i\ \text{and all}\ t\in\mathbb{R}\\
& f_i(t,x_{i-1},x_i,x_{i+1})\geq  \delta ,\quad \text{for any}\ x_i\leq  -C,|x_{i\pm 1}|\geq -x_i\ \text{and all}\ t\in\mathbb{R}.
\end{split}
\]

For any given $(t,x)\in \mathbb{R}\times\mathbb{R}^n$ define $\|g(t,x)\|=\max\{|g_1(t,x)|,\cdots,|g_n(t,x)|\}$. Let $M_g=\sup_{(t,x)\in\mathbb{R}\times\mathbb{R}^n}\|g(t,x)\|$.

Assume that

\begin{equation}\label{tri-small}
\end{equation}

Our main result concerning (<ref>) is the following

For any given $(x^0,\theta)\in \mathbb{R}^n\times \Theta$, the positive orbit $\{\Lambda^g_t(u_0,\theta),t>0\}$ is bounded in $\mathbb{R}^n\times \Theta$. Let $\omega^g(x^0,\theta)$ be the omega limit set generated by $\Lambda^g_t(x^0,\theta)$, then there exists $\epsilon_1>0$ (independent the choosing of $(x^0,\theta)\in \mathbb{R}^n\times \Theta$) such that for any $0<\epsilon <\epsilon_1$ in (<ref>), $\omega^g(x^0,\theta)$ contains at most two minimal sets and every minimal set is an almost $1$-cover of the base flow.

To prove Theorem <ref>, similar as the discussions for parabolic equations, we first need to check that (<ref>) satisfies the strong cone conditions. Assume that (D-T-1) and (<ref>) hold, we immediately have

Any solution $\phi^g(t,x,\theta)$ of (<ref>) will enter the box $\mathcal{B}=\{x\in\mathbb{R}^n:|x_i|\leq C\ \text{for all }i\}$ at some finite time and then remains there.

The proof of this lemma is very similar to that for <cit.>, we omit here.

We now check that (<ref>) satisfies assumptions (H1)-(H5).

Let $\mathcal{B}=\{x\in\mathbb{R}^n:|x_i|\leq  C\ \text{for all }i\}$ and $\mathcal{U}_0=\mathcal{\mathring{B}}\times \Theta$, $\mathcal{U}^e_0=\mathcal{\mathring{B}}\times\mathcal{\mathring{B}}\times \Theta$ and $K_0=\mathcal{A}$. Denote $z=(x^1,x^2,\theta)\in \mathcal{U}^e_0$, let $\phi(t,x^1,\theta)$ and $\phi(t,x^2,\theta)$ two solutions of (<ref>), and $\psi(t,x,\theta)=\phi(t,x^1,\theta)-\phi(t,x^2,\theta)$, then $\psi$ satisfies the linear tridiagonal systems in following form

\begin{equation}\label{linearity-tri-system}
\begin{split}
\dot{\psi}_1 &= a_{11}(t)\psi_1+a_{12}(t)\psi_2,\\
\dot{\psi}_i &= a_{i,i-1}(t)\psi_{i-1}+a_{ii}(t)\psi_i+a_{i,i+1}(t)\psi_{i+1},\qquad 2\leq i\leq n-1, \\
\dot{\psi}_n &= a_{n,n-1}(t)\psi_{n-1}+a_{nn}(t)\psi_n,
\end{split}
\end{equation}

where $\psi=(\psi_1,\cdots,\psi_n)$ and

a_{ij}(t)=\int_{0}^{1}\frac{\partial f_i}{\partial x_j}(\theta \cdot t,u_{i-1}(s,t), u_i(s,t),u_{i+1}(s,t))ds

with $u_l=s\phi_l(t,x^1,\theta)+(1-s)\phi_l(t,x^2,\theta),l=i-1,i,i+1$.
Then, by our assumptions, $a_{i,i+1}(t)\geq\varepsilon_0,\,  a_{i+1,i}(t)\geq\varepsilon_0$, for all $t\in \RR$ and $1\leq i\leq n-1$.

Let $\Phi(t,z)$ be the fundamental solution matrix of (<ref>). Then, we have

$\bullet$$\Phi(t_2,z\cdot t_1)\circ \Phi(t_1,z)=\Phi(t_1+t_2,z)$ for any $t_1,t_2>0$;

$\bullet$ the function  $\Phi(t,z)$ continuous depends on $(t,z)\in (0, + \infty )\times \mathcal{U}^e$.

