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# Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

Bram Lentjes Department of Mathematics, Hasselt University, Diepenbeek Campus,
3590 Diepenbeek, Belgium (bram.lentjes@uhasselt.be). Mattias Windmolders
Department of Mathematics, KU Leuven, 3000 Leuven, Belgium
(mattias.windmolders@student.kuleuven.be). Yuri A. Kuznetsov Department of
Mathematics, Utrecht University, 3508 TA Utrecht, The Netherlands and
Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The
Netherlands (i.a.kouznetsov@uu.nl).

###### Abstract

In this paper, we deal with a classical object, namely, a nonhyperbolic limit
cycle in a system of smooth autonomous ordinary differential equations. While
the existence of a center manifold near such a cycle was assumed in several
studies on cycle bifurcations based on periodic normal forms, no proofs were
available in the literature until recently. The main goal of this paper is to
give an elementary proof of the existence of a periodic smooth locally
invariant center manifold near a nonhyperbolic cycle in finite-dimensional
ordinary differential equations by using the Lyapunov-Perron method. In
addition, we provide several explicit examples of analytic vector fields
admitting (non)-unique, (non)-$C^{\infty}$-smooth and (non)-analytic periodic
center manifolds.

Center manifold theorem, nonhyperbolic cycles, ordinary differential equations

34C25, 34C45, 37G15

##  1 Introduction

Center manifold theory is without doubt one of the most well-known and
powerful techniques to study local bifurcations of dynamical systems [24, 40].
In its simplest form, center manifold theory allows us to analyze the behavior
of a complicated high-dimensional nonlinear dynamical system near a
bifurcation by reducing the system to a low-dimensional invariant manifold,
called the center manifold.

The center manifold theorem for finite-dimensional ordinary differential
equations (ODEs) near a nonhyperbolic equilibrium has first been proved in
[47, 37] and developed further in [29, 5]. Over the years, the existence of a
center manifold near a nonhyperbolic equilibrium has been established for
various other classes of dynamical systems by employing different techniques,
such as, for example, the graph transform [49, 48], the parametrization method
[53, 54] and the Lyapunov-Perron method [55, 27]. Mainly this last method has
been proven to be very successful in the setting of infinite-dimensional
dynamical systems. For example, the center manifold theorem for equilibria has
been obtained under various assumptions for ODEs in Banach spaces [57, 9, 55,
56, 27], partial differential equations [28, 38, 44, 45, 2], stochastic
dynamical systems [4, 23, 7, 8, 43], classical delay (differential) equations
[6, 21, 26, 22, 3], renewal equations [21, 22, 19], abstract delay
(differential) equations [36], impulsive delay differential equations [11],
mixed functional difference equations [31] and mixed functional differential
equations [32]. Various interesting and important qualitative properties of
center manifolds for equilibria can be found in [50] and an extensive
literature overview on such manifolds in various classes of dynamical systems
can be found in [46]. In all cases, the dimension of the center manifold is
equal to the number of the critical eigenvalues of the equilibrium, i.e. those
with zero real parts.

The natural question arises if the whole center manifold construction can be
repeated for nonhyperbolic periodic orbits (cycles) in various classes of
dynamical systems. While the literature for center manifolds for equilibria is
extensive, the same cannot be said for center manifolds near cycles. A first
proof on the existence and smoothness of a center manifold for periodic mixed
functional differential equations was given in [33] and has been later adapted
in [10, 11] to the setting of periodic impulsive delay differential equations.
Recently, in [42], the existence of a smooth periodic finite-dimensional
center manifold near a cycle for classical delay differential equations has
been established using the general sun-star calculus framework [14, 12, 13,
15, 20, 22], which expands its applicability to various other classes of delay
equations. Here, the dimension of the center manifold is also determined by
the number of the critical multipliers of the cycle, including the trivial
(equal to one) multiplier.

However, as the state space in all mentioned references on this topic is
infinite-dimensional, many proofs are rather involved as one must rely on non-
trivial functional analytic techniques. While the resulting center manifold
theorems could be applied to finite-dimensional ODEs without delays, this is
certainly a redundant overkill. The main goal of this paper is to directly
state and prove a center manifold theorem for cycles in finite-dimensional
ODEs, using only elementary tools. Essentially, the proofs below are rather
straightforward adaptations of those from [42] in a much simpler finite-
dimensional context. We already remark that our exposition is based on the
classical Lyapunov-Perron method as a variation of constants formula is easily
available in this setting.

To study stability and bifurcations of limit cycles in ODEs, one can
alternatively work with a Poincaré map on a cross-section to the cycle [24,
40]. In most cases, this is sufficient, but then we miss one dimension, i.e.
the phase coordinate along the cycle. It should also be noted immediately that
the existence of a smooth (non-unique) center manifold for the fixed point of
the Poincaré map on a cross-section to the cycle does not imply directly the
existence of a smooth center manifold in a tubular neighborhood of the
cycle.111Notice that one can deduce the existence of the unique stable and
unstable manifold near a hyperbolic cycle from the fixed point of the Poincaré
map, see [26, Theorem 10.3.2]. A motivation in keeping this phase dimension is
to directly obtain all information of the dynamics near the cycle.

We fill two important gaps in the literature. First, from a theoretical point
of view, the results from [34, 35] on the existence of a special coordinate
system on the center manifold that allows us to describe the local dynamics
near a bifurcating cycle in terms of so-called periodic normal forms, rely
heavily on the local invariance and smoothness properties of the center
manifold. However, no proof nor a reference towards the literature has been
provided which ensures the existence of a periodic sufficiently smooth center
manifold near a nonhyperbolic cycle. Second, from a more practical point of
view, many researchers use nowadays the well-known software package `MatCont`
[18] to study codimension one and two bifurcations of limit cycles in finite-
dimensional ODEs. In particular, if one is interested in determining the
nature (subcritical, supercritical or degenerate) of a bifurcation, one should
compute the critical normal form coefficients of an associated periodic normal
form. However, the computation of these coefficients in `MatCont` employs a
combination of the periodic normalization method [41, 16, 17], again based on
the smoothness and local invariance of the center manifold, with the special
coordinate system and the periodic normal forms mentioned above.

###  1.1 Statement of the main theorem

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a $C^{k+1}$-smooth vector
field for some $k\geq 1$ and consider the ordinary differential equation

$\dot{x}(t)=f(x(t)),$ (ODE)

where $x(t)\in\mathbb{R}^{n}$. Assume that (ODE) admits a $T$-periodic
solution $\gamma$ for some (minimal) $T>0$ and let
$\Gamma:=\gamma(\mathbb{R})$ denote the associated (limit) _cycle_. Consider
now the variational equation around $\Gamma$

$\dot{y}(t)=A(t)y(t),$ (1)

where $A(t):=Df(\gamma(t))$ and $y(t)\in\mathbb{R}^{n}$. The unique (global)
solution $y$ of (1) is generated by the _fundamental matrix_
$U(t,s)\in\mathbb{R}^{n\times n}$ as $y(t)=U(t,s)y_{0}$ for all
$(t,s)\in\mathbb{R}^{2}$, where $y_{0}\in\mathbb{R}^{n}$ is an initial
condition specified at time $s$. The eigenvalues of the matrix $U(s+T,s)$ are
called _Floquet multipliers_ (of $\Gamma$), and we say that $\Gamma$ is
_nonhyperbolic_ if there are at least $n_{0}+1\geq 2$ Floquet multipliers on
the unit circle that are counted with algebraic multiplicity. Let $E_{0}(s)$
denote the $(n_{0}+1)$-dimensional _center subspace_ (at time $s$) defined by
the direct sum of all generalized eigenspaces with a Floquet multiplier on the
unit circle and let
$E_{0}:=\\{(s,y_{0})\in\mathbb{R}\times\mathbb{R}^{n}:y_{0}\in E_{0}(s)\\}$
denote the _center bundle_. The main result on the existence of a periodic
smooth local invariant center manifold near the cycle $\Gamma$ is summarized
in thm:main and two illustrative examples of two-dimensional _local center
manifolds around $\Gamma$_, denoted by
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$, can be found in fig:CM.
Explicit minimal equations (model systems) of the form (ODE) admitting a
$2\pi$-periodic two-dimensional center manifold around a nonhyperbolic cycle
are given by ex:analytic (cylinder) and ex:mobius (Möbius band).

[Center Manifold Theorem for Cycles] Consider (ODE) with a $C^{k+1}$-smooth
right-hand side $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ for some $k\geq 1$. Let
$\gamma$ be a $T$-periodic solution of (ODE) such that the associated cycle
$\Gamma:=\gamma(\mathbb{R})$ is nonhyperbolic with $(n_{0}+1)$-dimensional
center subspace $E_{0}(s)$ at time $s\in\mathbb{R}$. Then there exists a
locally defined $T$-periodic $C^{k}$-smooth $(n_{0}+1)$-dimensional invariant
manifold $\mathcal{W}^{c}_{\operatorname{loc}}(\Gamma)$ defined around
$\Gamma$ and tangent to the center bundle $E_{0}$.

Figure 1: Illustration of two-dimensional local center manifolds around
$\Gamma$ [40]. The left figure represents the case when $-1$ is not a Floquet
multiplier and then $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is locally
diffeomorphic to a cylinder in a neighborhood of $\Gamma$. The right figure
represents the case when $-1$ is a Floquet multiplier and then
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is locally diffeomorphic to a
Möbius band in a neighborhood of $\Gamma$. The $(\tau,\xi)$-coordinate system
on the center manifold is the special coordinate system described in [34, 35].

###  1.2 Overview

The paper is organized as follows. In sec:Floquet we review some basic
principles of Floquet theory for ODEs and elaborate a bit more on spectral
decompositions. In sec:existence we use the theory from previous section to
prove the existence of a Lipschitz continuous center manifold for (nonlinear)
periodic ODEs. In sec:properties we prove that this center manifold is
periodic, sufficiently smooth, locally invariant and its tangent bundle is
precisely the center bundle. The technical proofs regarding smoothness of the
center manifold are relegated to Appendix A. Combining all these results
proves thm:main. In sec:examples we provide explicit examples of analytic
vector fields admitting (non)-unique, (non)-$C^{\infty}$-smooth and
(non)-analytic periodic center manifolds.

##  2 Floquet theory and spectral decompositions

Consider $\eqref{eq:ODE}$ admitting a $T$-periodic solution $\gamma$ with
associated cycle $\Gamma:=\gamma(\mathbb{R})$. The aim of this section is to
determine the stability of $\Gamma$ and characterize nonhyperbolicity using
Floquet theory. This (linear) theory will allow us to state and prove results
regarding spectral properties of our operators and spaces of interest.
Standard references for this entire section are the books [25, 35, 40] on ODEs
and [1] on Linear Algebra. All unreferenced claims relating to basic
properties of Floquet theory and spectral decompositions can be found here.

To study the stability of $\Gamma$, set $x=\gamma+y$ and notice that $y$
satisfies the (nonlinear) periodic ODE

$\dot{y}(t)=A(t)y(t)+R(t,y(t)),$ (2)

where $A(t):=Df(\gamma(t))$ and
$R(t,y(t)):=f(\gamma(t)+y(t))-f(\gamma(t))-A(t)y(t)$. Hence,
$A:\mathbb{R}\to\mathbb{R}^{n\times n}$ is a $T$-periodic $C^{k}$-smooth
function and $R:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is
$T$-periodic in the first component, $C^{k}$-smooth, $R(\cdot,0)=0$ and
$D_{2}R(\cdot,0)=0$, i.e. $R$ consists solely of nonlinear terms. Note that
the nonlinearity $R$ has one degree of smoothness less than the original
vector field $f$. For a starting time $s\in\mathbb{R}$ and initial condition
$y_{0}\in\mathbb{R}^{n}$ for (2), it follows from the $C^{k}$-smoothness of
$R$ and the Picard-Lindelöf theorem that (2) admits a unique (maximal)
solution for all $t\in\mathbb{R}$ sufficiently close to $s$. Hence, let for
such $t$ and $s$ the map $S(t,s,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote
the _(time-dependent) flow_ , also called _process_ [39], of (2). One can
verify by uniqueness of solutions that

$\displaystyle S(t,s,S(r,s,\cdot))=S(t+r,s,\cdot),\quad S(s,s,\cdot)=I,\quad
S(t+T,s+T,\cdot)=S(t,s,\cdot),$ (3)

for all $t,r\in\mathbb{R}$ sufficiently close to $s$. It is clear from this
construction that studying solutions of (ODE) near $\Gamma$ is equivalent to
studying solutions of (2) near the origin. Therefore, we start by
investigating the linearization of (2) around the origin:

$\dot{y}(t)=A(t)y(t).$ (4)

Observe that its (global) solutions are generated by the _fundamental matrix_
$U(t,s)\in\mathbb{R}^{n\times n}$ as $y(t)=U(t,s)y_{0}$ for all
$(t,s)\in\mathbb{R}^{2}$, whenever an initial condition
$y_{0}\in\mathbb{R}^{n}$ at starting time $s\in\mathbb{R}$ is specified.
Moreover, we have that the map $(t,s)\mapsto U(t,s)$ is $C^{k}$-smooth. Using
uniqueness of solutions for (4), $T$-periodicity of $A$, and the fact that
$U(s,s)=I$ for all $s\in\mathbb{R}$, one can easily verify that

$\displaystyle U(t,r)U(r,s)=U(t,s),\quad U(t,s)^{-1}=U(s,t),\quad
U(t+T,s+T)=U(t,s),$ (5)

for all $t,r,s\in\mathbb{R}$. There holds for any $(t,s)\in\mathbb{R}^{2}$
that

$\displaystyle\frac{\partial}{\partial
t}U(t,s)=A(t)U(t,s),\quad\frac{\partial}{\partial t}U(s,t)=-U(s,t)A(t).$

The first equality follows immediately from (4). To prove the second equality,
observe from (5) that $U(s,t)U(t,s)=I$ and so differentiating both sides with
respect to $t$ yields after rearranging

$\bigg{(}\frac{\partial}{\partial t}U(s,t)\bigg{)}U(t,s)=-U(s,t)A(t)U(t,s),$

which proves the claim. Combining the first and third equality of (5) together
with induction, one proves that $U(s+nT,s)=U(s+T,s)^{n}$ for all
$s\in\mathbb{R}$ and $n\in\mathbb{Z}$ and so $y(s+nT)=U(s+T,s)^{n}y_{0}$.
Hence, the long term behavior of the solution $y$ is determined by the
_monodromy matrix_ (at time $s$) $U(s+T,s)$ and especially its eigenvalues,
called _Floquet multipliers_. To develop a spectral theory for our problem of
interest, notice that one has to _complexify_ the state space $\mathbb{R}^{n}$
and all discussed operators defined on $\mathbb{R}^{n}$, i.e. one has to
extend the state space to $\mathbb{C}^{n}$ and extend all discussed operators
to $\mathbb{C}^{n}$, see [1, Chapter 9] for more information. However, for the
sake of simplicity, we will not introduce any additional notation for the
complexification of the operators.

Let us now study the spectrum $\sigma(U(s+T,s))$ of $U(s+T,s)$ in depth. It
follows from [40, Theorem 1.6] that the Floquet multipliers are independent of
the starting time $s$ and that $1$ is always a Floquet multiplier. To see this
last claim, differentiating $\dot{\gamma}(t)=f(\gamma(t))$ yields
$\ddot{\gamma}(t)=A(t)\dot{\gamma}(t)$ and so $\dot{\gamma}$ is a solution of
(4), i.e. $\dot{\gamma}(t)=U(t,s)\dot{\gamma}(s)$. Exploiting $T$-periodicity
of $\gamma$ yields
$\dot{\gamma}(s)=\dot{\gamma}(s+T)=U(s+T,s)\dot{\gamma}(s)$, which proves that
$1$ is an eigenvalue of $U(s+T,s)$ with associated eigenvector
$\dot{\gamma}(s)$.

Let $\lambda$ be a Floquet multiplier of algebraic multiplicity $m_{\lambda}$,
i.e. the $m_{\lambda}$-dimensional $U(s+T,s)$-invariant subspace
$E_{\lambda}(s):=\ker((U(s+T,s)-\lambda I)^{m_{\lambda}})$ of $\mathbb{C}^{n}$
is maximal, or equivalently, $m_{\lambda}$ is the order of a root of the
characteristic polynomial $\det(U(s+T,s)-\lambda I)$. This allows us to choose
a basis of $m_{\lambda}$ linearly independent (generalized) eigenvectors
$\zeta_{1}(s),\dots,\zeta_{m_{\lambda}}(s)$ of $E_{\lambda}(s)$. Moreover, let
$\pi_{\lambda}(s)$ be the projection from $\mathbb{C}^{n}$ to $E_{\lambda}(s)$
with kernel the orthogonal complement of $E_{\lambda}(s)$. Our next aim is to
extend $\zeta_{1}(s),\dots,\zeta_{m_{\lambda}}(s)$ and $\pi_{\lambda}(s)$
forward and backward in time. The results can be found in the following two
lemmas.

