# Time-preserving Structural Stability of Hyperbolic Differential Dynamics
with Noncompact Phase Spaces

Xiongping Dai  Department of Mathematics
Nanjing University
Nanjing, 210093, P. R. CHINA<EMAIL_ADDRESS>
(Date: May 17, 2008)

###### Abstract.

Let $S\colon\mathbb{E}\rightarrow\mathbb{R}^{n}$ where
$T_{w}\mathbb{E}=\mathbb{R}^{n}$ for all $w\in\mathbb{E}$, be a
$C^{1}$-differential system on an $n$-dimensional Euclidean $w$-space
$\mathbb{E}$, which naturally gives rise to a flow $\phi\colon(t,w)\mapsto
t_{\cdot}w$ on $\mathbb{E}$, and let $\Lambda$ be a $\phi$-invariant closed
subset containing no any singularities of $S$. If $\Lambda$ is compact and
hyperbolic, then Anosov’s theorem asserts that $S$ is structurally stable on
$\Lambda$ in the sense of topological equivalence; that is, for any
$C^{1}$-perturbation $V$ close to $S$, there is an $\varepsilon$-homeomorphism
$H\colon\Lambda\rightarrow\Lambda_{V}$ sending orbits $\phi(\mathbb{R},w)$ of
$S$ into orbits $\phi_{V}(\mathbb{R},H(w))$ of $V$ for all $w$ in $\Lambda$.
In this paper, using Liao theory Anosov’s result is generalized as follows:
Let $\psi_{V}\colon\mathbb{R}\times\Sigma\rightarrow\Sigma$ be the cross-
section flow of $V$ relative to $S$ locally defined on the Poincaré cross-
section bundle $\Sigma=\bigcup_{w\in\Lambda}\Sigma_{w}$ of $S$, where
$\Sigma_{w}=\left\\{w^{\prime}\in\mathbb{E}\,|\,\langle
S(w),w^{\prime}-w\rangle=0\right\\}$. If $S$ is hyperbolic on $\Lambda$ and
$V$ is $C^{1}$-close to $S$, then there is an $\varepsilon$-homeomorphism
$w\mapsto H(w)\in\Sigma_{w}$ from $\Lambda$ onto a closed set $\Lambda_{V}$
such that $\psi_{V}(t,H(w))=H(t_{\cdot}w)$ for all $w\in\Lambda$, where
$\Lambda$ need not be compact. Finally, an example is provided to illustrate
our theoretical outcome.

###### Key words and phrases:

Structural stability, hyperbolic differential system, Liao standard system

###### 2000 Mathematics Subject Classification:

Primary 37C10, 34D30; Secondary 37C20, 37D20

This project was supported by NSFC (No. 10671088) and 973 project (No.
2006CB805903)

##  1\. Introduction

In [6, 7], professor S.-T. Liao established the theory of standard systems of
differential equations for $C^{1}$-differential dynamical systems on compact
Riemannian manifolds. Then he systematically applied methods in the
qualitative theory of ODE to study stability problems of differentiable
dynamical systems via his theory [8]. We in [2, 3] generalized in part Liao’s
theory to differential systems on Euclidean spaces. Via the generalized, in
turn we can apply the approaches of ergodic theory and differentiable
dynamical systems to the study of the qualitative theory of ODE [3, 4]. In the
present paper, we continue to perfect Liao theory and give a further
application.

Assume, throughout this paper, that
$S\colon\mathbb{E}\rightarrow\mathbb{R}^{n}$ is a $C^{1}$-vector field on an
$n$-dimensional Euclidean $w$-space $\mathbb{E}$, where $n\geq 2$ and
$T_{w}\mathbb{E}=\mathbb{R}^{n}$ for all $w$, and the equation $\dot{w}=S(w)$
naturally induces a continuous-time dynamical system
$\phi\colon\mathbb{R}\times\mathbb{E}\rightarrow\mathbb{E};\ (t,w)\mapsto
t_{\cdot}w$ on the phase-space $\mathbb{E}$. Let

$\Sigma={\bigcup}_{w\in\mathbb{E}}\Sigma_{w}\quad\textrm{ where
}\Sigma_{w}=\left\\{w^{\prime}\in\mathbb{E}\,|\,\langle
S(w),w^{\prime}-w\rangle=0\right\\},$

be the cross-section bundle of $S$. Then, $S$ gives naturally rise to a formal
(local) Poincaré cross-section flow

$\psi\colon\mathbb{R}\times\Sigma\rightarrow\Sigma;\ (t,w+x)\mapsto
t_{\cdot}w+\psi_{t,w}x,$

where $w^{\prime}=w+x$ means $w^{\prime}\in\Sigma_{w}$ and where
$\psi_{t,w}\colon\Sigma_{w}-w\rightarrow\Sigma_{t_{\cdot}w}-t_{\cdot}w$ is
locally well defined for any $(t,w)\in\mathbb{R}\times\mathbb{E}$ by
$\psi_{t,w}x=\phi(t_{0},w+x)-t_{\cdot}w$, where $t_{0}$ is the first
$t^{\prime}>0$ when $t>0$ or the first $t^{\prime}<0$ when $t<0$ with
$\phi(t^{\prime},w+x)\in\Sigma_{t_{\cdot}w}$. Clearly, $\psi$ is a local skew-
product flow based on $\phi$ satisfying $\psi_{t,w}\textbf{0}=\textbf{0}$.

Let $\mathfrak{X}^{1}(\mathbb{E})$ be the space of all $C^{1}$-vector fields
on $\mathbb{E}$ endowed with the $C^{1}$-topology induced by the usual
$C^{1}$-norm $\|\cdot\|_{1}$. Then, for any
$V\in\mathfrak{X}^{1}(\mathbb{E})$, on $\Sigma$ we may also naturally define a
formal local skew-product flow

$\psi_{V}\colon\mathbb{R}\times\Sigma\rightarrow\Sigma;\ (t,w+x)\mapsto
t_{\cdot}w+\psi_{V;t,w}x.$

Note here that $\psi_{V;t,w}\textbf{0}$ need not equal 0 when $V\not=S$.

Let $\mathbb{T}_{w}=T_{\textbf{0}}\Sigma_{w}$ be the $(n-1)$-dimensional
tangent space to the hyperplane $\Sigma_{w}$ at $w+\textbf{0}$ for all
$w\in\mathbb{E}$ and $\mathbb{T}=\bigcup_{w\in\mathbb{E}}\mathbb{T}_{w}$
called the _transversal tangent bundle to $S$_ over $\mathbb{E}$. Clearly,
$\mathbb{T}_{w}=\Sigma_{w}-w=\\{x\in\mathbb{R}^{n}\,|\,\langle
S(w),x\rangle=0\\}$. Then, we can define naturally the linear skew-product
flow transversal to $S$

$\Psi\colon\mathbb{R}\times\mathbb{T}\rightarrow\mathbb{T};\
(t,(w,x))\mapsto(t_{\cdot}w,\Psi_{t,w}x),$

where $\Psi_{t,w}\colon\mathbb{T}_{w}\rightarrow\mathbb{T}_{t_{\cdot}w}$ is
defined as $\Psi_{t,w}=D_{\textbf{0}}\psi_{t,w}$ for any
$(t,w)\in\mathbb{R}\times\mathbb{E}$, associated with $S$.

Recall that a $\phi$-invariant closed subset $\Lambda$ is said to be
_hyperbolic_ , provided that there exist constants $C\geq 1,\lambda<0$ and a
continuous $\Psi$-invariant splitting

$\mathbb{T}_{w}=T_{w}^{s}\oplus T_{w}^{u}\quad w\in\Lambda$

such that

$\displaystyle\|\Psi_{t_{0}+t,w}x\|$ $\displaystyle\leq C^{-1}\exp(\lambda
t)\|\Psi_{t_{0},w}x\|\quad\forall\,x\in T_{w}^{s}$ and
$\displaystyle\|\Psi_{t_{0}+t,w}x\|$ $\displaystyle\geq C\exp(-\lambda
t)\|\Psi_{t_{0},w}x\|\quad\forall\,x\in T_{w}^{u}$

for any $t_{0}\in\mathbb{R}$ and for all $t>0$.

Then, Anosov’s structural stability theorem [1, 9] asserts that: If $\Lambda$
is a compact hyperbolic set for $S$, then for any $\varepsilon>0$ there is a
$C^{1}$-neighborhood $\mathcal{U}$ of $S$ in $\mathfrak{X}^{1}(\mathbb{E})$
such that, if $V\in\mathcal{U}$ then there exists a $\varepsilon$-topological
mapping $h$ from $\Lambda$ onto some subset $\Lambda_{V}$ of $\mathbb{E}$
which sends orbits of $S$ in $\Lambda$ into orbits of $V$ in $\Lambda_{V}$.

This important theorem was extended to axiom A differential systems [11, 13],
and to $C^{0}$-perturbations by considering the so-called semi-structural
stability independently by [5, 7]; for discrete versions, see [12, 14, 9, 15,
10]. On another direction, in this paper, we study the structural stability of
_noncompact_ hyperbolic set under _time-preserving_ conjugacy between the
induced cross-section flows. More precisely, using Liao theory we prove the
following.

######  Main Theorem.

Let $\Lambda$ be a hyperbolic set for $S$, not necessarily compact, satisfying
the following conditions:

  1. (U1)

The first derivative $S^{\prime}(w)$ is uniformly bounded on $\Lambda$;

  2. (U2)

$0<\inf_{w\in\Lambda}\|S(w)\|\leq\sup_{w\in\Lambda}\|S(w)\|<\infty$;

  3. (U3)

$S^{\prime}(w)$ is uniformly continuous at $\Lambda$; that is, to any
$\epsilon>0$ there is some $\delta>0$ so that for any
$\mathfrak{w}\in\Lambda$,
$\|S^{\prime}(w)-S^{\prime}(\mathfrak{w})\|<\epsilon$ whenever
$\|w-\mathfrak{w}\|<\delta$.