We now introduce an integer-valued Lyapunov function $\sigma$. Following [51], we define a continuous map $\sigma:\mathcal{R}\rightarrow\{0,1,2,\cdots,n-1\}$ where $\mathcal{R}=\{x=(x_1,\cdots,x_n)\in \mathbb{R}^{n}:x_1\neq 0,x_n\neq 0\ \text{and if}\ x_i=0\ \text{for some}\ i,2\leq i\leq n-1,\ \text{then}\ x_{i-1}x_{i+1}<0\}

\[
\sigma(x)=\#\{i|x_i=0\ \text{or}\ x_i x_{i+1}\leq 0\},
\]

where $\#$ denotes the cardinal of the set. Then, $\mathcal{R}$ is open and dense in $\mathbb{R}^{n}$ and it is the maximal domain on which $\sigma$ is continuous. For any given integer $0\leq k\leq n$, the sets

\[
C_k:=cl\{x\in \mathcal{R}:\ N(x)\leq k-1\}
\]

are $k$-cones, particularly, $C_0=\{0\}$ and $C_{n-1}=\mathbb{R}^n$. Moreover, by <cit.> or <cit.>, we have $\Phi(t,z)$ and $C_i$ satisfy (H1)-(H4). Define the map $\pi$ as:

\[
x=(x_1,\cdots,x_n)\longmapsto x_1,\quad \mathbb{R}^n\to \mathbb{R},
\]

and let $\mathcal{H}=\pi^{-1}(0)$, then $\dim \mathcal{H}=n-1$. Assume that $\Phi(t_0,z)x\in (C_i\setminus C_{i-1})\cap \mathcal{H}$, for some $t_0>0$, $z\in \mathcal{U}^e$, $x\in \mathbb{R}^n$, $i\in \{1,\cdots,n\}$, then $x\notin C_i$. We have checked (H5).
Consequently, we have the following lemma parallel to Proposition <ref>.

There exists open neighborhood $\mathcal{V}_0$ of $K_0$ (i.e., there exists a family of open sets $\mathcal{V}_0(\theta)$ of $K_0(\theta)$ such that $\mathcal{V}_0=\{(u,\theta)|u\in K_0(\theta),\,\theta\in\Theta\}$) and $\epsilon_0>0$ such that for any
continuous skew-product semiflow $\tilde\Lambda=(\tilde \phi,\theta)$ on $\mathcal { V }_0$ with $\tilde \phi(t,\cdot,\theta) \in C ^ { 1 } ( \mathcal {U}_0(\theta), \mathbb{R}^n ) , t \in \left[\frac{1}{2}, 1 \right]$ and

\begin{equation}\label{Tri-c1}
\|\phi(t,\cdot,\theta) - \tilde \phi(t,\cdot,\theta) \|_{C^1(\mathcal{U}(\theta),X)} < \epsilon_0 \quad \text { for all } \frac{1}{2} \leq t \leq 1, \ \theta\in \Theta
\end{equation}

such that the any minimal set of $\tilde \Lambda_t$ contained in $\mathcal{V}_0$ is an almost $1$-cover of the base flow $\Theta$; moreover, any $\omega$-limit set of $\tilde \omega(u_0,\theta)\subset \mathcal{V}_0$ for the skew-product semiflow $\tilde \Lambda_t$ contains at most two minimal sets.

Take $\epsilon_0>0$ in Lemma <ref> sufficient small, then for any $(u_0,\theta)\in \mathcal{U}_0$, $\tilde \omega(u_0,\theta)$ contains at most two minimal sets.

Now we prove Theorem <ref>.

Based on Lemma <ref>, the proof is very similar to that for parabolic equations, we omit the detail.

§ SOME REMARKS

1) Remark on the assumption of $C^1$-perturbations. Note that even for a finite dimension system arbitrary small $C^0$-perturbations could have a chaos (see an example in [16] for negative cyclic feedback systems), that is, the structure of $\omega$-limit sets may not keep under small $C^0$-perturbations. Therefore, $C^0$-perturbations are not sufficient to assure the stability of the structure of $\omega$-limit sets.