Let $\lambda$ be a Floquet multiplier, then the restriction
$U_{\lambda}(t,s):E_{\lambda}(s)\to E_{\lambda}(t)$ is well-defined and
invertible for all $(t,s)\in\mathbb{R}^{2}$. Moreover, there exist
$C^{k}$-smooth maps $\zeta_{i}:\mathbb{R}\to\mathbb{C}^{n}$ such that
$\zeta_{1}(t),\dots,\zeta_{m_{\lambda}}(t)$ is a basis of $E_{\lambda}(t)$ for
all $t\in\mathbb{R}$. One can verify easily from the equalities in (5) that

$(U(t+T,t)-\lambda I)U(t,s)=U(t,s)(U(s+T,s)-\lambda I).$

Therefore, for each $v\in E_{\lambda}(s)$, we get

$\displaystyle(U(t+T,t)-\lambda I)^{n}U(t,s)v=U(t,s)(U(s+T,s)-\lambda
I)^{n}v=0,$

which proves that $U_{\lambda}(t,s)v\in E_{\lambda}(t)$ since the Floquet
multipliers are independent of the starting time. As $U(t,s)$ is invertible,
its restriction $U_{\lambda}(t,s)$ is invertible as well and this proves the
first claim. To prove the second claim, let
$\zeta_{1}(s),\dots,\zeta_{m_{\lambda}}(s)$ be a basis of $E_{\lambda}(s)$ and
define for all $i\in\\{1,\dots,m_{\lambda}\\}$ the $C^{k}$-smooth maps
$\zeta_{i}:\mathbb{R}\to\mathbb{C}^{n}$ by
$\zeta_{i}(t):=U_{\lambda}(t,s)\zeta_{i}(s)$. By the first claim, it is clear
that $\zeta_{1}(t),\dots,\zeta_{m_{\lambda}}(t)$ is a basis of
$E_{\lambda}(t)$ for all $t\in\mathbb{R}$, and this completes the proof.

[Iooss & Adelmeyer, 1998] Let $\lambda$ be a Floquet multiplier, then there
exists a $T$-periodic $C^{k}$-smooth map
$\pi_{\lambda}:\mathbb{R}\to\mathbb{C}^{n\times n}$ such that
$\pi_{\lambda}(t)$ is the projection from $\mathbb{C}^{n}$ onto
$E_{\lambda}(t)$ for all $t\in\mathbb{R}$ and satisfies the periodic linear
ODE

$\dot{\pi}_{\lambda}(t)=A(t)\pi_{\lambda}(t)-\pi_{\lambda}(t)A(t).$ (6)

It will be convenient in the sequel to introduce the sets
$\Lambda_{-}:=\\{\lambda\in\sigma(U(s+T,s)):|\lambda|<1\\},\Lambda_{0}:=\\{\lambda\in\sigma(U(s+T,s)):|\lambda|=1\\}$
and $\Lambda_{+}:=\\{\lambda\in\sigma(U(s+T,s)):|\lambda|>1\\}$, where the
elements of these sets have to be counted with algebraic multiplicity. We say
that the cycle $\Gamma$ is _nonhyperbolic_ if there are at least $n_{0}+1\geq
2$ Floquet multipliers on the unit circle that are counted with algebraic
multiplicity, i.e. the cardinality of $\Lambda_{0}$ is at least $2$.

The following properties hold.

  1. $1.$

For each $s\in\mathbb{R}$, Euclidean $n$-space admits a direct sum
decomposition

$\mathbb{R}^{n}=E_{-}(s)\oplus E_{0}(s)\oplus E_{+}(s)$ (7)

in a _stable subspace_ , _center subspace_ and _unstable subspace_ (at time
$s$) respectively.

  2. $2.$

There exist three $T$-periodic $C^{k}$-smooth projectors
$\pi_{i}:\mathbb{R}\to\mathbb{R}^{n\times n}$ with
$\operatorname{ran}(\pi_{i}(s))=E_{i}(s)$ for all $s\in\mathbb{R}$ and
$i\in\\{-,0,+\\}$.

  3. $3.$

There exists a constant $N\geq 0$ such that
$\sup_{s\in\mathbb{R}}(\|\pi_{-}(s)\|+\|\pi_{0}(s)\|+\|\pi_{+}(s)\|)=N<\infty$.

  4. $4.$

The projections are mutually orthogonal: $\pi_{i}(s)\pi_{j}(s)=0$ for all
$s\in\mathbb{R}$ and $i\neq j$ with $i,j\in\\{-,0,+\\}$.

  5. $5.$

The projections commute with the fundamental matrix:
$U(t,s)\pi_{i}(s)=\pi_{i}(t)U(t,s)$ for all $(t,s)\in\mathbb{R}^{2}$ and
$i\in\\{-,0,+\\}$.

  6. $6.$

The restrictions $U_{i}(t,s):E_{i}(s)\to E_{i}(t)$ are well-defined and
invertible for all $(t,s)\in\mathbb{R}^{2}$ and $i\in\\{-,0,+\\}$.

  7. $7.$

The decomposition (7) is an exponential trichotomy on $\mathbb{R}$ meaning
that there exist $a<0<b$ such that for every $\varepsilon>0$ there exists a
$C_{\varepsilon}>0$ such that

$\displaystyle\|U_{-}(t,s)\|$ $\displaystyle\leq
C_{\varepsilon}e^{a(t-s)},\quad t\geq s,$ $\displaystyle\|U_{0}(t,s)\|$
$\displaystyle\leq C_{\varepsilon}e^{\varepsilon|t-s|},\quad
t,s\in\mathbb{R},$ $\displaystyle\|U_{+}(t,s)\|$ $\displaystyle\leq
C_{\varepsilon}e^{b(t-s)},\quad t\leq s.$

We verify the seven properties step by step.

1\. From the generalized eigenspace decomposition theorem, we have that

$\mathbb{R}^{n}=\bigoplus_{\lambda\in\sigma(U(s+T,s))}E_{\lambda}(s),$

and if we define $E_{i}(s):=\oplus_{\lambda\in\Lambda_{i}}E_{\lambda}(s)$ for
$i\in\\{-,0,+\\}$, the result follows. Notice that $E_{i}(s)$ can be regarded
as real vector space since, if $\lambda\in\Lambda_{i}$, then
$\overline{\lambda}\in\Lambda_{i}$ because $U$ is a real operator.

2\. Define for $i\in\\{-,0,+\\}$ the map $\pi_{i}$ by
$\pi_{i}(s):=\sum_{\lambda\in\Lambda_{i}}\pi_{\lambda}(s)$. It follows from
linearity and lemma:Iooss that $\pi_{i}$ is $T$-periodic and $C^{k}$-smooth.
By construction, the range of $\pi_{i}(s)$ is $E_{i}(s)$ for all
$s\in\mathbb{R}$. The same argument as in the first assertion shows that
$\pi_{i}$ is a real operator.

3\. Because $\pi_{i}$ and the norm $\|\cdot\|$ are continuous, we have that
the map $\|\pi_{i}(\cdot)\|:\mathbb{R}\to\mathbb{R}$ is $T$-periodic and
continuous. The claim follows now from applying three times the extreme value
theorem.

4\. For $y\in\mathbb{R}^{n}$ the direct sum (7) admits a unique decomposition
$y=y_{-}+y_{0}+y_{+}$ with $y_{i}\in E_{i}(s)$. Hence,
$\pi_{i}(s)\pi_{j}(s)y=\pi_{i}(s)y_{j}=0$ if $i\neq j$ for all
$s\in\mathbb{R}$.

5\. Differentiating $t\mapsto U(t,s)\pi_{i}(s)$ and $t\mapsto\pi_{i}(t)U(t,s)$
while using (6), one sees that they both satisfy (4). Since they coincide at
time $t=s$, we have by uniqueness that they must be equal.

6\. Define for $i\in\\{-,0,+\\}$ the map $U_{i}(t,s)$ by
$U_{i}(t,s):=\oplus_{\lambda\in\Lambda_{i}}U_{\lambda}(t,s)$ for all
$(t,s)\in\mathbb{R}^{2}$. The claim follows now from linearity and
lemma:basis.

7\. We will only prove the $U_{-}(t,s)$ estimate as the other ones can be
proven similarly. Since the spectrum of $U_{-}(s+T,s)$ lies inside the unit
disk, it follows from the spectral radius formula that

$\lim_{n\rightarrow\infty}\|U_{-}(s+T,s)^{n}\|^{\frac{1}{n}}=\max_{\lambda\in\sigma(U_{-}(s+T,s))}|\lambda|<1.$

Hence, there exists an $a<0$ and an integer $n>0$ such that

$\|U_{-}(s+T,s)^{n}\|<(1+aT)^{n},$

and by continuity of the map $t\mapsto U_{-}(t,s)$, there is some $L>0$ such
that $\sup_{s\leq t\leq s+T}\|U_{-}(t,s)\|\leq L$. Denote
$L_{n}:=L\max_{j=0,\dots,n-1}\|U_{-}(s+T,s)^{j}\|$ and let $m_{t}$ be the
largest positive integer such that $s+m_{t}nT\leq t$ and let $0\leq
m_{t}^{\star}\leq n-1$ be the largest integer such that
$s+m_{t}nT+m_{t}^{\star}T\leq t$. Using (5), one obtains

$\displaystyle U_{-}(t,s)=U_{-}(t-m_{t}nT-
m_{t}^{\star}T,s)U_{-}(s+T,s)^{m_{t}^{\star}}U_{-}(s+T,s)^{m_{t}n}.$

By the maximum property of $m_{t}^{\star}$: $s\leq t-m_{t}nT-
m_{t}^{\star}T\leq s+m_{t}nT+(m_{t}^{\star}+1)T-m_{t}nT-m_{t}^{\star}T=s+T$,
and so

$\displaystyle\|U_{-}(t,s)\|\leq L_{n}\|U_{-}(s+T,s)^{n}\|^{m_{t}}\leq
L_{n}(1+aT)^{m_{t}n}\leq L_{n}[(1+aT)^{\frac{1}{aT}}]^{a(t-s)}\leq
L_{n}e^{a(t-s)},$

where we used in the last equality the fact that the map
$x\mapsto(1+\frac{1}{x})^{x}$ is monotonically increasing on $(-\infty,0)$ and
converges to $e$ as $x\to-\infty$. For the other estimates, for a given
$\varepsilon>0$ and sufficiently large $m\in\mathbb{N}$, one finds that there
exists a $M_{\varepsilon}$ and $N_{m}$ such that $||U_{0}(t,s)||\leq
M_{\varepsilon}e^{\varepsilon|t-s|}$ for $(t,s)\in\mathbb{R}^{2}$ and
$||U_{+}(t,s)||\leq N_{m}e^{b(t-s)}$ for $t\leq s$. Choosing
$C_{\varepsilon}:=\max\\{L_{n},M_{\varepsilon},N_{m}\\}$ proves the claim.

##  3 Existence of a Lipschitz center manifold

The aim of this section is to prove the existence of a (local) center manifold
for (2) around the origin. The proof consists of four steps. In the first
step, we show that we can formulate (2) equivalently as an integral equation.
In the second step, we determine a pseudo-inverse for solutions of this
integral equation on a suitable Banach space. In the third step, we modify our
nonlinearity $R$ outside a ball of radius $\delta>0$ such that it becomes
Lipschitz continuous, and eventually a contraction when $\delta$ is chosen
small enough. In the last step, we construct a (family of) fixed point
operators using the pseudo-inverse and modified nonlinearity for a
sufficiently small $\delta$. The fixed points of these contractions constitute
the center manifold.

The ordinary differential equation (2) is equivalent to the integral equation

$u(t)=U(t,s)u(s)+\int_{s}^{t}U(t,\tau)R(\tau,u(\tau))d\tau.$ (8)

Any $u$ satisfying (8) is clearly differentiable, and it follows from the
Leibniz integral rule that

$\displaystyle\dot{u}(t)$
$\displaystyle=A(t)U(t,s)u(s)+U(t,t)R(t,u(t))+\int_{s}^{t}A(t)U(t,\tau)R(\tau,u(\tau))d\tau$
$\displaystyle=A(t)\left[U(t,s)u(s)+\int_{s}^{t}U(t,\tau)R(\tau,u(\tau))d\tau\right]+R(t,u(t))=A(t)u(t)+R(t,u(t)),$

which proves that $u$ satisfies (2). Conversely, let $u$ satisfy (2) and let
$w(t)=U(s,t)u(t)$. Then

$\dot{w}(t)=\bigg{(}\frac{\partial}{\partial
t}U(s,t)\bigg{)}u(t)+U(s,t)\dot{u}(t).$

The second equality in lemma:partialUts together with the fact that $u$
satisfies (2) shows that $\dot{w}(t)=U(s,t)R(t,u(t))$. Integrating both sides
with respect to $t$ yields

$\displaystyle u(t)$
$\displaystyle=U(t,s)u(s)+U(t,s)\int_{s}^{t}U(s,\tau)R(\tau,u(\tau))d\tau=U(t,s)u(s)+\int_{s}^{t}U(t,\tau)R(\tau,u(\tau))d\tau,$

where we used (5) in the last equality.

Let $C_{b}(\mathbb{R},\mathbb{R}^{n})$ denote the Banach space of
$\mathbb{R}^{n}$-valued continuous bounded functions defined on $\mathbb{R}$
equipped with the supremum norm $\|\cdot\|_{\infty}$. If we want to study
solutions of (4) (or equivalently (8) with $R=0$) in the center subspace, it
turns that such solutions can be unbounded, and so we can not work in the
space $C_{b}(\mathbb{R},\mathbb{R}^{n})$. Instead, we must work in a function
space that allows limited (sub)exponential growth both at plus and minus
infinity. Therefore, define for any $\eta,s\in\mathbb{R}$ the normed space

$\operatorname{BC}_{s}^{\eta}:=\bigg{\\{}f\in
C(\mathbb{R},\mathbb{R}^{n}):\sup_{t\in\mathbb{R}}e^{-\eta|t-s|}\|f(t)\|<\infty\bigg{\\}},$

with the weighted supremum norm

$\|f\|_{\eta,s}:=\sup_{t\in\mathbb{R}}e^{-\eta|t-s|}\|f(t)\|.$

Since the linear map
$\iota:(C_{b}(\mathbb{R},\mathbb{R}^{n}),\|\cdot\|_{\infty})\rightarrow(\operatorname{BC}_{s}^{\eta},\|\cdot\|_{\eta,s})$
defined by $\iota(f)(t):=e^{\eta|t-s|}f(t)$ is an isometry, it is clear that
$(\operatorname{BC}_{s}^{\eta},\|\cdot\|_{\eta,s})$ is a Banach space. The
following result proves that all solutions of (4) on the center subspace
belong to $\operatorname{BC}_{s}^{\eta}$.

Let $\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$. Then

$\displaystyle E_{0}(s)=\\{y_{0}\in\mathbb{R}^{n}:\text{ there exists a
solution of \eqref{eq:A(t)y(t)}}\text{ through $y_{0}$ belonging to
$\operatorname{BC}_{s}^{\eta}$}\\}.$

Choose $y_{0}\in E_{0}(s)$ and define $y(t)=U_{0}(t,s)y_{0}$, which is indeed
a solution of (4) through $y_{0}$. Let $\varepsilon\in(0,\eta]$ be given. The
exponential trichotomy from prop:criteria shows that

$\displaystyle e^{-\eta|t-s|}\|y(t)\|$
$\displaystyle=e^{-\eta|t-s|}\|U_{0}(t,s)y_{0}\|\leq
C_{\varepsilon}e^{(\varepsilon-\eta)|t-s|}\|y_{0}\|\leq
C_{\varepsilon}\|y_{0}\|,\quad\forall t,s\in\mathbb{R}.$

Taking the supremum over $t\in\mathbb{R}$ yields
$y\in\operatorname{BC}_{s}^{\eta}$. Conversely, let $y_{0}\in\mathbb{R}^{n}$
be such that $y$, defined by $y(t)=U(t,s)y_{0}$, is in
$\operatorname{BC}_{s}^{\eta}$. For $t\geq\max\\{s,0\\}$ and
$\varepsilon\in(0,\eta]$, we get

$\displaystyle\|\pi_{+}(s)y_{0}\|$
$\displaystyle=\|U_{+}(s,t)\pi_{+}(t)y(t)\|\leq
C_{\varepsilon}e^{b(s-t)}N\|y(t)\|,$

which shows that

$e^{-\eta|t-s|}\|y(t)\|\geq\frac{e^{(b-\eta)(t-s)}}{C_{\varepsilon}N}\|\pi_{+}(s)y_{0}\|\to\infty,$

as $t\to\infty$ unless $\pi_{+}(s)y_{0}=0$ as
$y\in\operatorname{BC}_{s}^{\eta}$. Similarly, one can prove that
$\pi_{-}(s)y_{0}=0$ and so
$y_{0}=(\pi_{-}(s)+\pi_{0}(s)+\pi_{+}(s))y_{0}=\pi_{0}(s)y_{0}$, i.e.
$y_{0}\in E_{0}(s)$.