Then, for any $\varepsilon>0$ there is a $C^{1}$-neighborhood $\mathcal{U}$ of
$S$ in $\mathfrak{X}^{1}(\mathbb{E})$ such that for any $V\in\mathcal{U}$
there exists a $\varepsilon$-topological mapping $H$ from $\Lambda$ onto some
closed subset $\Lambda_{V}$ which sends orbits of $S$ in $\Lambda$ into orbits
of $V$ in $\Lambda_{V}$, such that $H(w)\in\Sigma_{w}$ and
$\psi_{V}(t,H(w))=H(t_{\cdot}w)$ for all $w\in\Lambda$ and for any
$t\in\mathbb{R}$.

Notice here that if $\Lambda$ is compact, then conditions (U1), (U2) and (U3)
hold automatically. So our result is an extension of the classical one. Even
for the compact case, the time-preserving property is still a new ingredient
in our main theorem.

To prove this result, we will introduce the reduced standard systems of
differential equations for perturbations of $S$ in $\S\ref{sec2}$ and recall
Liao’s exponential dichotomy in $\S\ref{sec3}$. Finally, we prove the Main
Theorem in $\S\ref{sec4}$, using simplified and extended Liao approach that is
completely different from Anosov’s geometrical approach [1] and Moser’s
functional approach [9] to differentiable dynamical systems on compact
Riemannian manifolds. And in $\S\ref{sec4}$, we will construct a differential
system which has a noncompact, structurally stable, hyperbolic subset.

##  2\. Liao standard system of differential equations

Let $S$ be any given $C^{1}$-differential system on $\mathbb{E}$ and $\Lambda$
a $\phi$-invariant closed subset in $\mathbb{E}$ satisfying conditions (U1),
(U2) and (U3) as in the Main Theorem stated in $\S\ref{sec1}$. Around a
regular orbit $\phi(\mathbb{R},w)$ we defined in [3] the reduced standard
systems for $S$ itself. However, we will introduce below the standard systems
for perturbations $V$ of $S$.

### 2.1.

As usual in Liao theory [3, 4], let
$\mathscr{F}_{n-1}^{*\sharp}(\Lambda)={\bigsqcup}_{w\in\Lambda}\mathscr{F}_{n-1,w}^{*\sharp}$
be the bundle of transversal orthonormal $(n-1)$-frames, where the fiber over
$w$ is defined as

$\mathscr{F}_{n-1,w}^{*\sharp}=\left\\{{\gamma}=(\vec{u}_{1},\ldots,\vec{u}_{n-1})\in\mathbb{T}_{w}\times\cdots\times\mathbb{T}_{w}\,|\langle\vec{u}_{i},\vec{u}_{j}\rangle=\delta^{ij}\textrm{
for }1\leq i,j\leq n-1\right\\},$

endowed with the naturally induced topology. Then, $S$ naturally generates a
skew-product flow over $\phi$

(2.1)
$\chi^{*\sharp}\colon\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\rightarrow\mathscr{F}_{n-1}^{*\sharp}(\Lambda);\
(t,(w,{\gamma}))\mapsto(t_{\cdot}w,\chi_{t,w}^{*\sharp}{\gamma}),$

where
$\chi_{t,w}^{*\sharp}\colon\mathscr{F}_{n-1,w}^{*\sharp}\rightarrow\mathscr{F}_{n-1,t_{\cdot}w}^{*\sharp}$
is defined by the standard Gram-Schmidt orthonormalization process; cf. [2, 3]
for the details.

Let $\textbf{e}=\\{\vec{e}_{1},\ldots,\vec{e}_{n-1}\\}$ where
$\vec{e}_{j}=(\stackrel{{\scriptstyle
j^{\textrm{th}}}}{{0,\ldots,0,1,0,\ldots,0}})^{\textrm{T}}\in\mathbb{R}^{n-1}$,
be the standard basis of $\mathbb{R}^{n-1}$ and we view $y\in\mathbb{R}^{n-1}$
with components $y^{1},\ldots,y^{n-1}$ as a column vector
$(y^{1},\ldots,y^{n-1})^{\textrm{T}}$ and
$\gamma\in\mathscr{F}_{n-1,w}^{*\sharp}$ as an $n$-by-$(n-1)$ matrix with
columns $\textrm{col}_{1}{\gamma},\ldots,\textrm{col}_{n-1}{\gamma}$
successively.

Given any orthonormal $(n-1)$-frame
$(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$, sometimes written
simply as $\gamma_{w}$, we define by linear extension the linear
transformation

(2.2)
$\mathcal{T}_{{\gamma}_{w}}^{*}\colon\mathbb{R}^{n-1}\rightarrow\mathbb{T}_{w}$

in the way

$\vec{e}_{j}\mapsto{\mathrm{col}}_{j}{\gamma}\quad(1\leq j\leq n-1).$

Since ${\gamma}$ is an orthonormal basis of $\mathbb{T}_{w}$,
$\mathcal{T}_{\gamma_{w}}^{*}$ is an isomorphism such that

$\mathcal{T}_{{\gamma}_{w}}^{*}(y)={\gamma}y=\sum_{j=1}^{n-1}y^{j}\textrm{col}_{j}\gamma\textrm{
and
}\|y\|_{\mathbb{R}^{n-1}}=\|{\gamma}y\|_{\mathbb{T}_{w}}\quad\forall\,y\in\mathbb{R}^{n-1}.$

Moreover, we now define

(2.3)
$C_{{\gamma}_{w}}^{*}(t)=\mathcal{T}_{\chi^{*\sharp}(t,{\gamma}_{w})}^{*-1}\circ\Psi_{t,w}\circ\mathcal{T}_{\gamma_{w}}^{*}\quad\forall\,t\in\mathbb{R},$

where
$\chi^{*\sharp}\colon\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\rightarrow\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$
as in (2.1). Then the commutativity holds:

(2.4)
$\begin{CD}\mathbb{R}^{n-1}@>{C_{\gamma_{w}}^{*}(t)}>{}>\mathbb{R}^{n-1}\\\
@V{\mathcal{T}_{{\gamma}_{w}}^{*}}V{}V@V{}V{\mathcal{T}_{\chi^{*\sharp}(t,{\gamma}_{w})}^{*}}V\\\
\mathbb{T}_{w}@>{\Psi_{t,w}}>{}>\mathbb{T}_{t_{\cdot}w}.\end{CD}$

We now think of $C_{{\gamma}_{w}}^{*}(t)$ as an $(n-1)\times(n-1)$-matrix
under the base e of $\mathbb{R}^{n-1}$. Clearly,
$t\mapsto\frac{d}{dt}C_{{\gamma}_{w}}^{*}(t)$ makes sense since ${S}$ is of
class $C^{1}$ and by (2.4) we have

(2.5)
$C_{{\gamma}_{w}}^{*}(t_{1}+t_{2})=C_{\chi^{*\sharp}(t_{1},{\gamma}_{w})}^{*}(t_{2})\circ
C_{{\gamma}_{w}}^{*}(t_{1})\quad\forall\,t_{1},t_{2}\in\mathbb{R}.$

Put

(2.6)
$R_{{\gamma}_{w}}^{*}(t)=\left\\{\frac{d}{dt}C_{{\gamma}_{w}}^{*}(t)\right\\}{C_{{\gamma}_{w}}^{*}(t)}^{-1}\quad\forall\,(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda).$

######  Definition 2.1.

The linear differential equation

$None$
$\dot{y}=R_{{\gamma}_{w}}^{*}(t)y\quad(t,y)\in\mathbb{R}\times\mathbb{R}^{n-1}$

for any $(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$, is called the
_reduced linearized system_ of ${S}$ under the moving frame
$\chi^{*\sharp}(t,{\gamma}_{w})$. See [2, 3].

These reduced linearized systems of ${S}$ possess the following properties.

######  Lemma 2.2 ([2, 3]).

The following statements hold:

  1. (1)

Uniform boundedness: $R_{{\gamma}_{w}}^{*}(t)$ is continuous in
$(t,(w,\gamma))$ in $\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$
with

$\eta_{\Lambda}:=\sup\left\\{\sum_{i,j}|R_{{\gamma}_{w}}^{*ij}(t)|;\,t\in\mathbb{R},(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\right\\}<\infty.$

  2. (2)

Upper triangularity: ${R}_{\gamma_{w}}^{*}(t)$ is upper-triangular with

${R}_{\gamma_{w}}^{*}(t)=\left[\begin{matrix}{\omega}_{1}^{*}(\chi^{*\sharp}(t,\gamma_{w}))&\cdots&*\\\
\vdots&\ddots&\vdots\\\
0&\cdots&{\omega}_{n-1}^{*}(\chi^{*\sharp}(t,\gamma_{w}))\end{matrix}\right]\quad\forall\,t\in\mathbb{R}$

where ${\omega}_{k}^{*}(w,\gamma),1\leq k\leq n-1$, called the “Liao
qualitative functions” of $S$, are uniformly continuous in
$(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

  3. (3)

Geometrical interpretation: Let $\vec{v}={\gamma}{y}\in\mathbb{T}_{w}$ for
${y}\in\mathbb{R}^{n-1}$. If $y(t)=y(t,{y})$ is the solution of
$(R_{\gamma_{w}}^{*})$ with $y(0)={y}$, then

$\Psi_{t,w}\vec{v}=\mathcal{T}_{\chi^{*\sharp}(t,\gamma_{w})}^{*}y(t)=(\chi_{t,w}^{*\sharp}\gamma)y(t).$

Conversely, letting
$x(t)=(x^{1}(t),\ldots,x^{n-1}(t))^{\mathrm{T}}\in\mathbb{R}^{n-1}$ be defined
by

$x^{i}(t)=\left\langle\Psi_{t,w}\vec{v},\mathrm{col}_{i}{\chi_{t,w}^{*\sharp}}\gamma\right\rangle_{t_{\cdot}w}\quad
i=1,\ldots,{n-1},$

we have $\dot{x}(t)={R}_{\gamma_{w}}^{*}(t)x(t)\textrm{ and }x(0)={y}$.
Particularly, $C_{{\gamma}_{w}}^{*}(t)$ is the fundamental matrix solution of
$(R_{{\gamma}_{w}}^{*})$.