2) Remark on the advantage of invariant cones method. Back to equation (<ref>), similar functions to the zero number function $Z$, called the discrete Lyapunov functionals (DL functionals), are widely exist in many other systems ([2, 13, 15, 16, 51, 26, 27, 28, 29, 32, 30]). However, the forms of DL functionals are usually different for different systems, and not easy to find; hence, the DL functional may not work for its' perturbed systems. To this extend, the invariant nested cones method has at least two advantages as below : a) we only need to investigate the abstract systems that of strong cone properties, and check the given systems satisfy strong cone conditions; b) if the un-perturbed system satisfies the strong cone conditions, then with small $C^1$-perturbations one can construct new invariant nested cones near the original invariant nested cones; and hence, the structure of $\omega$-limit sets of the system may persistent under these perturbations. Here, we do not need to know the exact forms of the DL functionals for perturbed systems.

4) Remark on main results. The results generalized the conclusions in Tereščák [54] for periodic systems to time almost-periodic systems. Nevertheless, it does not mean that these generalizations can be obtained directly from the approaches in Tereščák [54], since there are some essential difficulties in dealing with non-autonomous systems, for example, by using the Poincaré map, Tereščák [54] proved that the DL functional induced by nest invariant cones is constant on any $\omega$-limit sets that generated by Poincaré map. This property plays a very important role in characterizing the shape of $\omega$-limit sets. While for non-autonomous systems, due to the lacking of Poincaré map, even for some specific systems, like (<ref>) (see [49], the constancy of DL functional on a minimal set is under the assumption that the minimal is unique ergodic) or tridigonal competitive and cooperative systems, one does not get the constant properties on a minimal set not to mention that for $\omega$-limit set. Therefore, to investigate the dynamics for non-autonomous systems, we first need to make some of the proofs in [54] rigorous (although the conclusions in [54] are correct, some details of the proof are not strictly given), and then generalized some of these results to non-autonomous cases in the framework of skew-product semiflow. Second, considering that the constant properties of DL functional are difficult to obtain for non-autonomous systems, we establish the relationship between the index of invariant cones and the solution of the linear systems, and then develop some tools for the non-autonomous systems. For instance, we defined a partial order on a compact invariant set $E\subset\mathcal{V}_0$  (see related the lemmas in  section <ref>), and then used this partial order and some theories for non-autonomous systems to investigate the structure of $\omega$-limit set in the case that $d=1$ in (H5).

5) Remark on the applications. For system (<ref>) in the autonomous case, that is $f$ and $c$ are independent of $t$, if $\epsilon>0$ is not sufficiently small, the results in [12] showed that for general $c(x)$ and $\nu(x)$, the $\omega$-limit can have arbitrary high dimension. Indeed, even when $f$ is linear, for any positive integer $m$, by suitable choosing of $c$ and $\nu$, one can find a trajectory which is dense in a $m$-torus; moreover, Smale horseshoes and Bernoulli shift dynamics can happen. Only under severe restrictions on $c$, $\nu$ can (<ref>) have simple dynamics; for instance, if $c(x), \nu(x)\geq 0$, then it belongs to the monotone dynamical systems, and hence, by the theories in Hirsch [22], “most” bounded trajectories of  (<ref>) approaches a set of equilibria; particularly, if  $c(x), \nu(x)$ are positive, then “most” bounded trajectories of  (<ref>) asymptotic to a single equilibrium, see Poláčik [36]. In the case that both $f$ and $c$ of $t$ are $T$-periodic, then most orbits converge to linearly stable cycles, see [18, 37]. As for system (<ref>) is periodic, the conclusions in Tereščák in [54] imply that if the periodic system (<ref>) is dissipative, for $\epsilon>0$ is sufficiently small (without the restriction that $c(x), \nu(x)$ are positive), the $\omega$-limit set of any bounded solutions of  (<ref>) is a periodic orbit, this is so called the $C^1$-structural stability of the $\omega$-limit sets of (<ref>). Nevertheless, for almost periodic monotone systems, there is no convergence property for “most” bounded orbits,
in the current article, when $f$ and $c$ (<ref>) in of $t$ are uniformly almost periodic, also without positivity assumption on $c(t,x), \nu(x)$, the shape of $\omega$-limit sets and minimal sets of (<ref>) were given, when $0<\epsilon\ll 1$, which in fact proved that structure of $\omega$-limit sets of (<ref>) can keep under this term of non-local perturbations.

Note that most research results about (<ref>) are that $f$ is of special case, like $f$ is logistic source and the initial value is non-negative, while the dynamics of (<ref>) is still unknown when $f$ is general dependent and $\epsilon$ is not small. The results in the present paper tells us, if the chemotaxis term is small enough, then $\omega$-limit set of (<ref>) has the same structure as that of (<ref>).