###  3.1 Bounded solutions of the linear inhomogeneous equation

Let $f:\mathbb{R}\to\mathbb{R}^{n}$ be a continuous function and consider the
linear inhomogeneous integral equation

$u(t)=U(t,s)u(s)+\int_{s}^{t}U(t,\tau)f(\tau)d\tau,$ (9)

for all $(t,s)\in\mathbb{R}^{2}$. To prove existence of a center manifold, we
need a pseudo-inverse of (exponentially) bounded solutions of (9). To do this,
define (formally) for any $\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$ the
operator
$\mathcal{K}_{s}^{\eta}:\operatorname{BC}_{s}^{\eta}\to\operatorname{BC}_{s}^{\eta}$
as

$\displaystyle(\mathcal{K}_{s}^{\eta}f)(t):=\int_{s}^{t}U(t,\tau)\pi_{0}(\tau)f(\tau)d\tau+\int_{\infty}^{t}U(t,\tau)\pi_{+}(\tau)f(\tau)d\tau+\int_{-\infty}^{t}U(t,\tau)\pi_{-}(\tau)f(\tau)d\tau,$

and we have to check that this operator is well-defined. This will be proven
in the following proposition and also the fact that $\mathcal{K}_{s}^{\eta}$
is precisely the pseudo-inverse we are looking for.

Let $\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$. The following properties
hold.

  1. $1.$

$\mathcal{K}_{s}^{\eta}$ is a well-defined bounded linear operator. Moreover,
the operator norm $\|\mathcal{K}_{s}^{\eta}\|$ is bounded above independent of
$s$.

  2. $2.$

$\mathcal{K}_{s}^{\eta}f$ is the unique solution of (9) in
$\operatorname{BC}^{\eta}_{s}$ with vanishing $E_{0}(s)$-component at time
$s$.

We start by proving the first assertion. Clearly $\mathcal{K}_{s}^{\eta}$ is
linear. Let $\varepsilon\in(0,\eta]$ be given and notice that for a given
$f\in\operatorname{BC}^{\eta}_{s}$, we can write as $\mathcal{K}_{s}^{\eta}f$
as the sum of three integrals, i.e.
$\mathcal{K}_{s}^{\eta}f=I_{0}(\cdot,s)+I_{+}+I_{-}$. We estimate now the norm
of each integral step by step.

$I_{0}(\cdot,s)$: The straightforward estimate

$\|I_{0}(t,s)\|\leq
C_{\varepsilon}N\|f\|_{\eta,s}\frac{e^{\eta|t-s|}}{\eta-\varepsilon}<\infty,\quad\forall
t\in\mathbb{R},$

implies that the norm of $I_{0}(\cdot,s)$ is bounded above.

$I_{+}:$ Notice that

$\|I_{+}(t)\|\leq
C_{\varepsilon}N\|f\|_{\eta,s}e^{bt}\int_{t}^{\infty}e^{-b\tau+\eta|\tau-s|}d\tau,\forall
t\in\mathbb{R},$

and to prove norm boundedness of $I_{+}$, we have to evaluate the integral
above. A calculation shows that

$\displaystyle\int_{t}^{\infty}e^{-b\tau+\eta|\tau-s|}d\tau=\begin{dcases}\frac{e^{-bt}}{b-\eta}e^{\eta(t-s)},\quad&t\geq
s\\\
\frac{e^{-bt}}{b+\eta}e^{\eta(s-t)}-\frac{e^{-bs}}{b+\eta}+\frac{e^{-bs}}{b-\eta},\quad&t\leq
s.\end{dcases}$ (10)

We want to estimate the $t\leq s$ case. Notice that for real numbers
$\alpha\geq\beta$ we have

$(\alpha-\beta)\bigg{(}\frac{1}{b+\eta}-\frac{1}{b-\eta}\bigg{)}=\frac{-2\eta(\alpha-\beta)}{(b+\eta)(b-\eta)}\leq
0,$

since $\eta<b$ by assumption. Hence,

$\frac{\alpha}{b+\eta}+\frac{\beta}{b-\eta}\leq\frac{\alpha}{b-\eta}+\frac{\beta}{b+\eta}.$

We want to replace $\alpha$ by $e^{-bt+\eta s-\eta t}$ and $\beta$ by
$e^{-bs}$ and therefore we have to show that $-bt+\eta s-\eta t+bs\geq 0$
which is true because $-bt+\eta s-\eta t+bs=(s-t)(b+\eta)\geq 0$ since
$s-t\geq 0$. Filling this into (10) yields

$\int_{t}^{\infty}e^{-b\tau+\eta|\tau-s|}d\tau\leq\frac{e^{-bt}}{b-\eta}e^{\eta|t-s|},$

which shows that

$\|I_{+}(t)\|\leq
C_{\varepsilon}N\|f\|_{\eta,s}\frac{e^{\eta|t-s|}}{b-\eta}<\infty,\quad\forall
t\in\mathbb{R}.$

and so we conclude that $I_{+}$ is well-defined.

$I_{-}:$ A similar estimate as for the $I_{+}$-case shows that

$\|I_{-}(t)\|\leq
C_{\varepsilon}N\|f\|_{\eta,s}\frac{e^{\eta|t-s|}}{-a-\eta}<\infty,\quad\forall
t\in\mathbb{R},$

and so it follows that the operator norm

$\|\mathcal{K}_{s}^{\eta}\|\leq
C_{\varepsilon}N\left(\frac{1}{\eta-\varepsilon}+\frac{1}{b-\eta}+\frac{1}{-a-\eta}\right)<\infty,$

is bounded above independent of $s$. We conclude that $\mathcal{K}_{s}^{\eta}$
is a bounded linear operator on $\operatorname{BC}_{s}^{\eta}$.

Let us now prove the second assertion by first showing that
$\mathcal{K}_{s}^{\eta}$ is indeed a solution of (9). Let
$f\in\operatorname{BC}_{s}^{\eta}$ and set $u=\mathcal{K}_{s}^{\eta}f$. Then,
a straightforward computation shows that

$U(t,s)u(s)+\int_{s}^{t}U(t,\tau)f(\tau)d\tau=u(t),$

and so $u$ is indeed a solution of (9). Let us now prove that $u$ has
vanishing $E_{0}(s)$-component at time $s$, i.e. $\pi_{0}(s)u(s)=0$. The
mutual orthogonality of the projectors (prop:criteria) implies

$\displaystyle\pi_{0}(s)u(s)=\int_{\infty}^{s}U(s,\tau)\pi_{0}(\tau)\pi_{+}(\tau)f(\tau)d\tau+\int_{-\infty}^{s}U(s,\tau)\pi_{0}(\tau)\pi_{-}(\tau)f(\tau)d\tau=0.$

It only remains to show that $u$ is the unique solution of (9) in
$\operatorname{BC}_{s}^{\eta}$. Let $v\in\operatorname{BC}_{s}^{\eta}$ be
another solution of (9) with vanishing $E_{0}(s)$-component at time $s$. Then
the function $w:=u-v$ is an element of $\operatorname{BC}_{s}^{\eta}$ and
satisfies $w(t)=U(t,s)w(s)$ for all $(t,s)\in\mathbb{R}^{2}$. prop:X0s shows
us that $w(s)\in E_{0}(s)$ and notice that $\pi_{0}(s)w(s)=0$ since $u$ and
$v$ have both vanishing $E_{0}(s)$-component at time $s$. From prop:criteria
we know that $w(t)=U_{0}(t,s)w(s)$ is in $E_{0}(t)$ and

$\displaystyle\pi_{0}(t)w(t)=\pi_{0}(t)U_{0}(t,s)w(s)=U_{0}(t,s)\pi_{0}(s)w(s)=0,$

so $u=v$.

###  3.2 Modification of the nonlinearity

To prove the existence of a center manifold, a key step will be to use the
Banach fixed point theorem on some specific fixed point operator. This
operator we will be of course linked to the inhomogeneous equation (9).
However, we can not expect that any nonlinear operator
$R(t,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ for fixed $t\in\mathbb{R}$ will
impose a Lipschitz condition on the fixed point operator that will be
constructed. As we are only interested in the local behavior of solutions near
the origin of (2), we can modify the nonlinearity $R(t,\cdot)$ outside a ball
of radius $\delta>0$ such that eventually the fixed point operator will become
a contraction. To modify this nonlinearity, introduce a $C^{\infty}$-smooth
cut-off function $\xi:[0,\infty)\to\mathbb{R}$ as

$\xi(s)\in\begin{cases}\\{1\\},\quad&0\leq s\leq 1,\\\ [0,1],\quad&0\leq s\leq
2,\\\ \\{0\\},\quad&s\geq 2,\end{cases}$

and define for any $\delta>0$ the _$\delta$ -modification_ of $R$ as the
operator $R_{\delta}:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ with
action

$\displaystyle
R_{\delta}(t,u):=R(t,u)\xi\bigg{(}\frac{\|\pi_{0}(t)u\|}{N\delta}\bigg{)}\xi\bigg{(}\frac{\|(\pi_{-}(t)+\pi_{+}(t))u\|}{N\delta}\bigg{)}.$

Since $R$ is of the class $C^{k}$, the cut-off function $\xi$ is
$C^{\infty}$-smooth, the Euclidean norm $\|\cdot\|$ is $C^{\infty}$-smooth on
$\mathbb{R}^{n}\setminus\\{0\\}$ and the projectors $\pi_{-},\pi_{0},\pi_{+}$
are $C^{k}$-smooth (prop:criteria), it is clear that $R_{\delta}$ is
$C^{k}$-smooth. This $\delta$-modification of $R$ will ensure that the
nonlinearity becomes eventually globally Lipschitz, as will be proven in the
upcoming two statements.

There exist a $\delta_{1}>0$ and $l:[0,\delta_{1}]\rightarrow[0,\infty)$,
continuous at $0$, such that $l(0)=0$ and $l(\delta)=:l_{\delta}$ is a
Lipschitz constant for $R(t,\cdot)$ on the open ball $B(0,\delta)$ for every
$t\in\mathbb{R}$ and $\delta\in(0,\delta_{1}]$.

Recall that $R$ is of the class $C^{k}$ and that $R(t,0)=D_{2}R(t,0)=0$ for
all $t\in\mathbb{R}$. By continuity, choose $\delta_{1}>0$ such that
$\sup\\{\|D_{2}R(t,y)\|:y\in B(0,\delta_{1})\\}\leq 1$ and define the map $l$
as

$l(\delta):=\begin{dcases}0,&\delta=0,\\\ \sup\\{\|D_{2}R(t,y)\|:y\in
B(0,\delta)\\},&\delta\in(0,\delta_{1}].\end{dcases}$

By the mean value theorem, $l(\delta)$ is a Lipschitz constant for
$R(t,\cdot)$ on $B(0,\delta)$. Moreover, $l$ is monotonically increasing and
observe that for each $\varepsilon>0$, there exists a
$0<\delta_{\varepsilon}\leq\delta_{1}$ such that $\sup\\{\|D_{2}R(t,y)\|:y\in
B(0,\delta_{\varepsilon})\\}\leq\varepsilon$. Then for
$0<\delta\leq\delta_{\varepsilon}$ we have that $0\leq l(\delta)\leq
l(\delta_{\varepsilon})\leq\varepsilon$ and so the map $l$ is continuous at
zero.

For $\delta>0$ sufficiently small, $R_{\delta}(t,\cdot)$ is globally Lipschitz
continuous for all $t\in\mathbb{R}$ with Lipschitz constant
$L_{\delta}\rightarrow 0$ as $\delta\downarrow 0$.

Define for any $\delta>0$ and $t\in\mathbb{R}$ the maps
$\xi_{\delta},\Xi_{\delta,t}:\mathbb{R}^{n}\to\mathbb{R}$ by

$\displaystyle\xi_{\delta}(y):=\xi\left(\frac{\|y\|}{N\delta}\right),\quad\Xi_{\delta,t}(y):=\xi_{\delta}(\pi_{0}(t)y)\xi_{\delta}(\pi_{-}(t)y+\pi_{+}(t)y),$

and so $R_{\delta}(t,y)=\Xi_{\delta,t}(y)R(t,y)$. Note that
$\xi_{\delta},\Xi_{\delta,t}\leq 1$ and let $C\geq 0$ be a global Lipschitz
constant of $\xi$. Then by composition of Lipschitz functions, $\xi_{\delta}$
has a global Lipschitz constant $C/{N\delta}$. For $y,z\in\mathbb{R}^{n}$:

$\displaystyle|\Xi_{\delta,t}(y)-\Xi_{\delta,t}(z)|$
$\displaystyle=|[\xi_{\delta}(\pi_{0}(t)y)\xi_{\delta}(\pi_{-}(t)y+\pi_{+}(t)y)-\xi_{\delta}(\pi_{0}(t)y)\xi_{\delta}(\pi_{-}(t)z+\pi_{+}(t)z)]$
$\displaystyle-[\xi_{\delta}(\pi_{0}(t)z)\xi_{\delta}(\pi_{-}(t)z+\pi_{+}(t)z)-\xi_{\delta}(\pi_{0}(t)y)\xi_{\delta}(\pi_{-}(t)z+\pi_{+}(t)z)]|$
$\displaystyle\leq\xi_{\delta}(\pi_{0}(t)y)|\xi_{\delta}(\pi_{-}(t)y+\pi_{+}(t)y)-\xi_{\delta}(\pi_{-}(t)z+\pi_{+}(t)z)|$
$\displaystyle+\xi_{\delta}(\pi_{-}(t)z+\pi_{+}(t)z)|\xi_{\delta}(\pi_{0}(t)y-\xi_{\delta}(\pi_{0}(t)z)|$
$\displaystyle\leq\frac{2C}{\delta}\|y-z\|.$

Now, note that $\|y\|\leq\|\pi_{0}(t)y\|+\|(\pi_{-}(t)+\pi_{+}(t))y\|$ for all
$y\in\mathbb{R}^{n}$. If $\|y\|\geq 4N\delta$, then
$\max\\{\|\pi_{0}(t)y\|,\|(\pi_{-}(t)+\pi_{+}(t))y\|\\}\geq 2N\delta$, so that
$\Xi_{\delta,t}(y)=0$. Let $\delta_{1}>0$ be such as in lem:local lipschitz
and fix $\delta>0$ such that $4N\delta\leq\delta_{1}$. For
$y,z\in\mathbb{R}^{n}$:

$\displaystyle\|R_{\delta}(t,y)-R_{\delta}(t,z)\|$
$\displaystyle=\|[\Xi_{\delta,t}(y)R(t,y)-\Xi_{\delta,t}(y)R(t,z)]-[\Xi_{\delta,t}(z)R(t,z)-\Xi_{\delta,t}(y)R(t,z)]\|$
$\displaystyle\leq\Xi_{\delta,t}(y)\|R(t,y)-R(t,z)\|+|\Xi_{\delta,t}(y)-\Xi_{\delta,t}(z)|\|R(t,z)\|$
$\displaystyle\leq\begin{dcases}l_{\delta}(4N\delta)\|y-z\|+8CNl_{\delta}(4N\delta)\|y-z\|,&\|y\|,\|z\|<4N\delta,\\\
0,&\|y\|,\|z\|\geq 4N\delta,\\\ 8CNl_{\delta}(4N\delta)\|y-z\|,&\|y\|\geq
4N\delta,\|z\|<4N\delta,\\\ \end{dcases}$ $\displaystyle\leq
l_{\delta}(4N\delta)(1+8CN)\|y-z\|,$

Hence, $L_{\delta}:=l_{\delta}(4N\delta)(1+8CN)$ is a Lipschitz constant for
$R_{\delta}(t,\cdot)$ for all $t\in\mathbb{R}$.

For $\delta>0$ sufficiently small, $\|R_{\delta}(t,y)\|\leq
4NL_{\delta}\delta$ for all $(t,y)\in\mathbb{R}\times\mathbb{R}^{n}$. Note
that $R_{\delta}(t,0)=0$. This means that $\|R_{\delta}(t,y)\|\leq
L_{\delta}\|y\|$. Obviously the claim holds if $\|y\|\leq 4N\delta$. On the
other hand, if $\|y\|>4N\delta$, then $R_{\delta}(t,y)=0$ and so the proof is
complete.

Let us introduce for any $\eta\in(0,\min\\{-a,b\\}),s\in\mathbb{R}$ and a
given $\delta$-modification of $R$, the _substitution operator_
$\tilde{R}_{\delta}:\operatorname{BC}_{s}^{\eta}\to\operatorname{BC}_{s}^{\eta}$
as

$\tilde{R}_{\delta}(u):=R_{\delta}(\cdot,u(\cdot)),$

and we show that this operator inherits the same properties as $R_{\delta}$.