As a consequence of the above lemma, we have

######  Corollary 2.3.

Let $\Lambda$ be hyperbolic for $S$ associated to $\Psi$-invariant splitting
$\mathbb{T}_{\Lambda}=T_{\Lambda}^{s}\oplus T_{\Lambda}^{u}$. Then, there are
two constants $\boldsymbol{\eta}>0$ and $\textbf{d}>0$ such that: for any
$(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$, if
$\mathrm{col}_{i}\gamma\in T_{w}^{s}$ for $i=1,\ldots,\dim T_{w}^{s}$ then

$\displaystyle\int_{0}^{T}\omega_{k}^{*}(\chi^{*\sharp}(t_{0}+t,(w,\gamma)))\,dt\leq-\boldsymbol{\eta}T,\quad
1\leq k\leq\dim T_{w}^{s}$ and
$\displaystyle\int_{0}^{T}\omega_{k}^{*}(\chi^{*\sharp}(t_{0}+t,(w,\gamma)))\,dt\geq\boldsymbol{\eta}T,\quad\dim
T_{w}^{s}+1\leq k\leq n-1$

for any $t_{0}\in\mathbb{R}$ and for all $T\geq\textbf{d}$.

###### Proof.

The statement comes immediately from Lemma 2.2 and [7, Lemma 3.7]. ∎

### 2.2.

For a constant $c>0$, let
$\mathbb{R}_{c}^{n-1}=\\{y\in\mathbb{R}^{n-1};\,\|y\|<c\\}$. Fix any
$w\in\Lambda$. For any $\gamma\in\mathscr{F}_{n-1,w}^{*\sharp}$, we need the
$C^{1}$-mapping

$\mathcal{P}_{w,\gamma}^{*}\colon\mathbb{R}\times\mathbb{R}^{n-1}\rightarrow\mathbb{E}$

defined by

$\mathcal{P}_{w,\gamma}^{*}(t,y)=t_{\cdot}w+({\chi_{t,w}^{*\sharp}}\gamma)y\in\Sigma_{t_{\cdot}w}\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}^{n-1}.$

It is known [3, Lemma 5.1] that there is a constant $\mathfrak{c}>0$, which is
independent of $(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$, such that
$\mathcal{P}_{w,\gamma}^{*}$ is locally diffeomorphic on
$\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$. In fact, according to [3]
there is some $\epsilon>0$ so that for any $w\in\Lambda$,
$\mathcal{P}_{w,\gamma}^{*}$ is diffeomorphic from
$(-\epsilon,\epsilon)\times\mathbb{R}_{\mathfrak{c}}^{n-1}$ into $\mathbb{E}$.

Given any $(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$. Define a
$C^{0}$-vector field on $\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

$\widehat{S}_{w,{\gamma}}\colon\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\rightarrow\mathbb{R}^{n}$

with
$\widehat{S}_{w,{\gamma}}(t,\textbf{0})=(1,\textbf{0})^{\textrm{T}}\in\mathbb{R}\times\mathbb{R}^{n-1}$
in the following way:

$\left(D_{(t,y)}\mathcal{P}_{w,\gamma}^{*}\right)\widehat{S}_{w,{\gamma}}(t,y)=S(\mathcal{P}_{w,\gamma}^{*}(t,y))\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}.$

Since $\mathcal{P}_{w,\gamma}^{*}$ is locally $C^{1}$-diffeomorphic,
$\widehat{S}_{w,{\gamma}}(t,y)$ is well defined. We now consider the
autonomous system

(2.7a) $\displaystyle\frac{d}{d\mathbbm{t}}\left(\begin{matrix}t\\\
y\end{matrix}\right)$
$\displaystyle=\widehat{S}_{w,{\gamma}}(t,y)\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$
and write (2.7b) $\displaystyle\widehat{S}_{w,{\gamma}}(t,y)$
$\displaystyle=\left(\widehat{S}_{w,{\gamma}}^{0}(t,y),\ldots,\widehat{S}_{w,{\gamma}}^{n-1}(t,y)\right)^{\textrm{T}}\in\mathbb{R}\times\mathbb{R}^{n-1}.$

Next, put

(2.8)
${S}_{w,{\gamma}}^{*}(t,y)=\left(\frac{\widehat{S}_{\mathfrak{w},{\gamma}}^{1}(t,y)}{\widehat{S}_{w,{\gamma}}^{0}(t,y)},\ldots,\frac{\widehat{S}_{w,{\gamma}}^{n-1}(t,y)}{\widehat{S}_{w,{\gamma}}^{0}(t,y)}\right)^{\textrm{T}}\in\mathbb{R}^{n-1}\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}.$

######  Definition 2.4 ([3]).

The non-autonomous differential equation

$None$
$\dot{y}={S}_{w,{\gamma}}^{*}(t,y)\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

is called the _reduced standard system of $S$ under the base
$(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$_.

Is is easy to see that

(2.9)
${S}_{w,{\gamma}}^{*}(t+t_{1},y)={S}_{\chi^{*\sharp}(t,(w,{\gamma}))}^{*}(t_{1},y)\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}.$

For convenience of our later discussion, we write

(2.10)
$\mathcal{P}_{w,\gamma}^{*}(t,y)=t_{\cdot}w+\widehat{\mathcal{P}}_{w,\gamma}^{*}(t,y),\textrm{
where }\widehat{\mathcal{P}}_{w,\gamma}^{*}(t,y)\in\mathbb{T}_{t_{\cdot}w}.$

The following is important for our later arguments.

######  Lemma 2.5 ([3]).

Under the conditions (U1), (U2) and (U3), the following statements hold: for
any $(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$

  1. (1)

${S}_{w,{\gamma}}^{*}(t,\mathbf{0})=\mathbf{0}\in\mathbb{R}^{n-1}$ for all
$t\in\mathbb{R}$, and ${S}_{w,{\gamma}}^{*}(t,y)$ is continuous with respect
to $(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$.

  2. (2)

For any $(\bar{t},\bar{y})\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$,
let
$\bar{w}=\bar{t}_{\cdot}w+\bar{x}={\mathcal{P}}_{w,\gamma}^{*}(\bar{t},\bar{y})\in\Sigma_{\bar{t}_{\cdot}w}$
and

$y^{*}(t)=y_{w,{\gamma}}^{*}(t;\bar{t},\bar{y})\quad
t\in(r^{\prime},r^{\prime\prime})\textrm{ where
}\bar{t}\in(r^{\prime},r^{\prime\prime}),$

be the solution of $({S}_{w,{\gamma}}^{*})$ with $y^{*}(\bar{t})=\bar{y}$.
Then

$\psi(t-\bar{t},\bar{w})={\mathcal{P}}_{w,\gamma}^{*}(t,y^{*}(t))\in\Sigma_{t_{\cdot}w}\quad(r^{\prime}<t<r^{\prime\prime}).$

  3. (3)

${S}_{w,{\gamma}}^{*}(t,y)$ is of class $C^{1}$ with respect to
$y\in\mathbb{R}_{\mathfrak{c}}^{n-1}$ such that

${\partial{S}_{w,{\gamma}}^{*}(t,y)}/{\partial
y}\to{R}_{w,{\gamma}}^{*}(t)\textrm{ as }y\to\mathbf{0}$

uniformly for
$(t,(w,\gamma))\in\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

From here on, for any $w\in\Lambda$ we will rewrite ($S_{w,\gamma}^{*}$) as

(2.11a)
$\displaystyle\dot{y}=R_{w,\gamma}^{*}(t)y+S_{\textsl{rem}(w,\gamma)}^{*}(t,y)\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$
where (2.11b) $\displaystyle
S_{\textsl{rem}(w,\gamma)}^{*}(t,y)=S_{w,\gamma}^{*}(t,y)-R_{w,\gamma}^{*}(t)y.$

Then, we have the following result.

######  Lemma 2.6 ([3]).

Under the conditions (U1), (U2) and (U3), to any $\kappa>0$, there is some
$\xi\in(0,\mathfrak{c}]$ so that

$\|S_{\textsl{rem}(w,\gamma)}^{*}(t,y)-S_{\textsl{rem}(w,\gamma)}^{*}(t,y^{\prime})\|\leq\kappa\|y-y^{\prime}\|\quad\textrm{whenever
}y,y^{\prime}\in\mathbb{R}_{\xi}^{n-1}$

holds uniformly for
$(t,(w,\gamma))\in\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

### 2.3.

In what follows, we let $V\colon\mathbb{E}\rightarrow\mathbb{R}^{n}$ be an
arbitrarily given another $C^{1}$ vector field on $\mathbb{E}$. Note here that
$(\mathscr{F}_{n-1}^{*\sharp}(\Lambda),\chi^{*\sharp})$ still corresponds to
$S$.

In order to introduce the standard systems of $V$ associated with $S$, let us
consider firstly a simple lemma.