Consider (<ref>) with Dirichlet boundary conditions,  by suitable modifications in section <ref>, one also obtain similar results as those for (<ref>) with Neumman boundary conditions.

§ PROOF OF PROPOSITION <REF>

To prove Proposition <ref>, we always assume that (H1)-(H4) are satisfied in this section.

The sketch of the proof of Proposition <ref> is similar to <cit.> for discrete systems, and can be split into the following four steps:

1)  check that the assumptions in Lemma  <ref> are satisfied;

2) find a suitable integer $N_0$ and construct finite nested cones $\mathcal{C}_i$ near the original cones $C_i$ ($0\leq i \leq N_0$);

3) give a detailed proof for Proposition <ref> (1) and (2);

4) prove Proposition <ref> (3).

§.§ Exponential separation property

Define the skew-product semiflow $\Lambda^{e}_t:X\times X\times \Theta\to X\times X\times \Theta$ as follows

\Lambda^e_t(x, y,\theta)=(\phi(t,x,\theta), \phi(t,y,\theta),\theta), \quad(t,x, y,\theta)\in \mathbb{R}^+\times X\times X\times \Theta.

Let $Z=\bigcup_{\theta\in\Theta}\{(x,y,\theta)|x,y\in K_0(\theta)\}$, then $Z$ is a compact invariant of $\Lambda^e_t$. For any $z=(x,y,\theta)\in Z$, denote $z\cdot t=(\phi(t,x,\theta), \phi(t,y,\theta),\theta\cdot t)$,
$\{T(t,z) :(t, z) \in (0,\infty)\times Z\}-$ the family of compact operators in $L(X, X)$ satisfy

\begin{equation}\label{difference-equation}
T(t,x,y,\theta)(x-y)=\phi(t,x,\theta)-\phi(t,y,\theta) \text { for all }(t,x, y,\theta) \in [0,1]\times Z.
\end{equation}

$C_{1} \subset C_{2} \subset \ldots \subset$
$C_{i} \subset \ldots$ the family of nested $k_{i}$-cones with $k_{i+1}>k_{i}$ for all $1 \leq i<N, C_{N}=X$.

For any $v \in X\setminus\{0\}$ there is an $j \in \mathbb{N}$ such that

\begin{equation}\label{map-into}
T(t,z)v \in C_{j} \backslash\{0\},\quad t\in [\frac{1}{2},1].
\end{equation}

Define the map $\bar\phi$ as the following

\bar\phi(t,x, y,\theta)=\left(\phi(t,x,\theta), \phi(t,y,\theta)\right), \quad(t,x, y,\theta) \in(0,\infty)\times Z.

Noticing that $K_{0}$ is an invariant subset of $\Lambda_t$, the function $\phi(t,\cdot,\cdot)$ is continuous and surjective on $K_0$. For any two distinct points $(x,\theta), (y,\theta)\in K_0$, since $v=x-y\neq 0$, equations (<ref>), (<ref>) imply that $\phi(t,\cdot,\cdot)$ is injective on $K_{0}$. Hence $\bar\phi(t,\cdot,\cdot,\cdot)$ on $Z$ is a homeomorphism. These ensure that all the assumptions in Lemma  <ref> with respect to $C_i$ ($i=1,2,\cdots$) are satisfied.

Hereafter, $V_z^i$ and $\operatorname{Anih}(L_z^i)$ ($i=1,2,\cdots$) are assumed to be the corresponding invariant spaces with respect to the exponential separation along $Z$ associated with the $k_i$-cone $C_i$ in Definition <ref>. Then, we have the following lemma with these invariant spaces.

Assume that (H1)-(H4) are satisfied. Then $V _ {z} ^ { i } \subset V _ { z } ^ { i + 1 } , L _ { z} ^ { i } \subset L _ {z} ^ { i + 1 }$
for all $z\in Z , 1 \leq i < N$. Moreover

(V_z^{ i + 1} \cap \operatorname{Anih}( L_z^ { i })) \oplus V _ {z} ^ { i } = V_z^ { i + 1 },\quad z \in Z , 1 \leq i < N

and each bundle $(V _ {z} ^ { i + 1 } \cap \operatorname{Anih}(L_z^ { i }))\times Z, 1 \leq i <N$ is a compact subbundle of $X\times Z$ invariant under the linear skew-product semiflow $(T,\sigma)$ on $X\times Z$.

It can be proved by using similar deductions as those in <cit.>.