Let $\eta\in(0,\min\\{-a,b\\}),s\in\mathbb{R}$ and $\delta>0$ be sufficiently
small. Then the substitution operator $\tilde{R}_{\delta}$ is well-defined and
inherits the Lipschitz properties of $R_{\delta}$. It follows from prop:global
lipschitz that

$\|\tilde{R}_{\delta}(u)(t)\|=\|R_{\delta}(t,u(t))\|\leq L_{\delta}\|u(t)\|,$

for all $u\in\operatorname{BC}_{s}^{\eta}$. Hence,
$\|\tilde{R}_{\delta}(u)\|_{\eta,s}\leq L_{\delta}\|u\|_{\eta,s}<\infty$, i.e.
$\tilde{R}_{\delta}(u)\in\operatorname{BC}_{s}^{\eta}$. The Lipschitz property
follows immediately from prop:global lipschitz since

$\|\tilde{R}_{\delta}(u)-\tilde{R}_{\delta}(v)\|_{\eta,s}\leq
L_{\delta}\|u-v\|_{\eta,s},$

for all $u,v\in\operatorname{BC}_{s}^{\eta}$, and so
$\|\tilde{R}_{\delta}(u)\|_{\eta,s}\leq 4NL_{\delta}\delta$. Define for any
$\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$ the linear operator
$U_{s}^{\eta}:E_{0}(s)\rightarrow\operatorname{BC}_{s}^{\eta}$ by

$(U_{s}^{\eta}y_{0})(t):=U(t,s)y_{0}.$

Let $\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$. Then the operator
$U_{s}^{\eta}$ is well-defined and bounded. Let $\varepsilon\in(0,\eta]$ be
given. It follows from prop:criteria that

$\|U_{s}^{\eta}y_{0}\|_{\eta,s}\leq
C_{\varepsilon}\|y_{0}\|\sup_{t\in\mathbb{R}}e^{(\varepsilon-\eta)|t-s|}=C_{\varepsilon}\|y_{0}\|,$

for all $y_{0}\in E_{0}(s)$, and so $U_{s}^{\eta}$ is well-defined and
bounded.

###  3.3 Existence of a Lipschitz center manifold

Our next goal is to define a parameterized fixed point operator such that its
fixed points correspond to (exponentially) bounded solutions on $\mathbb{R}$
of the modified equation

$u(t)=U(t,s)u(s)+\int_{s}^{t}U(t,\tau)R_{\delta}(\tau,u(\tau))d\tau,$ (11)

for all $(t,s)\in\mathbb{R}^{2}$ and some small $\delta>0$. For a given
$\eta\in(0,\min\\{-a,b\\}),s\in\mathbb{R}$ and sufficiently small $\delta>0$,
we define the fixed point operator
$\mathcal{G}_{s}^{\eta}:\operatorname{BC}_{s}^{\eta}\times
E_{0}(s)\to\operatorname{BC}_{s}^{\eta}$ as

$\mathcal{G}_{s}^{\eta}(u,y_{0}):=U_{s}^{\eta}y_{0}+\mathcal{K}_{s}^{\eta}(\tilde{R}_{\delta}(u)).$

It follows from prop:ketas, lemma:substitution and lemma:Uetas that
$\mathcal{G}_{s}^{\eta}$ is well-defined. We first show that
$\mathcal{G}_{s}^{\eta}(\cdot,y_{0})$ admits a unique fixed point and is
globally Lipschitz for all $y_{0}\in E_{0}(s)$.

Let $\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$. If $\delta>0$ is
sufficiently small, then the following statements hold.

  1. $1.$

For every $y_{0}\in E_{0}(s)$, the map $\mathcal{G}_{s}^{\eta}(\cdot,y_{0})$
has a unique fixed point $\hat{u}_{s}^{\eta}(y_{0})$.

  2. $2.$

The map $\hat{u}_{s}^{\eta}:E_{0}(s)\rightarrow\operatorname{BC}_{s}^{\eta}$
is globally Lipschitz and $\hat{u}_{s}^{\eta}(0)=0$.

Fix $\varepsilon\in(0,\eta]$. For $u,v\in\operatorname{BC}_{s}^{\eta}$ and
$y_{0},z_{0}\in E_{0}(s)$, we have

$\displaystyle\|\mathcal{G}_{s}^{\eta}(u,y_{0})-\mathcal{G}_{s}^{\eta}(v,z_{0})\|_{\eta,s}$
$\displaystyle\leq\sup_{t\in\mathbb{R}}e^{-\eta|t-s|}\|U_{0}(t,s)(y_{0}-z_{0})\|+L_{\delta}\|\mathcal{K}_{s}^{\eta}\|\|u-v\|_{\eta,s}$
$\displaystyle\leq
C_{\varepsilon}\|y_{0}-z_{0}\|+L_{\delta}\|\mathcal{K}_{s}^{\eta}\|\|u-v\|_{\eta,s}.$

To prove the first assertion, set $y_{0}=z_{0}$ and choose $\delta>0$ small
enough such that $L_{\delta}\|\mathcal{K}_{s}^{\eta}\|\leq\frac{1}{2}$
(prop:global lipschitz) since then

$\|\mathcal{G}_{s}^{\eta}(u,y_{0})-\mathcal{G}_{s}^{\eta}(v,y_{0})\|_{\eta,s}\leq\frac{1}{2}\|u-v\|_{\eta,s}.$

Since $\operatorname{BC}_{s}^{\eta}$ is a Banach space, the contracting
mapping principle applies and so the contraction
$\mathcal{G}_{s}^{\eta}(\cdot,y_{0})$ has a unique fixed point, say
$\hat{u}_{s}^{\eta}(y_{0})$.

To prove the second assertion, let $\hat{u}_{s}^{\eta}(y_{0})$ and
$\hat{u}_{s}^{\eta}(z_{0})$ be the unique fixed points of
$\mathcal{G}_{s}^{\eta}(\cdot,y_{0})$ and
$\mathcal{G}_{s}^{\eta}(\cdot,z_{0})$ respectively. Then,

$\displaystyle\|\hat{u}_{s}^{\eta}(y_{0})-\hat{u}_{s}^{\eta}(z_{0})\|_{\eta,s}$
$\displaystyle=\|\mathcal{G}_{s}^{\eta}(\hat{u}_{s}^{\eta}(y_{0}),y_{0})-\mathcal{G}_{s}^{\eta}(\hat{u}_{s}^{\eta}(z_{0}),z_{0})\|_{\eta,s}$
$\displaystyle\leq
C_{\varepsilon}\|y_{0}-z_{0}\|+\frac{1}{2}\|\hat{u}_{s}^{\eta}(y_{0})-\hat{u}_{s}^{\eta}(z_{0})\|_{\eta,s}.$

This implies that
$\|\hat{u}_{s}^{\eta}(y_{0})-\hat{u}_{s}^{\eta}(z_{0})\|_{\eta,s}\leq
2C_{\varepsilon}\|y_{0}-z_{0}\|$, and so $\hat{u}_{s}^{\eta}$ is globally
Lipschitz. Since
$\hat{u}_{s}^{\eta}(0)=\mathcal{G}_{s}^{\eta}(\hat{u}_{s}^{\eta}(0),0)=0,$ the
second assertion follows.

In order to construct a center manifold, define the _center bundle_
$E_{0}:=\\{(s,y_{0})\in\mathbb{R}\times\mathbb{R}^{n}:y_{0}\in E_{0}(s)\\}$
and the map $\mathcal{C}:E_{0}\rightarrow\mathbb{R}^{n}$ by

$\mathcal{C}(s,y_{0}):=\hat{u}_{s}^{\eta}(y_{0})(s).$ (12)

A _global center manifold_ of (11) is defined as the image
$\mathcal{W}^{c}:=\mathcal{C}(E_{0})$, whose $s$-_fibers_ are defined as
$\mathcal{W}^{c}(s):=\\{\mathcal{C}(s,y_{0})\in\mathbb{R}^{n}:y_{0}\in
E_{0}(s)\\}.$

Recall from thm:contraction that for a fixed $s\in\mathbb{R}$, the map
$\hat{u}_{s}^{\eta}$ is globally Lipschitz. Hence, the map
$\mathcal{C}(s,\cdot):E_{0}(s)\to\mathbb{R}^{n}$ is globally Lipschitz, where
the Lipschitz constant depends on $s$, i.e. $\mathcal{C}$ is only _fiberwise
Lipschitz_. The following result shows that the Lipschitz constant can be
chosen independently of the fiber, and so we can say that $\mathcal{W}^{c}$ is
a Lipschitz global center manifold of (11).

There exists a constant $L>0$ such that
$\|\mathcal{C}(s,y_{0})-\mathcal{C}(s,z_{0})\|\leq L\|y_{0}-z_{0}\|$ for all
$(s,y_{0}),(s,z_{0})\in E_{0}$. Let $(s,y_{0}),(s,z_{0})\in E_{0}$ be given.
It follows from lemma:substitution and thm:contraction that

$\displaystyle\|\mathcal{C}(s,y_{0})-\mathcal{C}(s,z_{0})\|$
$\displaystyle=\|\mathcal{G}_{s}^{\eta}(\hat{u}_{s}^{\eta}(y_{0}),y_{0})(s)-\mathcal{G}_{s}^{\eta}(\hat{u}_{s}^{\eta}(z_{0}),z_{0})(s)\|$
$\displaystyle\leq\|y_{0}-z_{0}\|+\|\mathcal{K}_{s}^{\eta}\|\|\tilde{R}_{\delta}(\hat{u}_{s}^{\eta}(y_{0}))-\tilde{R}_{\delta}(\hat{u}_{s}^{\eta}(z_{0}))\|_{\eta,s}$
$\displaystyle\leq\|y_{0}-z_{0}\|+L_{\delta}\|\mathcal{K}_{s}^{\eta}\|\|\hat{u}_{s}^{\eta}(y_{0})-\hat{u}_{s}^{\eta}(z_{0})\|_{\eta,s}$
$\displaystyle\leq(1+2C_{\varepsilon}L_{\delta}\|\mathcal{K}_{s}^{\eta}\|)\|y_{0}-z_{0}\|.$

Hence, $L:=1+2C_{\varepsilon}L_{\delta}\|\mathcal{K}_{s}^{\eta}\|$ is a
Lipschitz constant that is independent of $s$ by prop:ketas. Recall from the
definition of the $\delta$-modification of $R$ that $R_{\delta}=R$ on
$\mathbb{R}\times B(0,\delta)$. Hence, the modified integral equation (11) is
equivalent to the original integral equation (8), and by
lemma:integralequivalence to the ordinary differential equation (2), on
$B(0,\delta)$. A _local center manifold_ of (2) is defined as the image

$\mathcal{W}^{c}_{\operatorname{loc}}:=\mathcal{C}(\\{(s,y_{0})\in
E_{0}:\mathcal{C}(s,y_{0})\in B(0,\delta)\\}).$

In the definitions of the center manifolds and their associated fiber bundles
(def:Wc and def:Wcloc), we used the map $\mathcal{C}$ to explicitly construct
these objects. However, sometimes one likes to think of the center manifold as
the graph of a function. To obtain such a representation, define the map
$\mathcal{H}:E_{0}\to\mathbb{R}^{n}$ as
$\mathcal{H}(s,y_{0}):=(I-\pi_{0}(s))\mathcal{C}(s,y_{0})$ and notice from
prop:ketas that we have the decomposition
$\mathcal{C}(s,y_{0})=y_{0}+\mathcal{H}(s,y_{0})$ in the nonhyperbolic and
hyperbolic part respectively. Hence, we can write for example

$\displaystyle\mathcal{W}_{c}(s)=\\{y_{0}+\mathcal{H}(s,y_{0}):y_{0}\in
E_{0}(s)\\}\cong\\{(y_{0},\mathcal{H}(s,y_{0})):y_{0}\in
E_{0}(s)\\}=\mbox{graph}(\mathcal{H}(s,\cdot)),$ (13)

and since $E_{0}(s)$ and $E_{+}(s)\oplus E_{-}(s)$ have only zero in their
intersection (prop:criteria), this identification makes sense. Notice that the
map $\mathcal{H}$, identified as a graph in (13), is strictly speaking a map
that takes values in $E_{+}(s)\oplus E_{-}(s)$. Similar graph-like
representations can be obtained for $\mathcal{W}^{c}$ and
$\mathcal{W}_{\operatorname{loc}}^{c}$.

##  4 Properties of the center manifold

In this section, we prove that $\mathcal{W}_{\operatorname{loc}}^{c}$ is
locally invariant and consists of slow dynamics. Moreover, we prove that the
center manifold inherits the same order of smoothness as the nonlinearity $R$
and its tangent bundle is precisely the center bundle $E_{0}$. Lastly, we
prove that the center manifold is $T$-periodic in a neighborhood of the
origin. At the end of the section, we combine all the results to prove
thm:main.

Our first aim is to prove the local invariance property of
$\mathcal{W}_{\operatorname{loc}}^{c}$. Therefore, let
$S_{\delta}(t,s,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote the (time-
dependent) flow of

$\dot{y}(t)=A(t)y(t)+R_{\delta}(t,y(t)).$ (14)

Moreover, lemma:integralequivalence still holds when $R$ is replaced by
$R_{\delta}$ and so the ordinary differential equation (14) is equivalent to
the integral equation (11). By (local) uniqueness of solutions, we have that
(3) still holds with $S$ replaced by $S_{\delta}$. The following result is the
nonlinear analogue of prop:X0s and is a preliminary result to prove in
prop:invariance the local invariance property of the center manifold.

Let $\eta\in(0,\min\\{-a,b\\})$ and $s\in\mathbb{R}$. Then

$\displaystyle\mathcal{W}^{c}(s)=\\{y_{0}\in\mathbb{R}^{n}:\text{ there exists
a solution of \eqref{eq:yRdelta}}\text{ through $y_{0}$ belonging to
$\operatorname{BC}_{s}^{\eta}$}\\}.$

Choose $y_{0}\in\mathcal{W}^{c}(s)$, then
$y_{0}=\mathcal{C}(s,z_{0})=\hat{u}_{s}^{\eta}(z_{0})(s)$ for some $z_{0}\in
E_{0}(s)$. prop:ketas shows that $\mathcal{K}_{s}^{\eta}\tilde{R}_{\delta}(u)$
is the unique solution of (9) with $f=\tilde{R}_{\delta}(u)$. Since
$u=\hat{u}_{s}^{\eta}(z_{0})$ is a fixed point of
$\mathcal{G}_{s}^{\eta}(\cdot,z_{0})$, we get

$\displaystyle u(t)$
$\displaystyle=U(t,s)z_{0}+(\mathcal{K}_{s}^{\eta}\tilde{R}_{\delta}(u))(t)$
$\displaystyle=U(t,s)z_{0}+U(t,s)(\mathcal{K}_{s}^{\eta}\tilde{R}_{\delta}(u))(s)+\int_{s}^{t}U(t,\tau){R}_{\delta}(\tau,u(\tau))d\tau$
$\displaystyle=U(t,s)u(s)+\int_{s}^{t}U(t,\tau){R}_{\delta}(\tau,u(\tau))d\tau.$

for all $(t,s)\in\mathbb{R}^{2}$. Hence, $u=\hat{u}_{s}^{\eta}(z_{0})$ is a
solution of (11), and so (14), through $u(s)=y_{0}$ which belongs to
$\operatorname{BC}_{s}^{\eta}$. Conversely, let $y_{0}\in\mathbb{R}^{n}$ such
that there exists a solution $u$ of (14), and so (11), in
$\operatorname{BC}_{s}^{\eta}$ satisfying $u(s)=y_{0}$. It follows from
prop:ketas that

$u(t)=U(t,s)\pi_{0}(s)u(s)+(\mathcal{K}_{s}^{\eta}\tilde{R}_{\delta}(u))(t).$

Hence, $u=\mathcal{G}_{s}^{\eta}(u,\pi_{0}(s)u(s))$ so
$y_{0}=u(s)=\mathcal{C}(s,\pi_{0}(s)y_{0})\in\mathcal{W}^{c}(s)$ by uniqueness
of the fixed point.

The local center manifold $\mathcal{W}_{\operatorname{loc}}^{c}$ has the
following properties.

  1. $1.$

$\mathcal{W}^{c}_{\operatorname{loc}}$ is _locally invariant_ : if
$(s,y_{0})\in\mathbb{R}\times\mathcal{W}^{c}_{\operatorname{loc}}$ and
$t_{-},t_{+}\in\mathbb{R}$ with $s\in(t_{-},t_{+})$ such that $S(t,s,y_{0})\in
B(0,\delta)$ for all $t\in(t_{-},t_{+})$, then
$S(t,s,y_{0})\in\mathcal{W}^{c}_{\operatorname{loc}}$.

  2. $2.$

$\mathcal{W}_{\operatorname{loc}}^{c}$ contains every solution of (2) that
exists on $\mathbb{R}$ and remains sufficiently small for all positive and
negative time: if $u:\mathbb{R}\rightarrow B(0,\delta)$ is a solution of (2),
then $u(t)\in\mathcal{W}^{c}_{\operatorname{loc}}$ for all $t\in\mathbb{R}$.