######  Lemma 2.7.

Let $h\colon\widehat{N}\rightarrow N$ be a map of class $C^{1}$ from a $C^{1}$
manifold $\widehat{N}$ into another $C^{1}$ manifold $N$. Let $\widehat{X}$
and $X$ be $C^{0}$ vector fields on $\widehat{N}$ and $N$, respectively. If
$(Dh)\widehat{X}=X$ then for any $\hat{p}\in\widehat{N}$, $h$ maps the
integral curve $\phi_{\hat{\textsl{x}}}(t,\hat{p})$ of $\widehat{X}$ into an
integral curve $\phi_{\textsl{x}}(t,h(\hat{p}))$ of $X$ such that
$\phi_{\textsl{x}}(t,h(\hat{p}))=h(\phi_{\hat{\textsl{x}}}(t,\hat{p}))$.

###### Proof.

Let $h(\hat{p})=p$. Define a $C^{1}$ curve in $N$ by $C\colon t\mapsto
h(\phi_{\hat{\textsl{x}}}(t,\hat{p}))$. Since

$\frac{d}{dt}\phi_{\hat{\textsl{x}}}(t,\hat{p})=\widehat{X}(\phi_{\hat{\textsl{x}}}(t,\hat{p}))\textrm{
and
}(Dh)\widehat{X}(\phi_{\hat{\textsl{x}}}(t,\hat{p}))=X(C(t))=\frac{d}{dt}C(t),$

we get that $C(t)$ is an integral curve of $X$ satisfying the initial
condition $C(0)=p$. Now put $\phi_{\textsl{x}}(t,p)=C(t)$, which satisfies the
requirement of Lemma 2.7. ∎

Particularly, we will be interesting to the case where
$\widehat{N}=\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1},N=\mathbb{E}$ and
$h=\mathcal{P}_{w,\gamma}^{*}$ and $X=V$ for any given
$(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$. Correspondingly, there
$\widehat{X}$ is right the so-called lifting system that we are going to
define.

######  Definition 2.8.

Given any $(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$. Define a
$C^{0}$-vector field

$\widehat{V}_{w,\gamma}\colon\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\rightarrow\mathbb{R}^{n}$

in the following way:

$\left(D_{(t,y)}\mathcal{P}_{w,\gamma}^{*}\right)\widehat{V}_{w,\gamma}(t,y)=V(\mathcal{P}_{w,{\gamma}}^{*}(t,y))\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}.$

Then, the autonomous differential equation

(2.12a) $\displaystyle\frac{d}{d\mathbbm{t}}\left(\begin{matrix}t\\\
y\end{matrix}\right)$
$\displaystyle=\widehat{V}_{w,{\gamma}}(t,y)\quad\mathbbm{t}\in\mathbb{R},(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

is referred to as a _lifting_ of $V$ under the moving frames
$(\chi^{*\sharp}(t,(w,{\gamma})))_{t\in\mathbb{R}}$.

Write

$\widehat{V}_{w,{\gamma}}(t,y)=\left(\widehat{V}_{w,{\gamma}}^{0}(t,y),\ldots,\widehat{V}_{w,{\gamma}}^{n-1}(t,y)\right)^{\textrm{T}}\in\mathbb{R}\times\mathbb{R}^{n-1}.$

Clearly, it follows from
$\mathcal{P}_{w,\gamma}^{*}(t,y)=\mathcal{P}_{\chi^{*\sharp}(t,(w,\gamma))}^{*}(0,y)$
that

(2.13)
$\widehat{V}_{w,{\gamma}}(t,y)=\widehat{V}_{\chi^{*\sharp}(t,(w,{\gamma}))}(0,y)\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}.$

Although $\mathcal{P}_{w,{\gamma}}^{*}$ is only $C^{1}$, we can obtain more
about the regularity of $\widehat{V}_{w,{\gamma}}(t,y)$ with respect to
$y\in\mathbb{R}_{\mathfrak{c}}^{n-1}$ as long as $V$ is $C^{1}$.

######  Lemma 2.9.

Given any $(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$, the lifting
$\widehat{V}_{w,{\gamma}}(t,y)$ is of class $C^{1}$ in $y$; precisely, for
$1\leq i\leq{n-1}$, $\partial\widehat{V}_{w,{\gamma}}(t,y)/{\partial y^{i}}$
makes sense and is continuous with respect to $(t,y,(w,{\gamma}))$ in
$\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

###### Proof.

The statement comes immediately from the regularity of
$\mathcal{P}_{w,\gamma}^{*}(t,y)$, as the argument of [3, Lemma 5.3]. ∎

Next, let

$\\{S,V\\}_{\Lambda}^{1}=\sup_{\begin{subarray}{c}(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\\\
(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\end{subarray}}\left\\{\|\widehat{S}_{w,\gamma}(t,y)-\widehat{V}_{w,\gamma}(t,y)\|+\|\frac{\partial}{\partial
y}[\widehat{S}_{w,\gamma}(t,y)-\widehat{V}_{w,\gamma}(t,y)]\|\right\\}.$

From (2.13) we get

$\\{S,V\\}_{\Lambda}^{1}=\sup_{\begin{subarray}{c}y\in\mathbb{R}_{\mathfrak{c}}^{n-1}\\\
(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\end{subarray}}\left\\{\|\widehat{S}_{w,\gamma}(0,y)-\widehat{V}_{w,\gamma}(0,y)\|+\|\frac{\partial}{\partial
y}[\widehat{S}_{w,\gamma}(0,y)-\widehat{V}_{w,\gamma}(0,y)]\|\right\\}.$

Then, we have

######  Lemma 2.10.

There exists some constant $\flat_{\Lambda}>0$ such that

$\|S-V\|_{1}\geq\flat_{\Lambda}\\{S,V\\}_{\Lambda}^{1}\quad\forall\,V\in\mathfrak{X}^{1}(\mathbb{E}).$

###### Proof.

For any $(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$ let

$J_{w,\gamma}(y)=\left.\frac{\partial\mathcal{P}_{w,\gamma}^{*}(t,y)}{\partial(t,y)}\right|_{(0,y)}$

be the $n$-by-$n$ Jacobi matrix of $\mathcal{P}_{w,\gamma}^{*}(t,y)$ at
$(0,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$. Then

$\displaystyle\mathcal{P}_{w,\gamma}^{*}(0,y)$ $\displaystyle=w+\gamma y$ and
$\displaystyle J_{w,\gamma}(y)$
$\displaystyle=\left[S(w)+\left.\frac{d}{dt}\right|_{t=0}({\chi_{t,w}^{*\sharp}}\gamma)y,\,\gamma\right]_{n\times
n}.$

Thus, for any $y\in\mathbb{R}_{\mathfrak{c}}^{n-1}$ we have

$\widehat{S}_{w,\gamma}(0,y)-\widehat{V}_{w,\gamma}(0,y)={J_{w,\gamma}(y)}^{-1}(S-V)(w+\gamma
y)$

Moreover, from condition (U1) we can prove by the argument of [3, Lemma 5.3]
that $\left.\frac{d}{dt}\right|_{t=0}{\chi_{t,w}^{*\sharp}}\gamma$, viewed as
an $n$-by-$(n-1)$ matrix, is uniformly continuous and bounded for any
$(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$. Therefore, there is some
constant $\flat_{\Lambda}>0$ which satisfies the requirement of Lemma 2.10. ∎

From Lemma 2.10, condition (U1) and
$\widehat{S}_{w,\gamma}^{0}(t,\textbf{0})=1$, we may assume, without any loss
of generality replacing $\mathfrak{c}$ by a more small positive constant if
necessary, that

  * •

$\exists\,\mathcal{N}_{\textsl{s}}$, a $C^{1}$-neighborhood of $S$ in
$\mathfrak{X}^{1}(\mathbb{E})$ such that: for any
$V\in\mathcal{N}_{\textsl{s}}$

  * •

$\frac{1}{2}\leq\widehat{V}_{w,\gamma}^{0}(t,y)\leq\frac{4}{2}$ for any
$(t,y,{(w,\gamma)})\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

Thus, the following definition makes sense.

######  Definition 2.11.

Given any
$(V,(w,\gamma))\in\mathcal{N}_{\textsl{s}}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$,
set

$V_{w,\gamma}^{*}(t,y)=\left(\frac{\widehat{V}_{w,\gamma}^{1}(t,y)}{\widehat{V}_{w,\gamma}^{0}(t,y)},\ldots,\frac{\widehat{V}_{w,\gamma}^{n-1}(t,y)}{\widehat{V}_{w,\gamma}^{0}(t,x)}\right)^{\textrm{T}}\in\mathbb{R}^{n-1}\quad\forall\,(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}.$

The non-autonomous differential equation

$None$
$\dot{y}=V_{w,\gamma}^{*}(t,y)\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

is referred to as the _standard system of $V$ associated to $(S,(w,\gamma))$_.

From (2.13) we have

(2.14)
$V_{w,\gamma}^{*}(t+t^{\prime},y)=V_{\chi^{*\sharp}(t,(w,\gamma))}^{*}(t^{\prime},y)\quad\forall\,t,t^{\prime}\in\mathbb{R}\textrm{
and }y\in\mathbb{R}_{\mathfrak{c}}^{n-1}.$

In what follows, we write $(V_{w,\gamma}^{*})$ as

$None$
$\dot{y}=R_{w,\gamma}^{*}(t)y+V_{\textsl{rem}(w,\gamma)}^{*}(t,y)\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

where

(2.15a) $\displaystyle V_{\textsl{rem}(w,{\gamma})}^{*}(t,y)$
$\displaystyle=V_{w,\gamma}^{*}(t,y)-R_{w,\gamma}^{*}(t)y$ such that (2.15b)
$\displaystyle V_{\textsl{rem}(w,{\gamma})}^{*}(t+t^{\prime},y)$
$\displaystyle=V_{\textsl{rem}(\chi^{*\sharp}(t,(w,\gamma)))}^{*}(t^{\prime},y)\quad\forall\,t,t^{\prime}\in\mathbb{R}.$

Similar to Lemma 2.5, we obtain the following result.