§.§ Construct a new family of nested invariant $k_i$-cones $\mathcal{C}_i$

The following lemma means that for an infinite Banach space $X$, if one can construct infinite many $k_i$-cones such that the linear operator $T(t,z)$ on $X$ satisfies (H1)-(H4), then for any given decaying exponent, one can find a sufficient large $k_i$-cone, and a $k_i$-codimension subspace out side this cone, such that all the points belong to this subspace will asymptotic to zero at given decaying exponent under the action of $T(t,z)$.

Assume that (H1)-(H4) are satisfied and $N=\infty$. Then for any $0<\lambda_1<1$ there is an $i _ { 0 } \in \mathbb { N }$ such that

\begin{equation}\label{contract-equa1}
\| T(t,z) u \| \leq\lambda_1^t \| u \|
\end{equation}

for any $(t,z) \in [\frac{1}{2},\infty)\times Z$  and $u \in \operatorname{Anih}( L _ { z} ^ { i _ { 0 } }) \backslash \{ 0 \} $.

Note that $Z$ is a compact invariant set with respect to $\Lambda^e_t$ and $T(t_1+t_2,z)=T(t_2,z\cdot t_1)\circ T(t_1,z)$ for any $t_1,t_2\in (0,+\infty)$, $z\in Z$. It is sufficient to prove (<ref>) for $t\in[\frac{1}{2},1]$. Let

\tau=\max\{\sup_{{(t,z)}\in [0,\frac{1}{2}]\times Z}\|T(t,z)\|,1\}

and fix some $0<\lambda_1<1$, let $\tilde \lambda_{1}=\dfrac{\lambda^2_1}{\tau^2}$.
For the special case $t=\frac{1}{2}$, it can be proved by using the same deduction in <cit.>. To be more precise, there exists $i_0\in \mathbb{N}$ such that

\|T(\frac{1}{2},z)u\|\leq \tilde \lambda^{\frac{1}{2}}_1\|u\|,\quad \forall z\in Z,\ u\in \operatorname{Anih}(L^{i_0}_z).

For any given $t\in(\frac{1}{2},1]$, one has

\|T(t,z)u\|\leq \|T(t-\frac{1}{2},z\cdot\frac{1}{2})\|\|T(\frac{1}{2},z)u\|\leq \tau\tilde\lambda^{\frac{1}{2}}_1\|u\|=\tau\frac{\lambda_1}{\tau}\|u\|\leq \lambda^t_1\|u\|.

By Lemma <ref>, we know that $V_{z}^{i}\times Z, L_{z}^{i}\times Z$ are continuous bundles and there are constants $M_{i}>0,0<\gamma_{i}<1$ ($i=1,2, \ldots$) for the corresponding exponential separations. Denote by $P_{z}^{i}$ the projection of $X$ on $V_{z}^{i}$ along $\operatorname{Anih}(L_{z}^{i})$ and the complementary projection $Q_{z}^{i} :=I-P_{z}^{i}$ for each $z \in Z, i=1,2, \ldots$, where $I$ is the identity map. Let also $V_{z}^{0}=\{0\}, L_{z}^{0}=\{0\}, P_{z}^{0}=0, Q_{z}^{0}=I$ for all
$z \in Z$. If $N=\infty$, then for fixed $0<\lambda_{1}<1$,  by Lemma <ref> there is an
$i_{0} \in \mathbb{N}$ such that

\begin{equation}\label{contrac-condition}
\text{if } P_{z}^{i_{0}} u=0 \quad \text { then } \quad\|T(t,z) u\| \leq \lambda_{1}^t\|u\|,\ t\in[\frac{1}{2},\infty).
\end{equation}

Therefore, we define $N_{0}$ as the following:

N,\quad  N<\infty,\\
i_0,\quad  N=\infty.
\end{cases}

For each $s\geq 0$, $z \in Z$ and $1 \leq i \leq N_0$ define the sets

\begin{equation*}
C_{z}^{i}(s)=\{u \in X :\|Q_{z}^{i} u\| \leq s\|P_{z}^{i} u\|\},\quad
\overline C_{z}^{i}(s)=C_{z}^{i}(s)\cap S
\end{equation*}

\begin{equation*}
D_{z}^{i}(s)=\{u \in X :\|P_{z}^{i} u\| \leq s\|Q_{z}^{i} u\|\},\quad
\overline D_{z}^{i}(s)=D_{z}^{i}(s)\cap S
\end{equation*}

where $S=\{u|\|u\|=1,u\in X\}$. Define for $1 \leq i \leq N_{0}$, $s \geq 0$ the sets

\begin{equation*}
\begin{array}{l}{\mathcal{C}_{i}(s) :=\bigcup_{z \in Z}C_{z}^{i}(s)} \\ {\mathcal{D}_{i}(s) :=\bigcup_{z \in Z}D_{z}^{i}(s)}.
\end{array}
\end{equation*}