  3. $3.$

If $(s,y_{0})\in\mathbb{R}\times\mathcal{W}^{c}_{\operatorname{loc}}$, then
$S(t,s,y_{0})=\hat{u}_{t}^{\eta}(\pi_{0}(t)S(t,s,y_{0}))(t)=\mathcal{C}(t,\pi_{0}(t)S(t,s,y_{0}))$
for all $t\in(t_{-},t_{+})$.

  4. $4.$

$0\in\mathcal{W}^{c}_{\operatorname{loc}}$ and $\mathcal{C}(t,0)=0$ for all
$t\in\mathbb{R}$.

We prove the four assertions step by step.

1\. By prop:W^c(s), choose a solution $u\in\operatorname{BC}_{s}^{\eta}$ of
(14) such that $u(s)=y_{0}$. Note that $u(s)=S_{\delta}(s,s,y_{0})$, so by
uniqueness $u(t)=S_{\delta}(t,s,y_{0})$ for all $t\in(t_{-},t_{+})$. Then
$S_{\delta}(t,s,y_{0})\in\mathcal{W}^{c}(t)\subset\mathcal{W}^{c}$. Since
$S_{\delta}(t,s,y_{0})\in B(0,\delta)$, it follows that
$S(t,s,y_{0})=S_{\delta}(t,s,y_{0})\in\mathcal{W}^{c}_{\operatorname{loc}}$.

2\. Recall that (2) and (14) are equal on $B(0,\delta)$. If $u$ is such a
solution, then $u\in\operatorname{BC}_{s}^{\eta}$. The assumption that $u$
takes values in $B(0,\delta)$ and prop:W^c(s) together imply with the first
assertion the result.

3\. In the proof of prop:W^c(s) it is shown that
$y_{0}=\mathcal{C}(s,\pi_{0}(s)y_{0})$ for any $y_{0}\in\mathcal{W}^{c}(s)$.
So it is certainly true for $y_{0}\in\mathcal{W}^{c}_{\operatorname{loc}}$
that
$S_{\delta}(s,s,y_{0})=y_{0}=\hat{u}_{s}^{\eta}(\pi_{0}(s)S_{\delta}(s,s,y_{0}))(s)=\mathcal{C}(s,\pi_{0}(s)S_{\delta}(s,s,y_{0}))$.
Because $\mathcal{W}^{c}_{\operatorname{loc}}$ is locally invariant, we have
that $S_{\delta}(t,s,y_{0})\in\mathcal{W}^{c}_{\operatorname{loc}}$ for all
$t\in\mathbb{R}$ sufficiently close to $s$ and by uniqueness of solutions,
$S_{\delta}(t,s,y_{0})=\hat{u}_{t}^{\eta}(\pi_{0}(t)S_{\delta}(t,s,y_{0}))=\mathcal{C}(t,\pi_{0}(t)S_{\delta}(t,s,y_{0}))$.
Since we are on the local center manifold, we can replace $S_{\delta}$ with
$S$.

4\. Notice that $\mathcal{C}(t,0)=\hat{u}_{t}^{\eta}(0)(t)=0$ for all
$t\in\mathbb{R}$, where the last equality follows from thm:contraction.
Clearly, $0=\mathcal{C}(t,0)\in\mathcal{W}_{\operatorname{loc}}^{c}$ and so
the proof is complete.

It is now possible to explain the fact that the dynamics on the center
manifold is rather slow. Indeed, the local invariance of
$\mathcal{W}_{\operatorname{loc}}^{c}$ (prop:invariance) in combination with
prop:W^c(s) shows that solutions on the center manifold are in
$\operatorname{BC}_{s}^{\eta}$ for some sufficiently small $\eta>0$, i.e.
their asymptotic behavior forward and backward in time can only be a limited
exponential.

The next step is to show that the map $\mathcal{C}$ inherits the same order of
smoothness as the time-dependent nonlinear perturbation $R$. Proving
additional smoothness of center manifolds requires work. A well-known
technique to increase smoothness of center manifolds is via the theory of
contraction on scales of Banach spaces [57]. Since this part of the theory is
rather technical, it is delegated to Appendix A. The main result is presented
in thm:smoothnessC and simply states that the map $\mathcal{C}$ is
$C^{k}$-smooth and so $\mathcal{W}^{c}$ and
$\mathcal{W}_{\operatorname{loc}}^{c}$ are both $C^{k}$-smooth manifolds in
$\mathbb{R}^{n}$. The additional regularity of the center manifold allows us
to study its tangent bundle.

The tangent bundle of $\mathcal{W}^{c}$ and
$\mathcal{W}_{\operatorname{loc}}^{c}$ is $E_{0}$:
$D_{2}\mathcal{C}(s,0)y_{0}=y_{0}$ for all $(s,y_{0})\in E_{0}$. Let
$\eta\in[\eta_{-},\eta_{+}]\subset(0,\min\\{-a,b\\})$ such that
$k\eta_{-}<\eta_{+}$. Differentiating

$\hat{u}_{s}^{\eta_{-}}(y_{0})=U_{s}^{\eta_{-}}y_{0}+\mathcal{K}_{s}^{{\eta_{-}}}\circ\tilde{R}_{\delta}(\hat{u}_{s}^{\eta_{-}}(y_{0}))$

with respect to $y_{0}$ yields

$D\hat{u}_{s}^{\eta_{-}}(y_{0})=U_{s}^{\eta_{-}}+\mathcal{K}_{s}^{{\eta_{-}}}\circ\tilde{R}_{\delta}^{{(1)}}(\hat{u}_{s}^{\eta_{-}}(y_{0}))\circ
D\hat{u}_{s}^{\eta_{-}}(y_{0}).$

Setting $y_{0}=0$ and recalling the fact that
$\hat{u}_{s}^{\eta_{-}}(0)=\tilde{R}_{\delta}^{(1)}(0)=0$ shows that
$D\hat{u}_{s}^{\eta_{-}}(0)=U_{s}^{\eta_{-}}$. If
$\operatorname{ev}_{s}:\operatorname{BC}_{s}^{\eta}\to\mathbb{R}^{n}:f\mapsto
f(s)$ denotes the bounded linear evolution operator (at time $s$), then

$D_{2}\mathcal{C}(s,0)=\operatorname{ev}_{s}(D(\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}_{s}^{\eta_{-}})(0))=\operatorname{ev}_{s}(U_{s}^{\eta})=I,$

which proves the claim. Since our original system (2) is $T$-periodic, it is
not surprising that the center manifold itself is $T$-periodic in a
neighborhood of zero. To prove this, let us define for all $s\in\mathbb{R}$
and sufficiently small $\delta>0$ the map $N_{s}:E_{0}(s)\to E_{0}(s)$ by

$N_{s}(y_{0}):=\pi_{0}(s)S_{\delta}(s+T,s,\mathcal{C}(s,y_{0})).$

The function $N_{s}$ is invertible in a neighborhood of the origin. Moreover,
this neighborhood can be written as $U\cap E_{0}(s)$ for some open
neighborhood $U\subset\mathbb{R}^{n}$ of zero, independent of $s$. Recall the
well-known standard result: $D[S_{\delta}(t,s,\cdot)](0)=U(t,s)$ for all
$(t,s)\in\mathbb{R}^{2}$, see for instance [30, Section 17.6]. The
differential of $N_{s}$ in $0$ is given by

$\displaystyle DN_{s}(0)$ $\displaystyle=\pi_{0}(s)\circ
D[S_{\delta}(s+T,s,\cdot)](\mathcal{C}(s,0))\circ D[\mathcal{C}(s,\cdot)](0)$
$\displaystyle=\pi_{0}(s)\circ D[S_{\delta}(s+T,s,\cdot)](0)$
$\displaystyle=\pi_{0}(s)U_{0}(s+T,s)=U_{0}(s+T,s),$

where we used prop:criteria, prop:bundle, and the fact that
$U_{0}(s+T,s)y_{0}\in E_{0}(s+T)=E_{0}(s)$. It follows from prop:criteria that
$DN_{s}(0)=U_{0}(s+T,s)$ is a bounded linear isomorphism and so $N_{s}$ is
locally invertible by the inverse function theorem. To prove that the
neighborhood may be written as claimed, let us first observe that for a given
$\varepsilon>0$ there holds

$\displaystyle\|DN_{s}(y_{0})-DN_{s}(0)\|$
$\displaystyle\leq\|U_{0}(s+T,s)\pi_{0}(s)\|\|D\mathcal{C}(s,y_{0})-D\mathcal{C}(s,0)\|$
$\displaystyle\leq NC_{\varepsilon}e^{\varepsilon
T}\|D\mathcal{C}(s,y_{0})-D\mathcal{C}(s,0)\|$ $\displaystyle\leq
NC_{\varepsilon}e^{\varepsilon T}L(1)||y_{0}||\to 0,\quad\mbox{ as }y_{0}\to
0,$

due to prop:criteria and cor:LipschitzC. Hence, $DN_{s}(y_{0})$ is uniformly
convergent (in the variable $s$) as $y_{0}\to 0$ and so the implicit function
may be defined on a neighborhood that does not depend on $s$.

There exists a $\delta>0$ such that
$\mathcal{C}(s+T,y_{0})=\mathcal{C}(s,y_{0})$ for all $(s,y_{0})\in E_{0}$
satisfying $\|y_{0}\|\leq\delta$. Let $(s,y_{0})\in E_{0}$ be given. By
lemma:invertibility, choose $\delta>0$ such that if $\|y_{0}\|\leq\delta$, it
is possible to write $y_{0}=N_{s}(z_{0})$. It follows from prop:criteria,
prop:invariance and (3) that

$\displaystyle\mathcal{C}(s+T,y_{0})$
$\displaystyle=\mathcal{C}(s+T,\pi_{0}(s)S_{\delta}(s+T,s,\mathcal{C}(s,z_{0})))$
$\displaystyle=S_{\delta}(s+T,s,\mathcal{C}(s,z_{0}))$
$\displaystyle=S_{\delta}(s,s-T,\mathcal{C}(s,z_{0}))$
$\displaystyle=\mathcal{C}(s,\pi_{0}(s)S_{\delta}(s,s-T,\mathcal{C}(s,z_{0})))$
$\displaystyle=\mathcal{C}(s,\pi_{0}(s)S_{\delta}(s+T,s,\mathcal{C}(s,z_{0})))=\mathcal{C}(s,y_{0}).$

This proves the $T$-periodicity of the center manifold.

Recall that (2) was just a time-dependent translation of (ODE) via the given
periodic solution. Hence, if $x$ is a solution of (ODE) then $y=x-\gamma$ is a
solution of (2) and so

$\displaystyle\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma):=\\{\gamma(s)+\mathcal{C}(s,y_{0})\in\mathbb{R}^{n}:(s,y_{0})\in
E_{0}\mbox{ and }\mathcal{C}(s,y_{0})\in B(0,\delta)\\}$ (15)

is a $T$-periodic $C^{k}$-smooth $(n_{0}+1)$-dimensional manifold in
$\mathbb{R}^{n}$ defined in the vicinity of $\Gamma$ for a sufficiently small
$\delta>0$. To see this, recall that $\gamma$ is $T$-periodic and
$C^{k}$-smooth together with the fact that $\mathcal{C}$ is $T$-periodic in
the first component and $C^{k}$-smooth. Recall from prop:invariance that
$\mathcal{C}(t,0)=0$ and so
$\Gamma\subset\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$. We call
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ a _local center manifold around
$\Gamma$_ and notice that this manifold inherits all the properties of
$\mathcal{W}_{\operatorname{loc}}^{c}$, which proves thm:main.

##  5 Examples and counterexamples

It is widely known that center manifolds for equilibria have interesting
qualitative properties [24, 40, 50]. For example, such center manifolds are
not necessarily unique and are not necessarily of the class $C^{\infty}$ even
if the vector field is $C^{\infty}$-smooth. Of course, there always exists for
$C^{\infty}$-smooth systems an open neighborhood $U_{k}$ around the
equilibrium such that a center manifold is $C^{k}$-smooth on $U_{k}$. However,
the neighborhood $U_{k}$ may shrink towards the equilibrium as $k\to\infty$,
see [52, 51, 50] for explicit examples. When the vector field is analytic,
there is the possibility of the existence of a non-analytic
$C^{\infty}$-smooth center manifold, see [37, 50] for explicit examples. It is
studied in [50] under which conditions a unique, $C^{\infty}$-smooth or
analytic center manifold exists.

The aim of this section is to provide several explicit examples illustrating
similar behavior for the periodic center manifolds. For example, we provide in
ex:analytic an analytic $2\pi$-periodic two-dimensional center manifold near a
nonhyperbolic cycle that is a cylinder. To complete the periodic two-
dimensional center manifold theory, we provide in ex:mobius a system that
admits a $2\pi$-periodic two-dimensional center manifold near a nonhyperbolic
cycle that is a Möbius band. Both examples are minimal polynomial vector
fields admitting a cylinder or Möbius band as a periodic center manifold.

To illustrate the existence of a non-unique and non-analytic
$C^{\infty}$-smooth periodic center manifold, we will study the analytic
nonlinear periodically driven system

$\begin{dcases}\dot{x}=-x^{2},\\\ \dot{y}=-y+\sin(t)x^{2}.\end{dcases}$ (16)

This vector field is a modification of the vector field used in [37] to
illustrate the non-unique behavior of center manifolds for equilibria. Note
that the system (16) is not of the form (ODE) but already written in the style
of (2) where $A$ is autonomous and $R$ is $2\pi$-periodic in the first
argument. The assumption of an autonomous linear part is not a restriction.
Indeed, the Floquet normal form $U(t,0)=Q(t)e^{Bt}$, where $Q$ is
$T$-periodic, $Q(0)=I$, $Q(t)$ is invertible for all $t\in\mathbb{R}$, and the
matrix $B\in\mathbb{C}^{n\times n}$ that satisfies $U(T,0)=e^{BT}$ shows that
a general system of the form (2) is equivalent to the nonlinear periodically
driven system

$\dot{z}(t)=Bz(t)+G(t,z(t)),$ (17)

where $G(t,z(t)):=Q(t)^{-1}R(t,Q(t)z(t))$, and $z$ satisfies the Lyapunov-
Floquet transformation $y(t)=Q(t)z(t)$. Clearly (17) has an autonomous linear
part and $G$ is $T$-periodic in the first component since $R$ and $Q$ are both
$T$-periodic. Notice that the whole periodic center manifold construction from
previous subsections still applies for systems of the form (17). The reason we
study systems of the form (17) instead of a general system of the form (ODE)
is to keep the calculations rather simple. Indeed, if one would like to cook
up an explicit non-trivial example of a periodic center manifold near a
nonhyperbolic cycle, one needs to be able to compute explicitly the periodic
solution, the fundamental matrix and its associated Floquet multipliers, which
is rather difficult for general systems of the form (ODE). We remark that the
computations for the periodic center manifolds of simple periodically driven
systems considered in this section are rather tedious compared with their
equilibrium analogues.

In ex:nonanalytic we show that (16) admits a non-analytic $2\pi$-periodic
center manifold. Next, we show in ex:nonunique that (16) admits a
$2\pi$-periodic center manifold that is _locally (non)-unique_ , i.e. there
exist subneighborhoods of any neighborhood of $\mathbb{R}\times\\{0\\}$ where
the center manifold is unique and others where it is not unique. This freedom
of different center manifolds allows us to choose in ex:Cinftysmooth a
particular $2\pi$-periodic center manifold for (16) that is
$C^{\infty}$-smooth. Hence, we have shown that analytic vector fields can
admit non-analytic $C^{\infty}$-smooth periodic center manifolds. To complete
this list of examples, we will show in ex:nonCinfitysmooth that the
$C^{\infty}$-smooth (analytic) nonlinear periodically driven system

$\begin{dcases}\dot{x}=xz-x^{3},\\\ \dot{y}=y+(1+\sin(t))x^{2},\\\
\dot{z}=0.\end{dcases}$ (18)

admits a $2\pi$-periodic non-$C^{\infty}$-smooth center manifold. This vector
field is a modification of the vector field used in [52] to illustrate the
non-$C^{\infty}$-smoothness of center manifolds near equilibria. Hence, we
have proven that there exists analytic vector fields admitting locally
(non)-unique, (non)-$C^{\infty}$-smooth and (non)-analytic periodic center
manifolds.