######  Theorem 2.12.

Given any $V\in\mathcal{N}_{\textsl{s}}$, the following statements hold:

  1. (1)

$V_{\textsl{rem}(w,{\gamma})}^{*}(t,y)$ and ${\partial
V_{\textsl{rem}(w,\gamma)}^{*}(t,y)}/{\partial y}$ are continuous in
$(t,y,{(w,\gamma)})\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

  2. (2)

Given any ${(w,\gamma)}\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$. If
$y^{*}(t)=y_{V;w,\gamma}^{*}(t;t_{0},y))$ where
$t^{\prime}<t,t_{0}<t^{\prime\prime}$, is the solution of $(V_{w,\gamma}^{*})$
with $y^{*}(t_{0})=y$, then

$\psi_{V}(t-t_{0},{\mathcal{P}}_{w,\gamma}^{*}(t_{0},y))={\mathcal{P}}_{w,\gamma}^{*}(t,y^{*}(t))\in\Sigma_{t_{\cdot}w}.$

Moreover, similar to Lemma 2.6 we have the following important result.

######  Theorem 2.13.

The following three statements hold.

  1. (1)

Given any
$(V,{(w,\gamma)})\in\mathcal{N}_{\textsl{s}}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$,
there is some $L>0$ such that

$\|V_{\textsl{rem}(w,\gamma)}^{*}(t,y)-V_{\textsl{rem}(w,\gamma)}^{*}(t,y^{\prime})\|\leq
L\|y-y^{\prime}\|$

for any $t\in\mathbb{R}$ and for any
$y,y^{\prime}\in\mathbb{R}_{\mathfrak{c}}^{n-1}$.

  2. (2)

To any $\eta>0$ there exists a $C^{1}$-neighborhood
$\mathcal{U}_{\textsl{s}}^{\prime}\subset\mathcal{N}_{\textsl{s}}$ of $S$ and
$\xi^{\prime}\in(0,\mathfrak{c}]$ such that:
$\forall\,V\in\mathcal{U}_{\textsl{s}}^{\prime}$

$\sup_{(t,y)\in\mathbb{R}\times\mathbb{R}_{\xi^{\prime}}^{n-1}}\|V_{\textsl{rem}(w,\gamma)}^{*}(t,y)\|\leq\eta\xi^{\prime}\quad\forall\,(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda).$

  3. (3)

To any given $\kappa>0$ there corresponds a $C^{1}$-neighborhood
$\mathcal{U}_{\textsl{s}}^{\prime\prime}\subset\mathcal{N}_{\textsl{s}}$ of
$S$ and a constant $\xi^{\prime\prime}\in(0,\mathfrak{c}]$ such that:
$\forall\,V\in\mathcal{U}_{\textsl{s}}^{\prime\prime}$

$\|V_{\textsl{rem}(w,\gamma)}^{*}(t,y)-V_{\textsl{rem}(w,\gamma)}^{*}(t,y^{\prime})\|\leq\kappa\|y-y^{\prime}\|\quad\forall\,y,y^{\prime}\in\mathbb{R}_{\xi^{\prime\prime}}^{n-1}$

uniformly for
$(t,(w,\gamma))\in\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$.

###### Proof.

By (2.15a), Lemma 2.10 and Lemma 2.2 and condition (U1)

$L:=\sup_{(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}}\left\\{\|\partial
V_{w,\gamma}^{*}(t,y)/\partial y\|+\|R_{w,\gamma}^{*}(t)\|\right\\}<+\infty$

which satisfies the requirement of the statement (1).

Given any $\eta>0$. For any $V\in\mathcal{N}_{\textsl{s}}$ and for any
$(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$ one can write

$V_{\textsl{rem}(w,\gamma)}^{*}(t,y)=\left(V_{w,\gamma}^{*}(t,y)-{S}_{w,\gamma}^{*}(t,y)\right)+\left({S}_{w,\gamma}^{*}(t,y)-{R}_{w,\gamma}^{*}(t)y\right)$

for any $(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$. Then, from
Lemma 2.10 and Lemma 2.6 there exists a $C^{1}$-neighborhood
$\mathcal{U}_{\textsl{s}}^{\prime}\subset\mathcal{N}_{\textsl{s}}$ of $S$ and
a constant $\xi^{\prime}\in(0,\mathfrak{c}]$ such that

$\sup_{(t,y)\in\mathbb{R}\times\mathbb{R}_{\xi^{\prime}}^{n-1}}\|V_{w,\gamma}^{*}(t,y)\|\leq\eta\xi^{\prime}\quad\forall\,(w,{\gamma})\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\textrm{
and }V\in\mathcal{U}_{\textsl{s}}^{\prime}.$

This shows the statement (2).

Now given any $\kappa>0$. Next, for any $V\in\mathcal{N}_{\textsl{s}}$
consider

$\frac{\partial}{\partial
y}V_{\textsl{rem}(w,\gamma)}^{*}(t,y)=\frac{\partial}{\partial
y}\left(V_{w,\gamma}^{*}(t,y)-{S}_{w,\gamma}^{*}(t,y)\right)+\left(\frac{\partial}{\partial
y}{S}_{w,\gamma}^{*}(t,y)-{R}_{w,\gamma}^{*}(t)\right).$

From Lemma 2.10 we obtain that

$\|\frac{\partial}{\partial
y}\left({V}_{w,\gamma}^{*}(t,y)-{S}_{w,\gamma}^{*}(t,y)\right)\|\to 0\textrm{
as }\|V-S\|_{1}\to 0$

uniformly for
$(t,y,(w,{\gamma}))\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$
and, from Lemma 2.2 there exists $\xi^{\prime\prime}\in(0,\mathfrak{c}]$ so
that

$\|\frac{\partial}{\partial
y}{S}_{w,\gamma}^{*}(t,y)-{R}_{w,\gamma}^{*}(t)\|\leq\frac{\kappa}{2}\quad\forall\,(t,(w,\gamma))\in\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\textrm{
and }y\in\mathbb{R}_{\xi^{\prime\prime}}^{n-1}.$

Hence, there is a $C^{1}$-neighborhood
$\mathcal{U}_{\textsl{s}}^{\prime\prime}\subset\mathcal{N}_{\textsl{s}}$ of
$S$ such that: $\forall\,V\in\mathcal{U}_{\textsl{s}}^{\prime\prime}$

$\|\frac{\partial}{\partial
y}V_{\textsl{rem}(w,\gamma)}^{*}(t,y)\|\leq\kappa\quad\forall\,(t,(w,\gamma))\in\mathbb{R}\times\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\textrm{
and }y\in\mathbb{R}_{\xi^{\prime\prime}}^{n-1}.$

This implies the statement (3) by the mean value theorem.

Thus, Theorem 2.13 is proved. ∎

##  3\. Exponential dichotomy

In this section, we will introduce the exponential dichotomy due to Liao [7],
by which we consider in part the relationship between the phase portraits of
linear differential equations and their small perturbations on Euclidean
spaces. Here we shall deal with families of ordinary differential equations,
nor only a single equations.

Given a positive integer $p$. For convenience of our later discussion, let
$M_{p\times p}^{\vartriangle}$ be the set of continuous matrix-valued
functions $A\colon\mathbb{R}\rightarrow\textrm{gl}(p,\mathbb{R})$ such that

  1. (a)

$A(t)$ is triangular with $A_{ij}(t)=0$ for $1\leq j<i\leq p$;

  2. (b)

$A$ is uniformly bounded on $\mathbb{R}$ with
$\eta_{A}:=\sup_{t\in\mathbb{R}}\|A(t)\|<\infty$;

  3. (c)

$A$ is hyperbolic with index $p_{-}$ in the following sense:

$\xi_{A}:=\sup_{t\in\mathbb{R}}\left\\{\sum_{k=1}^{p_{-}}\int_{-\infty}^{t}e^{\int_{s}^{t}A_{kk}(\tau)\,d\tau}\,ds+\sum_{k=1+p_{-}}^{p}\int_{t}^{\infty}e^{\int_{s}^{t}A_{kk}(\tau)\,d\tau}\,ds\right\\}<\infty.$

In addition, let $M_{p\times 1}$ be the set of continuous functions
$f\colon\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}$ such that

  1. (d)

$f(t,z)$ is bounded on $\mathbb{R}\times\mathbb{R}^{p}$ with
$\eta_{f}:=\sup_{(t,z)\in\mathbb{R}\times\mathbb{R}^{p}}\|f(t,z)\|<\infty$;

  2. (e)

$f(t,z)$ is Lipschitz in $z$ with a Lipschitz constant $L_{f}$:

$\|f(t,z)-f(t,z^{\prime})\|\leq L_{f}\|z-z^{\prime}\|$

for all $t\in\mathbb{R}$ and for any $z,z^{\prime}\in\mathbb{R}^{p}$.

For any $(A,f)\in M_{p\times p}^{\vartriangle}\times M_{p\times 1}$, we will
study the equations

(3.1) $\dot{z}=A(t)z+f(t,z),\quad(t,z)\in\mathbb{R}\times\mathbb{R}^{p}$

and

(3.2) $\dot{z}=A(t)z,\quad(t,z)\in\mathbb{R}\times\mathbb{R}^{p}.$

For any $(s,u)\in\mathbb{R}\times\mathbb{R}^{p}$, let $z_{A,f}(t;s,u)$ and
$z_{A}(t;s,u)$ denote the solutions of (3.1) and (3.2) with
$z_{A,f}(s;s,u)=u=z_{A}(s;s,u)$, respectively.