It is not hard to see that if $0 \leq s_{1}<s_{2}$ and $1 \leq i \leq N_{0}$ then $\mathcal{C}_{i}(s_{1}) \subset \mathcal{C}_{i}(s_{2})$ as well as $\mathcal{D}_{i}(s_{1}) \subset$

The following lemma shows that for $0<s\ll 1$, $\mathcal{C}_{i}(s)$ are $k_{i}$-cones.

Assume that (H1)-(H4) are satisfied. Then, there exists a $0 <\delta_{0}\leq 1$ such that

\begin{equation}\label{inter-zero}
\mathcal{C}_{i}(s) \cap \mathcal{D}_{i}(s)=\{0\},  \quad 0 \leq s \leq \delta_{0},\ 1 \leq i \leq N_{0}.
\end{equation}

Moreover, each $\mathcal{C}_{i}(s)$ is a $k_{i}$-cone.

Suppose on the contrary that (<ref>) is not established for some fixed $1\leq i_0\leq N_0$. Then for any $n\in\mathbb{N}$, there exists $u_n\in \mathcal{C}_{i_0}(\frac{1}{n})\cap\mathcal{D}_{i_0}(\frac{1}{n})$ with $\|u_n\|=1$. By the definitions of $\mathcal{C}_i(\frac{1}{n})$ and $\mathcal{D}_i(\frac{1}{n})$, there are $z^1_n, z^2_n\in Z$ such that

\begin{equation}\label{two-estimates}
\|Q^{i_0}_{z^1_n}u_n\|\leq \frac{1}{n}\|P^{i_0}_{z^1_n}u_n\|,\quad \|P^{i_0}_{z^2_n}u_n\|\leq \frac{1}{n}\|Q^{i_0}_{z^2_n}u_n\|.
\end{equation}

Let $w^j_n=Q^{i_0}_{z^j_n}u_n$, $v^j_n=P^{i_0}_{z^j_n}u_n$, $j=1,2$. Then, $w^j_n\in \operatorname{Anih}(L^{i_0}_{z^j_n})$,$v^j_n\in V^{i_0}_{z^j_n}$.
Recall that $Z$ is compact, for simplicity, we assume that $z^j_n\to z^j$, as $n\to\infty$ ($j=1,2$). Therefore, $w^1_n\to 0$ and $u_n\to v^1\in V^{i_0}_{z^1}$, as $n\to\infty$. Meanwhile, we can also obtain that $v^2_n\to 0$ and $u_n\to w^2\in \operatorname{Anih}(L^{i_0}_{z^2})$. Hence, $v^1=w^2\in V^{i_0}_{z^1}\cap \operatorname{Anih}(L^{i_0}_{z^2})\setminus\{0\}$, a contradiction to that $V^{i_0}_{z^1}\cap \operatorname{Anih}(L^{i_0}_{z^2})\subset C_{i_0}\cap \operatorname{Anih}(L^{i_0}_{z^2})=\{0\}$. Thus, equation (<ref>) is established for some $\delta_0>0$.

We now turn to prove that for such $0<s \leq \delta_{0}$, $\mathcal{C}_i(s)$ is $k_i$-cone. Noticing that for each $1 \leq i \leq N_{0}$ the bundles $V_{z}^{i}\times Z, L_{z}^{i}\times Z$ are compact and the following relations are satisfied

\begin{equation}\label{cone-subset}
\mathcal{C}_{i}(0)=\bigcup_{z \in Z} V_{z}^{i} \subset C_{i}, \mathcal{D}_{i}(0) \cap C_{i}=\bigcup_{z \in Z} \operatorname{Anih}\left(L_{z}^{i}\right) \cap C_{i}=\{0\}.
\end{equation}