The analytic system

$\begin{dcases}\dot{x}_{1}=x_{1}-x_{2}-x_{1}(x_{1}^{2}+x_{2}^{2}),\\\
\dot{x}_{2}=x_{1}+x_{2}-x_{2}(x_{1}^{2}+x_{2}^{2}),\\\
\dot{x}_{3}=0,\end{dcases}$ (19)

admits an analytic $2\pi$-periodic two-dimensional center manifold that is a
cylinder. Notice that (19) admits a $2\pi$-periodic solution
$\gamma(t)=(\cos(t),\sin(t),0)$. The system around $\Gamma=\gamma(\mathbb{R})$
can be written in coordinates $x=\gamma+y$ as

$\begin{dcases}\dot{y}_{1}=-2\cos^{2}(t)y_{1}-(1+2\sin(t)\cos(t))y_{2}-3\cos^{2}(t)y_{1}^{2}-2\sin(t)y_{1}y_{2}-\cos(t)y_{2}^{2}-y_{1}^{3}-y_{1}y_{2}^{2},\\\
\dot{y}_{2}=(1-2\sin(t)\cos(t))y_{1}-2\sin^{2}(t)y_{2}-\sin(t)y_{1}^{2}-2\cos(t)y_{1}y_{2}-3\sin^{2}(t)y_{2}^{2}-y_{1}^{2}y_{2}-y_{2}^{3},\\\
\dot{y}_{3}=0,\end{dcases}$ (20)

where $y:=(y_{1},y_{2},y_{3})$ and $x:=(x_{1},x_{2},x_{3})$. The linearization
around the origin of (20) reads

$A(t)=\begin{pmatrix}-2\cos^{2}(t)&-2\sin(t)\cos(t)-1&0\\\
-2\sin(t)\cos(t)+1&-2\sin^{2}(t)&0\\\ 0&0&0\end{pmatrix}$

and so the solution of the variational equation around $\Gamma$ is generated
by the fundamental matrix $U(t,s)=V(t)V(s)^{-1}$ where

$V(t)=\begin{pmatrix}e^{-2t}\cos(t)&-\sin(t)&0\\\ e^{-2t}\sin(t)&\cos(t)&0\\\
0&0&1\end{pmatrix}.$

The Floquet multipliers are given by $\lambda_{1}=1,\lambda_{2}=e^{-4\pi}$ and
$\lambda_{3}=1$. Hence, the center subspace and stable subspace (at time $t$)
can be obtained as
$E_{0}(t)=\operatorname{span}\\{\zeta_{1}(t),\zeta_{3}(t)\\}$ and
$E_{-}(t)=\operatorname{span}\\{\zeta_{2}(t)\\}$ respectively, where
$\zeta_{1}(t)=(-\sin(t),\cos(t),0),\zeta_{2}(t)=(\cos(t),\sin(t),0)$ and
$\zeta_{3}(t)=(0,0,1)$. The center bundle $E_{0}$ parametrizes a cylinder as a
ruled surface since

$(x_{1}(t,v),x_{2}(t,v),x_{3}(t,v))=\gamma(t)+v\zeta_{3}(t),$

for all $t,v\in\mathbb{R}$. It follows from sec:properties that for any $k\geq
1$ there exists a $2\pi$-periodic $C^{k}$-smooth two-dimensional locally
invariant center manifold $\mathcal{W}_{\operatorname{loc}}^{c}$ for (20)
around the origin that is tangent to $E_{0}$. To obtain this center manifold,
let us transform (20) into eigenbasis, i.e. we perform the change of variables
from $y$ to $z:=(z_{1},z_{2},z_{3})$ as

$\displaystyle z_{1}$ $\displaystyle=-\sin(t)y_{1}+\cos(t)y_{2},$
$\displaystyle z_{2}$ $\displaystyle=\cos(t)y_{1}+\sin(t)y_{2},$ (21)
$\displaystyle z_{3}$ $\displaystyle=y_{3},$

to obtain the autonomous system

$\begin{dcases}\dot{z}_{1}=-z_{1}(z_{1}^{2}+z_{2}^{2}+2z_{2}),\\\
\dot{z}_{2}=-(z_{2}+1)(z_{1}^{2}+z_{2}^{2}+2z_{2}),\\\
\dot{z}_{3}=0.\end{dcases}$ (22)

The $z_{1}z_{3}$-plane corresponds to the center subspace while the
$z_{2}$-axis corresponds to the stable subspace. Therefore, the center
manifold is parametrized by $z_{2}(t)=\mathcal{H}(t,z_{1},z_{3})$, where
$\mathcal{H}$ is $2\pi$-periodic in the first variable and consists solely of
nonlinear terms in the last two variables. Because (22) is an autonomous
system, we have that $\mathcal{H}$ is constant in the first variable, and so
we can write $\mathcal{H}(t,z_{1},z_{3})=H(z_{1},z_{3})$ for all
$t\in\mathbb{R}$. Because $\mathcal{W}_{\operatorname{loc}}^{c}$ is locally
invariant, we must have that

$\displaystyle
z_{1}(z_{1}^{2}+H(z_{1},z_{3})^{2}+2H(z_{1},z_{3}))\frac{\partial}{\partial
z_{1}}H(z_{1},z_{3})=(H(z_{1},z_{3})+1)(z_{1}^{2}+H(z_{1},z_{3})^{2}+2H(z_{1},z_{3})).$

If $z_{1}^{2}+H(z_{1},z_{3})^{2}+2H(z_{1},z_{3})\neq 0$, then $H$ must satisfy

$\frac{\partial}{\partial
z_{1}}H(z_{1},z_{3})=\frac{1}{z_{1}}(H(z_{1},z_{3})+1),\quad H(0,0)=0,$

which has obviously no solution. Hence $z_{1}^{2}+(H(z_{1},z_{3})+1)^{2}=1$
and so $H(z_{1},z_{3})=\sqrt{1-z_{1}^{2}}-1$ since $H(0,0)=0$. Clearly $H$ is
analytic on $(-1,1)\times\mathbb{R}$ since

$H(z_{1},z_{3})=\sum_{k=1}^{\infty}(-1)^{k}\binom{\frac{1}{2}}{k}z_{1}^{2k},$

for all $(z_{1},z_{3})\in(-1,1)\times\mathbb{R}$ due to the Binomial series.
Hence, $\mathcal{H}$ is analytic on $\mathbb{R}\times(-1,1)\times\mathbb{R}$,
which proves the claim. Transforming the map $\mathcal{H}$ back into original
$y$-coordinates using (5) shows that the center manifold
$\mathcal{W}_{\operatorname{loc}}^{c}$ is parametrized as

$\displaystyle(y_{1}(t)+\cos(t))^{2}+(y_{2}(t)+\sin(t))^{2}=1,y_{3}(t)=c_{3},\quad
c_{3}\in\mathbb{R}.$ (23)

Writing this back into $x$-coordinates yields

$\displaystyle(x_{1}(t),x_{2}(t),x_{3}(t))=(y_{1}(t)+\cos(t),y_{2}(t)+\sin(t),c_{3}),\quad
c_{3}\in\mathbb{R},$

but due to (23) we have that $x_{1}(t)^{2}+x_{2}(t)^{2}=1$ and so

$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{1}^{2}+x_{2}^{2}=1\\}.$

Notice that $\mathcal{W}_{\operatorname{loc}}^{c}$ and
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ do not depend on a choice of
$\delta>0$. The reason is clear as for example the cylinder
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is an invariant manifold of
(19) since the function $V:\mathbb{R}^{3}\to\mathbb{R}$ defined by

$V(x_{1},x_{2},x_{3}):=x_{1}^{2}+x_{2}^{2}-1$

is constant along the trajectories whose points are contained in
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)=V^{-1}(\\{0\\})$.

The analytic system

$\begin{dcases}\dot{x}_{1}=-x_{2}+x_{1}\Phi(x_{1},x_{2}),\\\
\dot{x}_{2}=x_{1}+x_{2}\Phi(x_{1},x_{2}),\\\ \dot{x}_{3}=\frac{1}{4}(1-\sigma
x_{2})(x_{1}^{2}+x_{2}^{2}-1)+\frac{\sigma x_{3}}{2}(1+x_{1}),\end{dcases}$
(24)

where

$\Phi(x_{1},x_{2}):=\frac{\sigma}{4}(1-x_{1})(x_{1}^{2}+x_{2}^{2}-1)-\frac{x_{3}}{2}(1+\sigma
x_{2}),$

admits for $\sigma\neq 0$ a $2\pi$-periodic two-dimensional $C^{k}$-smooth
center manifold that is locally diffeomorphic to a Möbius band for every
$k\geq 1$. If $\sigma=0$, then the $2\pi$-periodic center manifold is the
whole state space $\mathbb{R}^{3}$ and (24) admits a family of invariant tori.

Notice that (19) admits for all $\sigma\in\mathbb{R}$ a $2\pi$-periodic
solution $\gamma_{\sigma}(t)=(\cos(t),\sin(t),0)$. Hence, the solution of the
variational equation around $\Gamma_{\sigma}$ is generated by the fundamental
matrix $U_{\sigma}(t,s)=V_{\sigma}(t)V_{\sigma}(s)^{-1}$ where

$V_{\sigma}(t)=\begin{pmatrix}\cos(t)\cos(\frac{t}{2})&-\sin(t)&-e^{\sigma
t}\sin(\frac{t}{2})\cos(t)\\\\[4.30554pt]
\sin(t)\cos(\frac{t}{2})&\cos(t)&-e^{\sigma
t}\sin(\frac{t}{2})\sin(t)\\\\[4.30554pt] \sin(\frac{t}{2})&0&e^{\sigma
t}\cos(\frac{t}{2})\end{pmatrix}.$

The Floquet multipliers are given by $\lambda_{1}=1,\lambda_{2}=-1$ and
$\lambda_{3,\sigma}=-e^{2\pi\sigma}$. Let $E_{0}^{\sigma}(t)$ and
$E_{\pm}^{\sigma}(t)$ denote the center and (un)stable subspace (at time $t$)
at parameter value $\sigma$ respectively. For the center subspace, we have
that $E_{0}^{\sigma}(t)=\operatorname{span}\\{\zeta_{1}(t),\zeta_{2}(t)\\}$
for $\sigma\neq 0$ and
$E_{0}^{0}(t)=\operatorname{span}\\{\zeta_{1}(t),\zeta_{2}(t),\zeta_{3}(t)\\}$.
For the (un)stable subspace we obtain
$E_{-}^{\sigma}(t)=\operatorname{span}\\{\zeta_{3}(t)\\}$ for $\sigma<0$ and
$E_{+}^{\sigma}(t)=\operatorname{span}\\{\zeta_{3}(t)\\}$ for $\sigma>0$ where

$\displaystyle\zeta_{1}(t)$ $\displaystyle=(-\sin(t),\cos(t),0),$
$\displaystyle\zeta_{2}(t)$
$\displaystyle=\bigg{(}\cos(t)\cos(\frac{t}{2}),\sin(t)\cos(\frac{t}{2}),\sin(\frac{t}{2})\bigg{)},$
$\displaystyle\zeta_{3}(t)$
$\displaystyle=\bigg{(}-\cos(t)\sin(\frac{t}{2}),-\sin(t)\sin(\frac{t}{2}),\cos(\frac{t}{2})\bigg{)}.$

Notice that the eigenvector $\zeta_{3}(t)$ is perpendicular to the plane
spanned by $\zeta_{1}(t)$ and $\zeta_{2}(t)$. Let us first discuss the case
$\sigma\neq 0$. Observe that the center bundle $E_{0}^{\sigma}$ at parameter
value $\sigma$ parametrizes locally a Möbius band as a ruled surface since

$(x_{1}(t,v),x_{2}(t,v),x_{3}(t,v))=\gamma_{\sigma}(t)+v\zeta_{2}(t),$

for all $t\in\mathbb{R}$ and $v\in[-1,1]$. thm:main provides us for any $k\geq
1$ a $2\pi$-periodic $C^{k}$-smooth two-dimensional locally invariant center
manifold $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma_{\sigma})$ for (24)
around $\Gamma_{\sigma}$ tangent to the center bundle $E_{0}^{\sigma}$ that is
locally diffeomorphic to a Möbius band, see fig:cmmobius.

When $\sigma=0$, it is clear that the Floquet multipliers are all on the unit
circle where $1$ is simple and $-1$ has algebraic multiplicity $2$. Hence, the
$2\pi$-periodic center manifold is $3$-dimensional, i.e. the whole state space
$\mathbb{R}^{3}$. Moreover, (24) admits a family of invariant tori
$\\{\mathbb{T}_{l}:l\geq 0\\}$ at $\sigma=0$ with major radius $1$ and minor
radius $r_{l}$ since the function $V_{l}:\mathbb{R}^{3}\to\mathbb{R}$ defined
by

$\displaystyle V_{l}(x_{1},x_{2},x_{3})$
$\displaystyle:=\bigg{(}\sqrt{x_{1}^{2}+x_{2}^{2}}-1\bigg{)}^{2}+x_{3}^{2}-r_{l}^{2}\bigg{(}\sqrt{x_{1}^{2}+x_{2}^{2}}\bigg{)}$

where

$r_{l}^{2}(u):=l+\frac{1}{2}u(u-4)+\ln(u)+\frac{3}{2},$

is constant along the trajectories whose points are contained in
$\mathbb{T}_{l}:=V_{l}^{-1}(\\{0\\})$. We claim that the family of tori is
rooted at the cycle, i.e. $\mathbb{T}_{0}=\Gamma_{0}$. It is clear that
solving $V_{0}(x_{1},x_{2},x_{3})=0$ is equivalent to

$\frac{1}{2}u^{2}-\ln(u)=\frac{1}{2}-z^{2},\quad
u=\sqrt{x_{1}^{2}+x_{2}^{2}},$

which has only one real solution at $z=0$ since
$\frac{1}{2}u^{2}-\ln(u)\geq\frac{1}{2}$ for all $u\geq 0$. Clearly this
solution corresponds to the cycle $\Gamma_{0}$. Examples of such invariant
tori can be found in fig:cmmobius.

Figure 2: The left figure represents several forward orbits on a local
$2\pi$-periodic two-dimensional center manifold (Möbius band) around
$\Gamma_{\sigma}$ for (24) at parameter value $\sigma=-1$. The right figure
represents two forward orbits on two different invariant tori for (24) at
parameter value $\sigma=0$. The forward orbits are obtained by numerical
integration and each orbit is represented by different color.

The analytic system (16) admits a non-analytic $2\pi$-periodic center
manifold. The fundamental matrix for the linearization around the origin of
(16) reads

$U(t,s)=\begin{pmatrix}1&0\\\ 0&e^{s-t}\\\ \end{pmatrix}$

for all $(t,s)\in\mathbb{R}^{2}$. The Floquet multipliers are given by
$\lambda_{1}=1$ and $\lambda_{2}=e^{-2\pi}$ and the center space (at time $t$)
is given by $E_{0}(t)=\operatorname{span}\\{(1,0)\\}$ while the stable space
(at time $t$) is given by $E_{-}(t)=\operatorname{span}\\{(0,1)\\}$. Hence,
the $x$-axis corresponds to the center space and so the center manifold can be
parametrized by $y(t)=\mathcal{H}(t,x)$, where $\mathcal{H}$ is
$2\pi$-periodic in the first argument and consists solely of nonlinear terms.
Because the center manifold is locally invariant, the map $\mathcal{H}$ must
satisfy

$\frac{\partial\mathcal{H}}{\partial
t}(t,x)-x^{2}\frac{\partial\mathcal{H}}{\partial
x}(t,x)=-\mathcal{H}(t,x)+\sin(t)x^{2}.$ (25)

Assume that $\mathcal{H}$ is analytic on an open neighborhood of
$\mathbb{R}\times\\{0\\}$, then we can write locally
$\mathcal{H}(t,x)=\sum_{n\geq 2}a_{n}(t)x^{n}$ for $2\pi$-periodic functions
$a_{n}$. Filling this expansion into (25) and comparing terms in $x^{n}$ shows
that the $2\pi$-periodic functions $a_{n}$ must satisfy

$\begin{dcases}\dot{a}_{2}(t)+a_{2}(t)=\sin(t),\quad&n=2,\\\
\dot{a}_{n}(t)+a_{n}(t)=(n-1)a_{n-1}(t),\quad&n\geq 3.\end{dcases}$

Hence, $a_{2}(t)=\alpha_{2}\sin(t)+\beta_{2}\cos(t)$, where
$\alpha_{2}=\frac{1}{2}$ and $\beta_{2}=-\frac{1}{2},$ and

$\displaystyle a_{n}(t)$
$\displaystyle=\frac{(n-1)e^{-t}}{e^{2\pi}-1}\bigg{(}\int_{t}^{2\pi}e^{\tau}a_{n-1}(\tau)d\tau+e^{2\pi}\int_{0}^{t}e^{\tau}a_{n-1}(\tau)d\tau\bigg{)}.$

Let us prove by induction for $n\geq 2$ that $a_{n}$ is a linear combination
of sines and cosines. If $n=2$, then the result is clear. Assume that the
claim holds for a certain $n\geq 3$, it follows from the induction hypothesis
and applying integration by parts twice on both integrals that

$\displaystyle a_{n}(t)$
$\displaystyle=\frac{(n-1)e^{-t}}{e^{2\pi}-1}\bigg{(}\int_{t}^{2\pi}e^{\tau}(\alpha_{n-1}\sin(\tau)+\beta_{n-1}\cos(\tau))d\tau+e^{2\pi}\int_{0}^{t}e^{\tau}(\alpha_{n-1}\sin(\tau)+\beta_{n-1}\cos(\tau))d\tau\bigg{)}$
$\displaystyle=\frac{n-1}{2}((\alpha_{n-1}+\beta_{n-1})\sin(t)+(-\alpha_{n-1}+\beta_{n-1})\cos(t)),$

which proves the claim. From the proof of the induction step, we obtain

$\begin{pmatrix}\alpha_{n}\\\
\beta_{n}\end{pmatrix}=\frac{n-1}{2}\begin{pmatrix}1&1\\\
-1&1\end{pmatrix}\begin{pmatrix}\alpha_{n-1}\\\ \beta_{n-1}\end{pmatrix},\quad
n\geq 3.$