The following result is important for the proof of our main theorem.

######  Theorem 3.1 ([7, Theorems 3.1 and 3.2]).

Let $(A,f)\in M_{p\times p}^{\vartriangle}\times M_{p\times 1}$ be any given.
Then, there is a unique surjective mapping

$\Delta_{A,f}\colon\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}\times\mathbb{R}^{p};\
(s,u)\mapsto(s,\Delta_{s}(u))$

which possesses the following properties:

  1. (i)

$\Delta_{A,f}$ maps the phase-portraits of (3.1) onto that of (3.2). In fact,

$\Delta_{A,f}(t,z_{A,f}(t;s,u))=(t,z_{A}(t;s,\Delta_{s}(u)));$

that is to say, the following commutativity holds:

$\begin{CD}\mathbb{R}^{p}@>{z_{A,f}(t;s,\cdot)}>{}>\mathbb{R}^{p}\\\
@V{\Delta_{s}}V{}V@V{}V{\Delta_{t}}V\\\
\mathbb{R}^{p}@>{z_{A}(t;s,\cdot)}>{}>\mathbb{R}^{p}.\end{CD}$

  2. (ii)

$\Delta_{A,f}$ is a $\varepsilon_{A,f}$-mapping, i.e.,
$\|(s,u)-\Delta_{A,f}(s,u)\|\leq\varepsilon_{A,f}$ for all
$(s,u)\in\mathbb{R}\times\mathbb{R}^{p}$, where

$\varepsilon_{A,f}=\eta_{f}\xi_{A}(1+2\eta_{A}\xi_{A})^{p};$

  3. (iii)

For any $(s,u),(s,u^{\prime})\in\mathbb{R}\times\mathbb{R}^{p}$,
$z_{A,f}(t;s,u)-z_{A}(t;s,u^{\prime})$ is bounded on $\mathbb{R}$ if and only
if $\Delta_{s}(u)=u^{\prime}$.

  4. (iv)

If

$L_{f}\leq\frac{1}{\xi_{A}(1+\eta_{A}\xi_{A})^{p}},$

then $\Delta_{A,f}$ is a self-homeomorphism of
$\mathbb{R}\times\mathbb{R}^{p}$.

Next, we endow $M_{p\times p}^{\vartriangle}\times M_{p\times 1}$ with the
compact-open topology. Let $(\mathbb{P},d)$ be a metric space with metric $d$
and $\eta_{\mathbb{P}}>0,\xi_{\mathbb{P}}>0,L_{\mathbb{P}}>0$ constants with
$L_{\mathbb{P}}\leq\frac{1}{\xi_{\mathbb{P}}(1+\eta_{\mathbb{P}}\xi_{\mathbb{P}})^{p}}$.
Let

$\mathfrak{S}\colon\mathbb{P}\rightarrow M_{p\times p}^{\vartriangle}\times
M_{p\times 1};\ \lambda\mapsto(A_{\lambda},f_{\lambda})$

be a continuous mapping such that
$\eta_{A_{\lambda}}\leq\eta_{\mathbb{P}},\xi_{A_{\lambda}}\leq\xi_{\mathbb{P}},L_{f_{\lambda}}\leq
L_{\mathbb{P}}$, and

$None$ $\dot{z}=A_{\lambda}(t)z,\quad(t,z)\in\mathbb{R}\times\mathbb{R}^{p}$

has no any nontrivial bounded solutions. We consider the bounded solutions of
the equations with parameter $\lambda$

$None$
$\dot{z}=A_{\lambda}(t)z+f_{\lambda}(t,z),\quad(t,z)\in\mathbb{R}\times\mathbb{R}^{p}.$

Define

$\Delta^{*}\colon\mathbb{P}\rightarrow\mathbb{R}^{p}$

in the way: for any $\lambda\in\mathbb{P}$

$\Delta_{\lambda}(0,\Delta^{*}(\lambda))=(0,\textbf{0})\in\mathbb{R}\times\mathbb{R}^{p}$

where
$\Delta_{\lambda}=\Delta_{A_{\lambda},f_{\lambda}}\colon\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}\times\mathbb{R}^{p}$
is determined by Theorem 3.1 for $(\ref{eq3.1})_{\lambda}$ and
$(\ref{eq3.2})_{\lambda}$.

We will need the following result, which will play a useful role in the later
proof of our main theorem in $\S\ref{sec4}$.

######  Theorem 3.2.

The mapping $\Delta^{*}\colon\mathbb{P}\rightarrow\mathbb{R}^{p}$ is
continuous.

###### Proof.

Let $\lambda_{0}\in\mathbb{P}$ and $\varepsilon>0$. Letting
$\textbf{x}_{0}=\Delta^{*}(\lambda_{0})\in\mathbb{R}^{p}$, we assert that
there exists some $\delta>0$ such that
$\|\Delta^{*}(\lambda)-\textbf{x}_{0}\|<\varepsilon$ whenever
$\lambda\in\mathbb{P}$ with $d(\lambda,\lambda_{0})<\delta$. If the assertion
were not true, there would be a sequence $\lambda_{j}\to\lambda_{0}$ in
$\mathbb{P}$ satisfying
$\|\Delta^{*}(\lambda_{j})-\textbf{x}_{0}\|\geq\varepsilon$ for all $j$. Since
for all $t\in\mathbb{R}$ we have

$\|z_{A_{\lambda_{j}},f_{\lambda_{j}}}(t;0,\Delta^{*}(\lambda_{j}))\|\leq\eta_{\mathbb{P}}\xi_{\mathbb{P}}(1+2\eta_{\mathbb{P}}\xi_{\mathbb{P}})^{p}\quad
j=1,2,\ldots$

by Theorem 3.1, we can assume $\Delta^{*}(\lambda_{j})\to\textbf{x}$ for some
$\textbf{x}\in\mathbb{R}^{p}$ and
$\|\textbf{x}-\textbf{x}_{0}\|\geq\varepsilon$. As $\mathfrak{S}$ is
continuous, it follows from a basic theorem of ODE that

$\lim_{j\to\infty}z_{A_{\lambda_{j}},f_{\lambda_{j}}}(t;0,\Delta^{*}(\lambda_{j}))=z_{A_{\lambda_{0}},f_{\lambda_{0}}}(t;0,\textbf{x})\quad\forall\,t\in\mathbb{R}$

which implies that $z_{A_{\lambda_{0}},f_{\lambda_{0}}}(t;0,\textbf{x})$ is a
bounded solution of $(\ref{eq3.1})_{\lambda_{0}}$. So,
$\textbf{x}=\textbf{x}_{0}$, it is a contradiction. ∎

##  4\. Structural stability of hyperbolic sets

In this section, we will prove our main theorem stated in the Introduction and
construct an explicit example.

We assume that $S\colon\mathbb{E}\rightarrow\mathbb{R}^{n}$ is a
$C^{1}$-vector field on the $n$-dimensional Euclidean $w$-space
$\mathbb{E},n\geq 2$, which gives rise to a flow $\phi\colon(t,w)\mapsto
t_{\cdot}w$. Let $\Lambda$ be a $\phi$-invariant closed subset, not
necessarily compact, of $\mathbb{E}$ such that

  1. (U1)

$S^{\prime}(w)$ is uniformly bounded on $\Lambda$;

  2. (U2)

$0<\inf_{w\in\Lambda}\|S(w)\|\leq\sup_{w\in\Lambda}\|S(w)\|<\infty$;

  3. (U3)

$S^{\prime}(w)$ is uniformly continuous at $\Lambda$; that is to say, to any
$\epsilon>0$ there is some $\delta>0$ so that for any
$\mathfrak{w}\in\Lambda$,
$\|S^{\prime}(w)-S^{\prime}(\mathfrak{w})\|<\epsilon$ whenever
$\|w-\mathfrak{w}\|<\delta$.

Now we prove the following structural stability theorem by using Liao methods.

######  Theorem 4.1.

Let $\Lambda$ be a hyperbolic set for $S$; that is to say, there exist
constants $C\geq 1,\lambda<0$ and a continuous $\Psi$-invariant splitting

$\mathbb{T}_{w}=T_{w}^{s}\oplus T_{w}^{u},\ \dim T_{w}^{s}=p_{-}(w)\quad
w\in\Lambda$

such that

$\displaystyle\|\Psi_{t_{0}+t,w}x\|$ $\displaystyle\leq C^{-1}\exp(\lambda
t)\|\Psi_{t_{0},w}x\|\quad\forall\,x\in T_{w}^{s}$ and
$\displaystyle\|\Psi_{t_{0}+t,w}x\|$ $\displaystyle\geq C\exp(-\lambda
t)\|\Psi_{t_{0},w}x\|\quad\forall\,x\in T_{w}^{u}$

for any $t_{0}\in\mathbb{R}$ and for all $t>0$. Then for any $\varepsilon>0$
there is a $C^{1}$-neighborhood $\mathcal{U}$ of $S$ in
$\mathfrak{X}^{1}(\mathbb{E})$ such that, if $V\in\mathcal{U}$ then there
exists a $\varepsilon$-topological mapping $H$ from $\Lambda$ onto some closed
subset $\Lambda_{V}$ which sends orbits of $S$ in $\Lambda$ into orbits of $V$
in $\Lambda_{V}$, such that $H(w)\in\Sigma_{w}$ and
$\psi_{V}(t,H(w))=H(t_{\cdot}w)\in\Sigma_{t_{\cdot}w}$ for all $w\in\Lambda$
and for any $t\in\mathbb{R}$.