These combine with (<ref>) imply that $\mathcal{C}_i(s)$ contains a $k_i$-dimension subspace of $X$ and any $k_i+1$-dimension subspace of $X$ cannot be contained in $\mathcal{C}_i$. Moreover,  $\lambda \mathcal{C}_{i}(s)=\mathcal{C}_{i}(s)$ for all $\lambda \in \mathbb{R} \backslash\{0\}$. The rest is to prove that $\mathcal{C}_{i}(s)$ is a closed subset of $X$. Let $u_n\in \mathcal{C}_i(s)$ be a convergent sequence with $u_n\to u$, then $\|u_n\|$ is bounded and $\|Q^i_{z_n}u_n\|\leq s\|P^i_{z_n}u_n\|$.
Recall that $\{P_{z}^{i}, Q_{z}^{i} :z \in Z\}$ is a compact set of continuous projections of
$X $, with out loss of generality, we assume that $P^i_{z_n}u_n\to P^i_zu$, $Q^i_{z_n}u_n\to Q^i_zu$, $z_n\to z$, as $n\to\infty$. Thus, $\|Q^i_{z}u\|\leq s\|P^i_{z}u\|$ and $u\in \mathcal{C}_i(s)$.

In the following, we list some symbols that will be used later.

* Let $c_P=\sup\{\|P^i_z\|, 1\leq i \leq N_0, \ z\in Z\}$, $c_Q=\sup\{\|Q^i_z\|, 1\leq i \leq N_0, \ z\in Z\}$. Denote

\[
\begin{split}
\zeta&=\max \{\sup \{\|T(t,z)\|: {\color{red} (t,z)} \in[\frac{1}{2},1]\times Z\}, 1\} \\
r&=\max\{c_P\cdot c_Q, 1\}.
\end{split}
\]

* Let $\delta_0$ be defined as in (<ref>) and take $0<\delta<\delta_{0}$, when $N=\infty$ we assume

\begin{equation}\label{coeff-relation}
\lambda_{0} :=2 \delta+\frac{\delta \zeta+\lambda_{1}^{\frac{1}{2}}}{1-\delta}<1.
\end{equation}

* Define

M=\max \left\{M_{i} : 1 \leq i \leq N_{0}\right\}, \gamma=\min \left\{\gamma_{i} : 1 \leq i \leq N_{0}\right\}>1 \quad \text { and } \quad c=\frac{1}{\delta}

where $M_i$, $\gamma_i$, $i=1,\cdots,N_0$ are the constants associated with the $k_i$-dimensional exponential separations.
Then, there is $T_{1}>0$ such that

\begin{equation}
c M e^{-\gamma t} \leq(8 r)^{-N_{0}} \delta \quad \text { for all } t \geq T_{1}.
\end{equation}

* Let $T_{0}>T_{1}+1$ be such that

\begin{equation}\label{large-const}
\left(\lambda_{0}-\delta\right)^{t-T_{1}}(\delta+\zeta)^{T_{1}} \leq \lambda_{0}^{t} \quad \text { for all } t \geq T_{0}.
\end{equation}

We now construct invariant $k_i$-cones $\mathcal{C}_i$.

\mathcal{C}_{i} :=\mathcal{C}_{i}((2 r)^{i-N_{0}} \delta), \quad 1 \leq i \leq N_{0}.

Let $1 \leq i<N_{0}$ and
$u \in \mathcal{C}_{i}((2 r)^{i-N_{0}} \delta)$. Then there is a $z \in Z$ such that

\|Q_{z}^{i} u\|\leq(2 r)^{i-N_{0}} \delta\|P_{z}^{i} u\| .

By virtue of Lemma <ref>,

Q_{z}^{i+1} u=Q_{z}^{i+1}Q_{z}^{i} u, \quad P_{z}^{i} u=P_{z}^{i} P_{z}^{i+1} u

for all $z \in Z, u \in X$ and $ 1 \leq i<N_{0}$. Therefore, for any $u \in \mathcal{C}_{i}((2 r)^{i-N_{0}} \delta)$,

\begin{equation*}
\begin{split}
&\|Q_{z}^{i+1} u\|=\|Q_{z}^{i+1} Q_{z}^{i} u\| \leq\|Q_{z}^{i+1}\|\|Q_{z}^{i} u\| \leq (2 r)^{i-N_{0}}\delta\|Q_{z}^{i+1}\| \|P_{z}^{i} u\|\\
&=(2 r)^{i-N_{0}} \delta\|Q_{z}^{i+1}\|\|P_{z}^{i} P_{z}^{i+1} u\| \leq(2 r)^{i-N_{0}} \delta\|Q_{z}^{i+1}\|\|P_{z}^{i}\|\|P_{z}^{i+1} u\| \\
&\leq r(2 r)^{i-N_{0}} \delta\|P_{z}^{i+1} u\|\leq (2 r)^{i+1-N_{0}} \delta\|P_{z}^{i+1} u\|.
\end{split}
\end{equation*}

That is, $u \in \mathcal{C}_{i+1}((2 r)^{i+1-N_{0}} \delta)$. In fact, we have

\mathcal{C}_{1} \subset \operatorname{int}(\mathcal{C}_{2}) \cup\{0\} \subset \mathcal{C}_{2} \subset \cdots \subset \mathcal{C}_{N_{0}-1} \subset \operatorname{int}(\mathcal{C}_{N_{0}}) \cup\{0\} \subset \mathcal{C}_{N_{0}}.