This is a linear system of difference equation and can be solved explicitly by
computing the diagonalization of the associated matrix. The final result reads

$\begin{pmatrix}\alpha_{n}\\\
\beta_{n}\end{pmatrix}=\frac{(n-1)!}{2^{\frac{n-1}{2}}}\begin{pmatrix}\cos(\frac{(n-1)\pi}{4})\\\
-\sin(\frac{(n-1)\pi}{4})\end{pmatrix}.$

Hence, the $2\pi$-periodic functions $a_{n}$ are given by

$\displaystyle a_{n}(t)$
$\displaystyle=\frac{(n-1)!}{2^{\frac{{n-1}}{2}}}\bigg{(}\cos(\frac{(n-1)\pi}{4})\sin(t)-\sin(\frac{(n-1)\pi}{4})\cos(t)\bigg{)},\quad
n\geq 2.$

Using the angle addition and subtractions formula for cosines yields the
center manifold expansion

$\mathcal{H}(t,x)=\sum_{n\geq
2}\frac{(n-1)!}{2^{\frac{{n-1}}{2}}}\sin(t-\frac{(n-1)\pi}{4})x^{n}.$ (26)

We will prove that the radius of convergence $R(t)$ at time $t\in\mathbb{R}$
of (26) is zero. Let us first observe that for any $n\geq 2$ one has

$\sup_{k\geq
n}\bigg{(}k!\bigg{|}\sin(t-\frac{k\pi}{4})\bigg{|}\bigg{)}^{\frac{1}{k}}\geq((4n)!|\sin(t)|)^{\frac{1}{4n}},$

for $t\neq l\pi$ and $\ l\in\mathbb{Z}$. To bound this supremum from below
when $t=l\pi$ for some $l\in\mathbb{Z}$, choose $m\in\mathbb{Z}$ such that
$r=2(l-1-m)\geq n$ because then

$\sup_{k\geq
n}\bigg{(}k!\bigg{|}\sin(l\pi-\frac{k\pi}{4})\bigg{|}\bigg{)}^{\frac{1}{k}}\geq(r!)^{\frac{1}{r}}\geq(n!)^{\frac{1}{n}}.$

The Cauchy-Hadamard theorem tells us that

$\displaystyle\frac{1}{R(t)}$
$\displaystyle=\limsup\limits_{n\to\infty}|a_{n}(t)|^{\frac{1}{n}}\geq\sqrt{2}\min\\{\lim_{n\to\infty}(n!|\sin(t)|)^{\frac{1}{n}},\lim_{n\to\infty}(n!)^{\frac{1}{n}}\\}=\infty,$

where in the first argument of the minimum it is assumed that $t\neq l\pi$ for
all $l\in\mathbb{Z}$. This proves $R(t)=0$ for all $t\in\mathbb{R}$, i.e.
$\mathcal{H}$ is not analytic.

The analytic system (16) admits a locally (non)-unique family of
$2\pi$-periodic center manifolds. Recall from ex:nonanalytic that the
parametrization $\mathcal{H}$ of the center manifold must satisfy (25) in an
open neighborhood of $\mathbb{R}\times\\{0\\}$. To construct the map
$\mathcal{H}$ explicitly, let us first introduce (formally) for arbitrary
constants $\alpha,\beta\in\mathbb{R}$ the family of functions
$I_{\alpha,\beta}:\mathbb{R}\times\mathbb{R}\setminus\\{0\\}\to\mathbb{R}$ as

$\displaystyle I_{\alpha,\beta}(t,x)$
$\displaystyle:=-\sqrt{2}e^{-\frac{1}{x}}\bigg{(}\cos(\frac{1}{x}-t)I_{\alpha}^{1}(x)+\sin(\frac{1}{x}-t)I_{\beta}^{2}(x)\bigg{)},$
(27)

where the functions $I_{\alpha}^{1}$ and $I_{\beta}^{2}$ are defined by

$\displaystyle
I_{\alpha}^{1}(x):=\int_{\alpha}^{x}\frac{e^{\frac{1}{s}}}{s}\sin(\frac{1}{s}+\frac{\pi}{4})ds,\quad
I_{\beta}^{2}(x):=\int_{\beta}^{x}\frac{e^{\frac{1}{s}}}{s}\sin(\frac{1}{s}-\frac{\pi}{4})ds.$
(28)

It turns out that the higher order derivatives of $I_{\alpha,\beta}(t,\cdot)$
will be important for the construction of the map $\mathcal{H}$. Therefore,
let us first determine all values of $\alpha$ and $\beta$ for which (27) is
well-defined on $\mathbb{R}\times(-\infty,0)$. Clearly $I_{\alpha,\beta}$ is
ill-defined on $\mathbb{R}\times(-\infty,0)$ whenever $\alpha,\beta>0$ due to
the singularities at zero for the functions defined in (28). Hence, we must
have that $\alpha,\beta\leq 0$. We will show that $\alpha=\beta=0$ are the
only values for which $I_{\alpha,\beta}$ is well-defined on
$\mathbb{R}\times(-\infty,0)$. Notice that

$\bigg{|}e^{-\frac{1}{x}}\cos(\frac{1}{x}-t)I_{0}^{1}(x)\bigg{|}\leq
e^{-\frac{1}{x}}\int_{x}^{0}\frac{e^{\frac{1}{s}}}{s}ds\to 0,$ (29)

as $x\uparrow 0$ by an application of L’Hôpital’s rule. A similar computation
for the second term in (27) shows that $I_{0,0}$ is well-defined. To show that
the function $I_{\alpha,\beta}$ is ill-defined on
$\mathbb{R}\times(-\infty,0)$ for all $\alpha,\beta<0$, consider for a fixed
$t\in\mathbb{R}$ the sequence $(x_{m})_{m\geq m_{0}}$ defined by
$x_{m}:=\frac{1}{t-m\pi}$, where the integer $m_{0}\geq 0$ is chosen large
enough to guarantee that $t-m_{0}\pi<0$. Hence,

$\displaystyle
e^{-\frac{1}{x_{m}}}\cos(\frac{1}{x_{m}}-t)I_{\alpha}^{1}(x_{m})=(-1)^{m}e^{-\frac{1}{x_{m}}}\int_{\alpha}^{x_{m}}\frac{e^{\frac{1}{s}}}{s}\sin(\frac{1}{s}+\frac{\pi}{4})ds.$
(30)

Because the integrand is continuous and bounded above on
$[\alpha,x_{m}]\subset[\alpha,0)$ for large enough $m\geq m_{0}$, it can be
extended from the left continuously at zero such that it attains the value
$M_{\alpha}<\infty$. If we set the integral in (30) to be $M_{\alpha}^{m}$,
then $M_{\alpha}^{m}\to M_{\alpha}$ when $m\to\infty$. Hence,

$e^{-\frac{1}{x_{m}}}\cos(\frac{1}{x_{m}}-t)I_{\alpha}^{1}(x_{m})=e^{-\frac{1}{x_{m}}}(-1)^{m}M_{\alpha}^{m}$

which is undetermined when $m\to\infty$. A similar reasoning shows that the
second term in (27) is ill-defined on $\mathbb{R}\times(-\infty,0)$ when
$\beta<0$ and so $I_{\alpha,\beta}$ is only well-defined on
$\mathbb{R}\times(-\infty,0)$ whenever $\alpha=\beta=0$. It can be proven
similarly as in (29) that $I_{\alpha,\beta}$ is well-defined on
$\mathbb{R}\times(0,\infty)$ and that $\lim_{x\downarrow
0}I_{\alpha,\beta}(\cdot,x)=0$ for all $\alpha,\beta>0$.

Our next goal is to determine the higher order partial derivatives of the
second component of $I_{\alpha,\beta}$ evaluated at zero. A straightforward
computation shows already that

$\frac{\partial}{\partial
x}I_{\alpha,\beta}(t,x)=\frac{\sqrt{2}}{x^{2}}\bigg{(}I_{\alpha,\beta}\bigg{(}t+\frac{\pi}{4},x\bigg{)}-x\sin(t+\frac{\pi}{4})\bigg{)}.$
(31)

Let us write $I_{\alpha,\beta}(t,x)=\sum_{n=0}^{N}b_{n}(t)x^{n}+R_{N}(t,x)$ as
a Taylor polynomial where $R_{N}$ is the remainder for some $N\in\mathbb{N}$.
Filling in this Taylor polynomial into (31), we see that $b_{0}$ is the zero
function, $b_{1}(t)=\sin(t)$ and
$b_{n}(t)=\frac{n-1}{\sqrt{2}}b_{n-1}(t-\frac{\pi}{4})$ for all $n=2,\dots,N$.
This recurrence relation shows that

$\displaystyle\frac{\partial^{n}}{\partial
x^{n}}I_{\alpha,\beta}(t,0)=n!b_{n}(t)=n!\frac{(n-1)!}{2^{\frac{n-1}{2}}}\sin(t-\frac{(n-1)\pi}{4}),$
(32)

for all $n=1,\dots,N$, where $N$ can be taken arbitrary large.

The construction of the map $\mathcal{H}$ will consists of two parts, namely
$x\in(-\infty,0]$ and $x\in[0,\infty)$. For the first part, let
$\phi:\mathbb{R}\to\mathbb{R}$ be any $2\pi$-periodic differentiable function
and observe that the map
$\mathcal{H}_{-}:\mathbb{R}\times(-\infty,0]\to\mathbb{R}$ defined by

$\mathcal{H}_{-}(t,x):=\begin{dcases}e^{-\frac{1}{x}}\phi\bigg{(}\frac{1}{x}-t\bigg{)}+I_{0,0}(t,x)-\sin(t)x,\quad&t\in\mathbb{R},\
x\in(-\infty,0),\\\ 0,\quad&t\in\mathbb{R},\ x=0,\end{dcases}$

satisfies the local invariance equation (25). However, for $\mathcal{H}_{-}$
to be a parametrization of a center manifold on $\mathbb{R}\times(-\infty,0]$,
we must have that $\lim_{x\uparrow 0}\mathcal{H}_{-}(\cdot,x)=0$. Since
$\lim_{x\uparrow 0}I_{0,0}(\cdot,x)$, it is clear that

$e^{-\frac{1}{x}}\phi\bigg{(}\frac{1}{x}-t\bigg{)}\to 0,\quad x\uparrow 0.$

We claim $\phi$ must be the zero function. Consider for fixed $t\in\mathbb{R}$
and $r\in\mathbb{Q}$ the sequence $(y_{m})_{m\geq m_{1}}$ defined by
$y_{m}:=\frac{1}{t-r-2m\pi}$ where the integer $m_{1}\geq 0$ is chosen large
enough to guarantee that $t-r-2m_{1}\pi<0$. The $2\pi$-periodicity of $\phi$
implies that

$e^{-\frac{1}{y_{m}}}\phi\bigg{(}\frac{1}{y_{m}}-t\bigg{)}=\phi(r)e^{-\frac{1}{y_{m}}}\to\infty,\quad
m\to\infty,$

unless $\phi(r)=0$. As $r\in\mathbb{Q}$ is arbitrary we have that $\phi$ is
the zero function on $\mathbb{Q}$ and because $\phi$ is (at least) continuous,
we have that $\phi$ is the zero function on $\mathbb{R}$. Moreover, we obtain
from (32) directly that $\lim_{x\uparrow 0}\frac{\partial}{\partial
x}\mathcal{H}_{-}(\cdot,x)=0$ and so $\mathcal{H}_{-}$ is indeed a
parametrization of a center manifold on $\mathbb{R}\times(-\infty,0]$. In
addition, it follows directly from (31) that $\mathcal{H}_{-}$ is
$C^{\infty}$-smooth on $\mathbb{R}\times(-\infty,0]$.

For the second part, let $\phi:\mathbb{R}\to\mathbb{R}$ be any $2\pi$-periodic
$C^{k}$-smooth function and observe that for any $\alpha,\beta>0$ and
sufficiently small $\delta_{k}>0$ the map
$\mathcal{H}_{+,\phi}^{\alpha,\beta}:\mathbb{R}\times[0,\delta_{k})\to\mathbb{R}$
defined by

$\mathcal{H}_{+,\phi}^{\alpha,\beta}(t,x):=\begin{dcases}e^{-\frac{1}{x}}\phi\bigg{(}\frac{1}{x}-t\bigg{)}+I_{\alpha,\beta}(t,x)-\sin(t)x,\quad&t\in\mathbb{R},\
x\in(0,\delta_{k}),\\\ 0,\quad&t\in\mathbb{R},\ x=0,\end{dcases}$

satisfies the local invariance equation (25). Since $\phi$ is $C^{k}$-smooth
and $2\pi$-periodic, we have for any $l=0,\dots,k$ that its $l$th derivative
is bounded above by some real number $0<M_{l}<\infty$. Hence,

$\bigg{|}e^{-\frac{1}{x}}\phi\bigg{(}\frac{1}{x}-t\bigg{)}\bigg{|}\leq
M_{0}e^{-\frac{1}{x}}\to 0,\quad x\downarrow 0,$

which already proves that $\lim_{x\downarrow
0}\mathcal{H}_{+,\phi}^{\alpha,\beta}(\cdot,x)=0$. To prove that
$\mathcal{H}_{+,\phi}^{\alpha,\beta}$ is tangent to the center bundle and
$C^{k}$-smooth (at the origin), note for any $l=0,\dots,k$ that

$\bigg{|}\frac{d^{l}}{dx^{l}}\bigg{[}e^{-\frac{1}{x}}\phi\bigg{(}\frac{1}{x}-t\bigg{)}\bigg{]}\bigg{|}\leq
e^{-\frac{1}{x}}\sum_{q=0}^{l}p_{q}\bigg{(}\frac{1}{|x|}\bigg{)}\to 0,$

as $x\downarrow 0$ due to the general Leibniz rule, Faà di Bruno’s formula and
the fact that $p_{q}$, dependent on $M_{0},\dots,M_{q}$, is a polynomial for
all $q=0,\dots,l$. Hence, $\mathcal{H}_{+,\phi}^{\alpha,\beta}$ is a
$C^{k}$-smooth function on $\mathbb{R}\times[0,\delta_{k})$ for some
$\delta_{k}>0$. As a consequence of the results derived above, the map
$\mathcal{H}_{\phi}^{\alpha,\beta}:\mathbb{R}\times(-\infty,\delta_{k})$
defined by

$\displaystyle\mathcal{H}_{\phi}^{\alpha,\beta}(t,x):=\begin{dcases}\mathcal{H}_{-}(t,x),&t\in\mathbb{R},\
x\in(-\infty,0],\\\ \mathcal{H}_{+,\phi}^{\alpha,\beta}(t,x),&t\in\mathbb{R},\
x\in[0,\delta_{k}),\\\ \end{dcases}$ (33)

parametrizes a locally (non)-unique family of $2\pi$-periodic $C^{k}$-smooth
center manifolds around $\mathbb{R}\times\\{0\\}$ of (16). Two different
$2\pi$-periodic center manifolds for (16) are visualized in fig:cmexample.

Figure 3: Two different $C^{\infty}$-smooth non-analytic $2\pi$-periodic
center manifolds around $\mathbb{R}\times\\{0\\}$ for the analytic system (16)
parametrized by the maps $\mathcal{H}_{0}^{1,1}$ and $\mathcal{H}_{0}^{2,3}$
respectively.

The analytic system (16) admits a $2\pi$-periodic $C^{\infty}$-smooth center
manifold. The map $\mathcal{H}_{0}^{1,1}$ from (33) provides us a
$2\pi$-periodic $C^{\infty}$-smooth center manifold for (16) on
$(-\infty,\delta_{\infty})$ with $\delta_{\infty}>0$ since
$\mathcal{H}_{0}^{1,1}$ is $C^{\infty}$-smooth in an open neighborhood of
$\mathbb{R}\times\\{0\\}$.