###### Proof.

Let

$\mathcal{A}=\left\\{(w,\gamma)\in\mathscr{F}_{n-1}^{*\sharp}(\Lambda)\,|\,\textrm{col}_{k}\gamma\in
T_{w}^{s}\textrm{ for }1\leq k\leq p_{-}(w)\right\\}.$

Clearly, $\mathcal{A}$ is a $\chi^{*\sharp}$-invariant closed subset of
$\mathscr{F}_{n-1}^{*\sharp}(\Lambda)$ with compact fibers $\mathcal{A}_{w}$.
For any $(w,\gamma)\in\mathcal{A}$, we consider the reduced linearized
equations

(4.1)
$\dot{y}=R_{w,\gamma}^{*}(t)y,\quad(t,y)\in\mathbb{R}\times\mathbb{R}^{n-1},$

which is defined as Definition 2.1, and consider the reduced standard system

(4.2)
$\dot{y}=R_{w,\gamma}^{*}(t)y+V_{\textsl{rem}(w,\gamma)}^{*}(t,y),\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

for any $V\in\mathfrak{X}^{1}(\mathbb{E})$ defined as in Definition 2.11,
associated with $S$. Then, we can take from Lemma 2.2 a constant
$\eta_{\Lambda}>0$ such that

$\sup_{t\in\mathbb{R},(w,\gamma)\in\mathcal{A}}\|R_{w,\gamma}^{*}(t)\|\leq\eta_{\Lambda}<\infty.$

Thus, it follows from Corollary 2.3 that there is another constant
$\xi_{\Lambda}>0$ such that

$\displaystyle\xi_{\Lambda}$
$\displaystyle=\sup_{\begin{subarray}{c}t\in\mathbb{R}\\\
(w,\gamma)\in\mathcal{A}\end{subarray}}\left\\{\sum_{k=1}^{p_{-}(w)}\int_{-\infty}^{t}e^{\int_{s}^{t}\omega_{k}^{*}(\chi^{*\sharp}(\tau,(w,\gamma)))\,d\tau}\,ds\right.$
$\displaystyle{\qquad\qquad}\qquad\left.+\sum_{k=1+p_{-}(w)}^{n-1}\int_{t}^{\infty}e^{\int_{s}^{t}\omega_{k}^{*}(\chi^{*\sharp}(\tau,(w,\gamma)))\,d\tau}\,ds\right\\}$
$\displaystyle<\infty.$

By Lemma 2.5(1), Theorem 2.13 and Theorem 3.1, there is no loss of generality
in assuming that for any $(w,\gamma)\in\mathcal{A}$ the reduced standard
systems of $S$

(4.3)
$\dot{y}=R_{w,\gamma}^{*}(t)y+S_{\textsl{rem}(w,\gamma)}^{*}(t,y),\quad(t,y)\in\mathbb{R}\times\mathbb{R}_{\mathfrak{c}}^{n-1}$

has no any nontrivial bounded global solutions on $\mathbb{R}$.

Let
$\mathfrak{N}_{\mathfrak{c}}(w)=\\{w+x\in\Sigma_{w};\,\|x\|\leq\mathfrak{c}\\}$.
Given any $\varepsilon>0$ small enough to satisfy that for any $w\in\Lambda$
and any $w^{\prime}\in\Lambda$ with $\|w-w^{\prime}\|<\varepsilon$, we have
$t_{\cdot}w^{\prime}\in\mathfrak{N}_{\mathfrak{c}}(w)$ for some
$|t|<2\varepsilon\diamondsuit_{\Lambda}^{-1}$, where
$\diamondsuit_{\Lambda}=\inf_{w\in\Lambda}\|S(w)\|>0$. On the other hand,
according to [3] we may assume that for any $w\in\Lambda$,
$\mathfrak{N}_{\mathfrak{c}}(w)\cap\mathfrak{N}_{\mathfrak{c}}(t_{\cdot}w)=\varnothing$
for all $|t|<2\varepsilon\diamondsuit_{\Lambda}^{-1}$.

Denote

$\rho_{\xi}=\frac{\xi}{4\xi_{\Lambda}(1+2\eta_{\Lambda}\xi_{\Lambda})^{n-1}}\quad\textrm{and}\quad\kappa=\frac{1}{4\xi_{\Lambda}(1+2\eta_{\Lambda}\xi_{\Lambda})^{n-1}}$

for any $\xi\in(0,\mathfrak{c}]$. Then, by Theorem 2.13 there exists a
$C^{1}$-neighborhood $\mathcal{U}$ of $S$ in $\mathfrak{X}^{1}(\mathbb{E})$
and a constant $\xi\in(0,\mathfrak{c}]$ with $\xi<1$ such that for any
$V\in\mathcal{U}$ we have

(4.4a) $\displaystyle\sup_{\begin{subarray}{c}(w,\gamma)\in\mathcal{A}\\\
(t,y)\in\mathbb{R}\times\mathbb{R}_{\xi}^{n-1}\end{subarray}}\|V_{\textsl{rem}(w,\gamma)}^{*}(t,y)\|$
$\displaystyle\leq\varepsilon\rho_{\xi}$ and (4.4b)
$\displaystyle\|V_{\textsl{rem}(w,\gamma)}^{*}(t,y)-V_{\textsl{rem}(w,\gamma)}^{*}(t,y^{\prime})\|$
$\displaystyle\leq\kappa\|y-y^{\prime}\|\quad\forall\,y,y^{\prime}\in\mathbb{R}_{\xi}^{n-1}$

uniformly for $(t,(w,\gamma))\in\mathbb{R}\times\mathcal{A}$.

Fix some $C^{\infty}$ bump function $b\colon[0,\infty)\rightarrow[0,1]$ with
$b|[0,1/2]\equiv 1$ and $b|[1,\infty)\equiv 0$. For any $V\in\mathcal{U}$ and
any $(w,\gamma)\in\mathcal{A}$, let

$\widetilde{V}_{\textsl{rem}(w,\gamma),\xi}(t,y)=\begin{cases}b(\|y\|/\xi)V_{\textsl{rem}(w,\gamma)}^{*}(t,y)&\textrm{
for }\|y\|\leq\mathfrak{c},\\\ \textbf{0}&\textrm{ for
}\|y\|\geq\mathfrak{c}.\end{cases}$

Next, we consider the adapted differential equations

(4.5)
$\dot{y}=R_{w,\gamma}^{*}(t)y+\widetilde{V}_{\textsl{rem}(w,\gamma),\xi}(t,y),\quad(t,y)\in\mathbb{R}\times\mathbb{R}^{n-1}.$

It is easily seen that $R_{w,\gamma}^{*}(t)\in
M_{(n-1)\times(n-1)}^{\vartriangle}$ and
$\widetilde{V}_{\textsl{rem}(w,\gamma),\xi}(t,y)\in M_{(n-1)\times 1}$ as in
$\S 3$ in the case $p=n-1$ and $p_{-}=p_{-}(w)$ for any
$(w,\gamma)\in\mathcal{A}$. Let $y_{V,(w,\gamma),\xi}(t;s,u)$ be the solution
of (4.5) such that $y_{V,(w,\gamma),\xi}(s;s,u)=u$ for any
$(s,u)\in\mathbb{R}\times\mathbb{R}^{n-1}$.

Given any $V\in\mathcal{U}$.

For any $(w,\gamma)\in\mathcal{A}$, it follows from Theorem 3.1 that there
_uniquely_ corresponds an $\textbf{x}\in\mathbb{R}^{n-1}$, writing
$h_{V,\xi}(w)=\gamma\textbf{x}\in\Sigma_{w}-w=\mathbb{T}_{w}$, such that
$y_{V,(w,\gamma),\xi}(t;0,\textbf{x})$ is bounded on $\mathbb{R}$ with

(4.6)
${\sup}_{t\in\mathbb{R}}\|y_{V,(w,\gamma),\xi}(t;0,\textbf{x})\|\leq\varepsilon\xi/4.$

So, $y_{V,(w,\gamma),\xi}(t;0,\textbf{x})$ is also the solution of (4.2).
According to Theorem 2.12(2) we easily see that such $h_{V,\xi}(w)$ is
independent of the choice of $\gamma$ in $\mathcal{A}_{w}$ and is such that
$\|h_{V,\xi}(w)\|\leq\min\\{\varepsilon,\xi\\}/4$ for $w\in\Lambda$. By
Theorem 3.1 and Theorem 2.12(2) again we have easily

(4.7)
$h_{V,\xi}(t_{\cdot}w)=\psi_{V;t,w}(h_{V,\xi}(w))\in\mathbb{T}_{t_{\cdot}w}\quad\forall\,t\in\mathbb{R},$

since
$\psi_{V;t,w}(h_{V,\xi}(w))=\widehat{\mathcal{P}}_{w,\gamma}^{*}(t,y_{V,(w,\gamma),\xi}(t;0,\textbf{x}))$
for any $w\in\Lambda$. Moreover, we can assert that the mapping $w\mapsto
w+h_{V,\xi}(w)$ is injective. In fact, if
$w+h_{V,\xi}(w)=w^{\prime}+h_{V,\xi}(w^{\prime})$ for some
$w,w^{\prime}\in\Lambda$, then
$\|t_{\cdot}w-t_{\cdot}w^{\prime}\|\leq\varepsilon/2$ for all
$t\in\mathbb{R}$. Since (4.3) has only one global bounded solution on
$\mathbb{R}$, there is some $t^{\prime}$ with
$|t^{\prime}|<2\varepsilon\diamondsuit_{\Lambda}^{-1}$ such that
$t^{\prime}_{\cdot}w^{\prime}=w$. Thus, $t^{\prime}=0$. Otherwise
$\mathfrak{N}_{\mathfrak{c}}(w)\cap\mathfrak{N}_{\mathfrak{c}}(w^{\prime})\not=\varnothing$,
it is a contradiction.