§.§ Preparation for the proof of  Proposition <ref>(1)

It is not hard to see that (<ref>) also implies that

\begin{equation}\label{cone-parame-subset}
\mathcal{C}_{i}(s) \subset \mathcal{C}_{i}(\delta_{0}) \subset\{u \in X :\|Q_{z}^{i} u\| \leq \frac{1}{\delta_{0}}\|P_{z}^{i}u\|\}
\end{equation}

for all $0<s \leq \delta_{0}, 1 \leq i \leq N_{0}$ and $z \in Z$.

By the definition of exponential separation

\begin{equation}\label{exponential-inequa}
\|Q_{z}^{i} v\| \leq c\|P_{z}^{i} v\| \quad \text { implies } \quad\|Q_{z\cdot t}^{i} T(t,z) v\| \leq(8 r)^{-N_{0}} \delta\|P_{z\cdot t}^{i} T(t,z) v\|
\end{equation}

for all $t \geq T_{1},z \in Z, v \in X, 1 \leq i \leq N_{0}$. This estimate together
with (<ref>) and (<ref>) ensure Proposition <ref> (1) correct for $\phi$ and $(x,\theta), (y,\theta)\in K_0(\theta)$ without perturbation.

The following lemma is about the persistent property of $\overline C_{z}^{i}(s)$.

Assume that $0\leq s_1<s_2$ and $1\leq i\leq N_0$. Let $\varrho_0=\min\{\frac{s_2-s_1}{2(c_Q+c_Ps_1)(1+s_1)},\frac{1}{2c_P(1+s_1)}\}$, then for any $z\in Z$ and $0<\varrho\leq\varrho_0$, one has

B(u,\varrho)\cap S\subset \overline C^i_z(s_2),\quad \text{for all} \ u\in \overline C^i_z(s_1)

where $B(u,\varrho)=\{u_0|\|u-u_0\|<\varrho,\ u_0\in X\}$.

Fix $u_1\in \overline C^i_z(s_1)$ and choose $u_2\in S$ with $\|u_2-u_1\|<\varrho_0$, then

\begin{equation}\label{cone-inequa}
\begin{split}
\|Q^i_zu_2\|&=\|Q^i_zu_2-Q^i_zu_1+Q^i_zu_1\|\\
&\leq \|Q^i_z(u_2-u_1)\|+\|Q^i_zu_1\|\\
&\leq c_Q\varrho_0+s_1\|P^i_zu_1\|\\
&\leq  c_Q\varrho_0+s_1(\|P^i_z(u_1-u_2)\|+\|P^i_zu_2\|)\\
& \leq c_Q\varrho_0+s_1c_P\varrho_0+s_1\|P^i_zu_2\|=(c_Q+s_1c_P)\varrho_0+s_1\|P^i_zu_2\|\\
& \leq \frac{s_2-s_1}{2(1+s_1)}+s_1\|P^i_zu_2\|.
\end{split}
\end{equation}

On the other hand, it is not hard to see that

\|P^i_zu_2\|\geq \|P^i_zu_1\|-\|P^i_z(u_2-u_1)\|\geq \frac{1}{1+s_1}-c_P\varrho_0\geq \frac{1}{2(1+s_1)}.

This combining with (<ref>) imply

\|Q^i_zu_2\|\leq s_2\|P^i_zu_2\|.

The proof of this lemma is completed.

We now consider the perturbed skew-product semiflows of $\Lambda_t$. Assume that there is another skew-product semiflow $\tilde \Lambda_t:X\times \Theta\rightarrow X\times \Theta$ of the following form

\begin{equation}\label{pertur-skew-product-semiflow}
\tilde \Lambda_{t}(x,\theta)=(\tilde \phi(t,x,\theta),\theta\cdot t),\quad t\geq 0,\, (x,\theta)\in X\times \Theta.
\end{equation}