Let $\Lambda_{V}=\\{w+h_{V,\xi}(w)\,|\,w\in\Lambda\\}$ and
$H_{V}\colon\Lambda\rightarrow\Lambda_{V};\,w\mapsto
w+h_{V,\xi}(w)\in\Sigma_{w}$. Clearly, $\|w-H_{V}(w)\|<\varepsilon$. It
remains to prove that $\Lambda_{V}$ is closed in $\mathbb{E}$ and $H_{V}$ is a
homeomorphism.

At first, we show that $\Lambda_{V}$ is closed in $\mathbb{E}$ and
$H_{V}^{-1}\colon\Lambda_{V}\rightarrow\Lambda$ continuous as well. Let
$w_{j}^{\prime}\to w^{\prime}$ with $w_{j}^{\prime}\in\Lambda_{V}$ and
$w_{j}=H_{V}^{-1}(w_{j}^{\prime})$ for $j=1,2,\ldots$. We have to prove
$w_{j}\to w$ for some $w\in\Lambda$ and $w^{\prime}=H_{V}(w)$. By the
definition of $H_{V}$, there is a sequence $(w_{j},\gamma_{j})$ in
$\mathcal{A}$ and a sequence $(\textbf{x}_{j})$ in $\mathbb{R}^{n-1}$ such
that

$w_{j}^{\prime}=w_{j}+h_{V,\xi}(w_{j})=w_{j}+\gamma_{j}\textbf{x}_{j}\quad\textrm{for
}j=1,2,\ldots.$

Since
$\gamma_{j}\in\mathcal{A}_{w_{j}}\subset\mathscr{F}_{n-1,w_{j}}^{*\sharp}\subset\mathscr{F}_{n-1}^{\sharp},\|\textbf{x}_{j}\|\leq\varepsilon\xi/4$
and $\mathscr{F}_{n-1}^{\sharp}$ is compact, without loss of generality we may
assume that $\gamma_{j}\to\gamma$ in $\mathscr{F}_{n-1}^{\sharp}$ and
$\textbf{x}_{j}\to\textbf{x}$ in $\mathbb{R}^{n-1}$. Let
$w=w^{\prime}-\gamma\textbf{x}$. Then $w_{j}\to w$ in $\Lambda$ and
$w^{\prime}=w+\gamma\textbf{x}$ and $(w,\gamma)\in\mathcal{A}$. In order to
prove $w^{\prime}=H_{V}(w)$, it is sufficient to prove that
$h_{V,\xi}(w)=\gamma\textbf{x}$. In fact, from Theorem 2.12(1) and a basic
theorem of ODE, we have

$\lim_{j\to\infty}\sup_{|t|<T,\|u\|\leq\mathfrak{c}}\|y_{V,(w_{j},\gamma_{j}),\xi}(t;0,u)-y_{V,(w,\gamma),\xi}(t;0,u)\|=0\quad\forall\,T>0.$

Thus, for all $t\in\mathbb{R}$ we have

$\lim_{j\to\infty}y_{V,(w_{j},\gamma_{j}),\xi}(t;0,\textbf{x}_{j})=y_{V,(w,\gamma),\xi}(t;0,\textbf{x}),$

which means $\|y_{V,(w,\gamma),\xi}(t;0,\textbf{x})\|\leq\varepsilon\xi/4$ for
all $t\in\mathbb{R}$. So, $h_{V,\xi}(w)=\gamma\textbf{x}$, as desired.

We can show that $H_{V}$ is continuous by Theorem 3.2.

Thus, the theorem is proved. ∎

######  Remark 4.2.

If $\Lambda=\mathbb{E}$, then $\Lambda_{V}=\mathbb{E}$ by a standard topology
argument. Indeed, letting $S^{n+1}=\mathbb{E}\cup\\{\infty\\}$, $H_{V}$ has a
continuous extension from the topological sphere $S^{n+1}$ to itself which
maps $\infty$ to $\infty$ and is homotopic to the identity. Thus, from
differential topology we know that $H_{V}(\mathbb{E})=\mathbb{E}$.

We conclude our arguments with an example.

######  Example 4.3.

Let $S(x,y,z)=(1,y,-z)^{\mathrm{T}}\in\mathbb{R}^{3}$ for any
$(x,y,z)\in\mathbb{E}^{3}$, which is a differential system on the
3-dimensional Euclidean $(x,y,z)$-space $\mathbb{E}^{3}$. Let
$\Lambda=\mathbb{R}\times\\{0\\}\times\\{0\\}$. Then, $S$ gives rise to the
$C^{1}$-flow $\phi\colon(t,(x,y,z))\mapsto(x+t,ye^{t},ze^{-t})$, and $S$ is
hyperbolic with $\mathbb{T}_{(x,0,0)}=T_{(x,0,0)}^{s}\oplus T_{(x,0,0)}^{u}$,
where
$T_{(x,0,0)}^{s}=\\{0\\}\times\\{0\\}\times\mathbb{R},T_{(x,0,0)}^{u}=\\{0\\}\times\mathbb{R}\times\\{0\\}$
for any $(x,0,0)\in\Lambda$, and $S$ satisfies conditions (U1), (U2) and (U3)
on $\Lambda$. Thus, $S$ is structurally stable on $\Lambda$ from the Main
Theorem. Particularly, for any $\varepsilon>0$, if
$V\in\mathfrak{X}^{1}(\mathbb{E}^{3})$ is $C^{1}$-close to $S$, then $V$ has
an integral curve which lies in the $\varepsilon$-tubular neighborhood of
$\mathbb{R}\times\\{(0,0)\\}$.

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\kern-3.50006pt{\rule[3.44444pt]{0.04303pt}{4.30554pt}\hskip
1.42271pt}}}^{bb}\langle{\sf K},{\sf S},{\sf
C}\rangle\mathrel{\rightarrow\mathrel{\mkern-14.0mu}\rightarrow}M$. It follows
that ${\sf E}^{bb}\mathbin{{\triangleq}}\langle\langle{\sf K},{\sf S},{\sf
C}\rangle\rangle$ is a self-evaluator: for all $M{\in}\Lambda$

${\sf E}^{bb}\mathopen{\mbox{\hskip
1.42271pt\rule[3.44444pt]{0.04303pt}{4.30554pt}
\kern-3.50006pt{\rule[7.74998pt]{2.58334pt}{0.04303pt}}}}{M}\mathclose{\mbox{\rule[7.74998pt]{2.58334pt}{0.04303pt}
\kern-3.50006pt{\rule[3.44444pt]{0.04303pt}{4.30554pt}\hskip
1.42271pt}}}^{bb}\equiv\langle\langle{\sf K},{\sf S},{\sf
C}\rangle\rangle\mathopen{\mbox{\hskip
1.42271pt\rule[3.44444pt]{0.04303pt}{4.30554pt}
\kern-3.50006pt{\rule[7.74998pt]{2.58334pt}{0.04303pt}}}}{M}\mathclose{\mbox{\rule[7.74998pt]{2.58334pt}{0.04303pt}
\kern-3.50006pt{\rule[3.44444pt]{0.04303pt}{4.30554pt}\hskip
1.42271pt}}}^{bb}\mathrel{\rightarrow}\mathopen{\mbox{\hskip
1.42271pt\rule[3.44444pt]{0.04303pt}{4.30554pt}
\kern-3.50006pt{\rule[7.74998pt]{2.58334pt}{0.04303pt}}}}{M}\mathclose{\mbox{\rule[7.74998pt]{2.58334pt}{0.04303pt}
\kern-3.50006pt{\rule[3.44444pt]{0.04303pt}{4.30554pt}\hskip
1.42271pt}}}^{bb}\langle{\sf K},{\sf S},{\sf
C}\rangle\mathrel{\rightarrow\mathrel{\mkern-14.0mu}\rightarrow}M.$

It is a remarkable coincidence that the term ${\sf
E}^{bb}\equiv\langle\langle{\sf K},{\sf S},{\sf C}\rangle\rangle$ abbreviates
the name “Kleene, Stephen Cole”, the full name of the inventor of self-
evaluation in $\lambda$-calculus. Corrado Böhm was fond of such tricks and had
for this and other reasons the nickname ‘il miracolo’.

## Coda

At a symposium in honor of Corrado Böhm’s ninety’s birthday, January 2013, at
Sapienza University, Rome, the jubilee treated the audience with an open
problem. Actually it is more a ‘Koan’ (not precisely stated) than a Problem
(with a precisely stated space of answers). But Koans are often the more
interesting problems in mathematics and computer science.

######  Problem/Koan 15 ((C. Böhm, 2013)).

Given $\beta$-normal forms $F\equiv\lambda x_{1}\cdots x_{n}.P$, and
$G\equiv\lambda x_{1}\cdots x_{n}.Q{\in}\Lambda^{o}$. By writing
$F^{d}\mathbin{{\triangleq}}\lambda
x.F(x{{{\mathbf{c}}}_{1}})\ldots(x{{{\mathbf{c}}}_{n}})$ and similarly for
$G^{d}$, these terms can be made unary. Trying to find closed terms ${M}$ from
solutions $N$ of the equation $F^{d}N=_{\beta}G^{d}N$? (Define a _deed_ to be
a closed nf of the form $\lambda x.xP_{1}\cdots P_{k}$. The $F^{d},G^{d}$ are
deeds up to $=_{\beta}$.)

## Acknowledgments

The author thanks Marko van Eekelen for explaining him many years ago the
method of bootstrapping (Section 1), Mariangiola Dezani for comments on the
paper, and Rinus Plasmeijer for discussions about Section 3. The referees
provided very useful remarks, improving the paper.

To the family of Corrado Böhm I am grateful for letting me spend wonderful
times with them, besides for fully enabling us to enjoy the combinators.

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