The analytic system (18) admits for any $k\geq 0$ a $2\pi$-periodic
$C^{k}$-smooth center manifold, but not a $2\pi$-periodic $C^{\infty}$-smooth
center manifold. The fundamental matrix for the linearization around the
origin of (18) reads

$U(t,s)=\begin{pmatrix}1&0&0\\\ 0&e^{t-s}&0\\\ 0&0&1\end{pmatrix}$

for all $(t,s)\in\mathbb{R}^{2}$. The Floquet multipliers are given by
$\lambda_{1}=1,\lambda_{2}=e^{2\pi}$ and $\lambda_{3}=1$. The center space (at
time $t$) is given by $E_{0}(t)=\operatorname{span}\\{(1,0,0),(0,0,1)\\}$
while the unstable space (at time $t$) is given by
$E_{+}(t)=\operatorname{span}\\{(0,1,0)\\}$. Hence, the $xz$-plane corresponds
to the center space and so the center manifold can be parametrized by
$y(t)=\mathcal{H}(t,x,z)$, where $\mathcal{H}$ is $2\pi$-periodic in the first
argument and only consists of nonlinear terms. It follows from thm:main that
there exists for any $k\geq 1$ an open neighborhood $U_{2k}$ of
$\mathbb{R}\times\\{0\\}\times\\{0\\}$ such that the map $\mathcal{H}$ is
$C^{2k}$-smooth on $U_{2k}$. Hence, one can write

$\mathcal{H}(t,x,z)=\sum_{n=2}^{2k}a_{n}(t,z)x^{n}+\mathcal{O}(x^{2k+1})$ (34)

in the neighborhood $U_{2k}$. Because the center manifold is locally
invariant, the map $\mathcal{H}$ must satisfy

$\displaystyle\frac{\partial\mathcal{H}}{\partial
t}(t,x,z)+x(z-x^{2})\frac{\partial\mathcal{H}}{\partial
x}(t,x,z)=\mathcal{H}(t,x,z)+(1+\sin(t))x^{2}$ (35)

Substituting (34) into (35) and comparing terms in $x^{n}$ for $n=2,\dots,2k$
shows that the functions $a_{n}$, which are $2\pi$-periodic in the first
component, must satisfy

$\begin{dcases}\frac{\partial a_{2}}{\partial
t}(t,z)+(2z-1)a_{2}(t,z)=1+\sin(t),\quad&n=2,\\\ \frac{\partial
a_{n}}{\partial
t}(t,z)+(nz-1)a_{n}(t,z)=(n-2)a_{n-2}(t,z),\quad&n=3,\dots,2k.\end{dcases}$
(36)

Because $\mathcal{H}$ consists only of nonlinear terms, we can assume that
$a_{1}=0$. Solving the $2\pi$-periodic boundary value problem (36) for $n=2$
yields

$\displaystyle
a_{2}(t,z)=\frac{1}{2z-1}+\frac{2z-1}{(2z-1)^{2}+1}\sin(t)-\frac{1}{(2z-1)^{2}+1}\cos(t),\quad
z\neq\frac{1}{2}.$ (37)

Furthermore, at $z=\frac{1}{2}$ one verifies easily from (36) that
$a_{2}(\cdot,\frac{1}{2})$ does not admit a $2\pi$-periodic solution.
Moreover, all odd coefficients $a_{1},a_{3},\dots,a_{2k-1}$ are zero and the
even coefficients $a_{2},a_{4},\dots,a_{2k}$ are recursively given by

$\displaystyle a_{2n}(t,z)$
$\displaystyle=\frac{2(n-1)e^{-(2nz-1)t}}{e^{2\pi(2nz-1)}-1}\bigg{(}\int_{t}^{2\pi}e^{(2nz-1)\tau}a_{2(n-1)}(\tau,z)d\tau$
$\displaystyle+e^{2\pi(2nz-1)}\int_{0}^{t}e^{(2nz-1)\tau}a_{2(n-1)}(\tau,z)d\tau\bigg{)},\quad
z\neq\frac{1}{2n}.$

To obtain a semi-explicit representation for $a_{2n}$, we will prove by
induction on $n=1,2,\dots,k$ that

$\displaystyle
a_{2n}(t,z)=2^{n-1}(n-1)!\prod_{l=1}^{n}\frac{1}{2lz-1}+\alpha_{n}(z)\sin(t)+\beta_{n}(z)\cos(t),\quad
z\neq\frac{1}{2n},$ (38)

where $\alpha_{n}$ and $\beta_{n}$ are well-defined rational functions on
$\mathbb{R}$. Clearly, the claim holds for $n=1$ due to (37). Assume that the
claim holds for a certain $n\geq 2$. Along the same lines of the induction
step in ex:nonanalytic, one derives

$\displaystyle a_{2n}(t,z)=2^{n-1}(n-1)!\prod_{l=1}^{n}\frac{1}{2lz-1}$
$\displaystyle+\frac{2(n-1)[(2nz-1)\alpha_{n-1}(z)+\beta_{n-1}(z)]}{(2nz-1)^{2}+1}\sin(t)$
$\displaystyle+\frac{2(n-1)[(2nz-1)\beta_{n-1}(z)-\alpha_{n-1}(z)]}{(2nz-1)^{2}+1}\cos(t).$

It remains to show that the coefficients in front of the sine and cosine are
well-defined rational functions on $\mathbb{R}$. Clearly,

$\displaystyle\begin{pmatrix}\alpha_{n}(z)\\\
\beta_{n}(z)\end{pmatrix}=\frac{2(n-1)}{(2nz-1)^{2}+1}\begin{pmatrix}2nz-1&1\\\
-1&2nz-1\end{pmatrix}\begin{pmatrix}\alpha_{n-1}(z)\\\
\beta_{n-1}(z)\end{pmatrix},\quad n\geq 3,$

with initial condition

$\alpha_{2}(z)=\frac{2z-1}{(2z-1)^{2}+1},\quad\beta_{2}(z)=\frac{1}{(2z-1)^{2}+1}.$

Solving this linear system of difference equations semi-explicitly yields

$\displaystyle\begin{pmatrix}\alpha_{n}(z)\\\
\beta_{n}(z)\end{pmatrix}=2^{n-1}(n-1)!\bigg{[}\prod_{l=2}^{n}\frac{1}{(2lz-1)^{2}+1}\begin{pmatrix}2lz-1&1\\\
-1&2lz-1\end{pmatrix}\bigg{]}\begin{pmatrix}\alpha_{2}(z)\\\
\beta_{2}(z)\end{pmatrix}.$

Hence, $\alpha_{n}$ and $\beta_{n}$ are both rational functions that are well-
defined on $\mathbb{R}$ since $(2lz-1)^{2}+1\geq 0$ for all $l=1,\dots,n$.
This concludes the induction step. On the other hand, if $z=\frac{1}{2n}$,
then one can verify rather easily from (36) that $a_{n}(\cdot,\frac{1}{2n})$
has no $2\pi$-periodic solution.

Using (38) in combination with (34), we see that $\mathcal{H}(t,x,\cdot)$ is
not $C^{2k}$-smooth on $(-\frac{1}{2k},\frac{1}{2k}]$ since
$a_{2k}(\cdot,\frac{1}{2k})$ is simply undefined. Suppose now that
$\mathcal{H}$ is $C^{\infty}$-smooth on
$\mathbb{R}\times\\{0\\}\times\\{0\\}$, then for fixed $t\in\mathbb{R}$ and
non-zero $x\in\mathbb{R}$ there exists an $\varepsilon>0$ such that
$\mathcal{H}(t,x,\cdot)$ is $C^{\infty}$-smooth on
$(-\varepsilon,\varepsilon)$. Now, if $k\geq 1$ is an integer that satisfies
$k>\frac{1}{2\varepsilon}$, then $\mathcal{H}(t,x,\cdot)$ is not
$C^{2k}$-smooth on $(-\varepsilon,\varepsilon)$. This contradicts the
assumption that (18) admits a $C^{\infty}$-smooth $2\pi$-periodic center
manifold at the origin. To illustrate the cascade of singularities of the
periodic center manifold towards the origin, second- and fourth-order
approximations at different time steps of $\mathcal{H}$ are presented in
fig:CMsingularity.

Figure 4: In (a) a second-order approximation of $\mathcal{H}(0,\cdot,\cdot)$
and in (b) a second-order approximation of $\mathcal{H}(\pi,\cdot,\cdot)$ . In
(c) a fourth-order approximation of $\mathcal{H}(0,\cdot,\cdot)$ and in (d) a
fourth-order approximation of $\mathcal{H}(\pi,\cdot,\cdot)$. The red vertical
planes indicate the singularities at $z=\frac{1}{4}$ and $z=\frac{1}{2}$.

##  6 Conclusion and outlook

We have proven the existence of a periodic smooth locally invariant center
manifold near a nonhyperbolic cycle in the setting of finite-dimensional
ordinary differential equations. Our results are based on rather simple
consequences of Floquet theory in combination with a fixed point argument on
the easily available variation of constants formula for periodic (nonlinear)
ODEs. In addition, we have provided several examples of (non)-unique,
(non)-$C^{\infty}$-smooth and (non)-analytic periodic center manifolds to
illustrate that periodic center manifolds admit similar interesting
qualitative properties as center manifolds for equilibria.

Despite our illustrations from sec:examples are very insightful on the nature
of periodic center manifolds, it is not clear under which conditions a
periodic center manifold is unique, non-unique or locally (non)-unique. To
answer the first question, we believe that one must generalize techniques from
[50] towards periodic center manifolds to state and prove a similar result as
in [50, Theorem 3.2]. Moreover, if a periodic center manifold is not uniquely
determined, how much can two periodic center manifolds differ from each other?
Such results have already been established in [50, Section 4] for center
manifolds for equilibria, but the question remains unanswered for periodic
center manifolds. However, we have already seen in ex:nonunique that periodic
center manifolds may differ from each other by a factor of
$e^{-\frac{1}{x}}\phi(\frac{1}{x}-t)$, where $\phi$ is any $T$-periodic (at
least) differentiable function. Furthermore, it is not clear under which
conditions a $C^{\infty}$-smooth or analytic periodic center manifold may
exist while this question is addressed and answered in [50, Section 5 and 6]
for center manifolds for equilibria. In particular, recall from ex:nonanalytic
that the $C^{\infty}$-smooth periodic center manifold is not analytic for all
$t\in\mathbb{R}$. However, is it possible to construct an example where a
$C^{\infty}$-smooth periodic center manifold may change periodically from non-
analytic to analytic? Or is there a possibility that the (non)-analyticitiy of
a periodic center manifold is time-independent?

## Acknowledgements

The authors would like to thank Profs. Renato Huzak and Peter De Maesschalck
(Hasselt University) for helpful discussions and suggestions.

##  Appendix A Smoothness of the center manifold

In this appendix, we will prove that the map $\mathcal{C}$ inherits the same
order of smoothness as the nonlinearity $R$. Our results are based on the
theory of contraction on scales of Banach spaces, see [22, Section IX.6 and
Appendix IV] and [57, 32, 33, 10, 42] for applications of this theory to
ordinary differential equations and (mixed) functional differential equations.
Our arguments here are based on the strategy developed in the mentioned
references and closely follow [42].

To prove additional smoothness of the map $\mathcal{C}$, let us first observe
that we are only interested in pairs $(s,y_{0})\in E_{0}$ due to (12).
Therefore, let us incorporate the starting time $s$ inside the domain of the
fixed point operator $\mathcal{G}_{s}^{\eta}$ from subsec: Lipschitz CM.
Hence, define for $\eta\in(0,\min\\{-a,b\\})$ and sufficiently small
$\delta>0$ the map $\mathcal{G}^{\eta}$ by

$\operatorname{BC}_{s}^{\eta}\times E_{0}\ni(u,s,y_{0})\mapsto
U_{s}^{\eta}y_{0}+\mathcal{K}_{s}^{\eta}(\tilde{R}_{\delta}(u))\in\operatorname{BC}_{s}^{\eta},$

and following the same steps from subsec: Lipschitz CM, we have that
$\mathcal{G}^{\eta}(\cdot,s,y_{0})$ has a unique fixed point
$\hat{u}^{\eta}:E_{0}\to\operatorname{BC}_{s}^{\eta}$ such that
$\hat{u}^{\eta}(s,\cdot)$ is globally Lipschitz and satisfies
$\hat{u}^{\eta}(s,0)=0$ for all $s\in\mathbb{R}$. It turns out that the space
$\operatorname{BC}_{s}^{\eta}$ is not really suited to increase smoothness of
the center manifold. The main idea is to work with another $\eta$-exponent
that makes a trade-off between ensuring smoothness while not losing the
contraction property. To make this construction, choose an interval
$[\eta_{-},\eta_{+}]\subset(0,\min\\{-a,b\\})$ such that $k\eta_{-}<\eta_{+}$
and $\delta>0$ small enough to guarantee that

$L_{\delta}\|\mathcal{K}_{s}^{\eta}\|_{\eta,s}<\frac{1}{4},\quad\forall\eta\in[\eta_{-},\eta_{+}],\
s\in\mathbb{R},$ (39)

which is possible since $L_{\delta}\to 0$ as $\delta\downarrow 0$ proven in
prop:global lipschitz.

From this construction, it is clear that we would like to switch back and
forth between, for example, the fixed points $\hat{u}^{\eta}(s,y_{0})$ and
$\hat{u}^{\eta_{-}}(s,y_{0})$. Therefore, introduce for any
$0<\eta_{1}\leq\eta_{2}<\min\\{-a,b\\}$ the linear embedding
$\mathcal{J}_{s}^{\eta_{2},\eta_{1}}:\operatorname{BC}_{s}^{\eta_{1}}\hookrightarrow\operatorname{BC}_{s}^{\eta_{2}}$
and notice that this map is bounded since

$\displaystyle\|u\|_{\eta_{2},s}=\sup_{t\in\mathbb{R}}e^{-\eta_{2}|t-s|}\|u(t)\|\leq\sup_{t\in\mathbb{R}}e^{-\eta_{1}|t-s|}\|u(t)\|=\|u\|_{\eta_{1},s}<\infty,$

for any $u\in\operatorname{BC}_{s}^{\eta_{1}}$. Hence,
$\mathcal{J}_{s}^{\eta_{2},\eta_{1}}$ is $C^{\infty}$-smooth and
$\operatorname{BC}_{s}^{\eta_{1}}$ can be considered as a subspace of
$\operatorname{BC}_{s}^{\eta_{2}}$. The following lemma shows we can switch
back and forth between the fixed points of interest. Let
$0<\eta_{1}\leq\eta_{2}<\min\\{-a,b\\}$ and $s\in\mathbb{R}$. Assume that
$\hat{u}^{\eta_{1}}(s,y_{0})$ is the fixed point of
$\mathcal{G}^{\eta_{1}}(\cdot,s,y_{0})$ for some $(s,y_{0})\in E_{0}$. Then
$\hat{u}^{\eta_{2}}(s,y_{0})=\mathcal{J}_{s}^{\eta_{2},\eta_{1}}\hat{u}^{\eta_{1}}(s,y_{0})$.

Note that the definition of the fixed point operator does not depend
explicitly on the choice of $\eta\in(0,\min\\{-a,b\\})$, since
$\mathcal{K}_{s}^{\eta_{1}}u=\mathcal{K}_{s}^{\eta_{2}}u$ for all
$u\in\operatorname{BC}^{\eta_{1}}_{s}$. Then by uniqueness of the fixed point
and since $\operatorname{BC}_{s}^{\eta_{1}}$ is continuously embedded in
$\operatorname{BC}_{s}^{\eta_{2}}$, it is clear that
$\hat{u}_{s}^{\eta_{2}}(y_{0})=\mathcal{J}_{s}^{\eta_{2},\eta_{1}}\hat{u}_{s}^{\eta_{1}}(y_{0})$.

A first step in increasing smoothness of the center manifold is to show that
$\tilde{R}_{\delta}$ is sufficiently smooth. Recall from subsec: modification
nonlinearity that $R_{\delta}$ is $C^{k}$-smooth. Consider now for any pair of
integers $p,q\geq 0$ with $p+q\leq k$ the map
$\tilde{R}_{\delta}^{(p,q)}(u)\in\mathcal{L}^{q}(C(\mathbb{R},\mathbb{R}^{n})):=\mathcal{L}^{q}(C(\mathbb{R},\mathbb{R}^{n}),C(\mathbb{R},\mathbb{R}^{n}))$
defined by

$\displaystyle\tilde{R}_{\delta}^{(p,q)}(u)(v_{1},\dots,v_{q})(t):=D_{1}^{p}D_{2}^{q}R_{\delta}(t,u(t))(v_{1}(t),\dots,v_{q}(t)).$

Here $\mathcal{L}^{q}(Y,Z)$ denotes the space of $q$-linear mappings from
$Y^{q}:=Y\times\dots\times Y$ into $Z$ for Banach spaces $Y$ and $Z$. The
following three lemmas, adapted from the literature towards the finite-
dimensional ODE-setting, will be crucial in the proof of thm:smoothnesscmt.

[[22, Lemma XII.7.6] and [33, Proposition 8.2]] Let $p,q\geq 0$ be positive
integers with $p+q\leq k$ and $\eta\geq q\mu>0$. Then for any $u\in
C(\mathbb{R},\mathbb{R}^{n})$ we have
$\tilde{R}_{\delta}^{(p,q)}(u)\in\mathcal{L}^{q}(\operatorname{BC}_{s}^{\mu},\operatorname{BC}_{s}^{\eta})$,
where the norm is bounded by

$\|\tilde{R}_{\delta}^{(p,q)}\|\leq\sup_{t\in\mathbb{R}}e^{-(\eta-q\mu)|t-s|}\|D_{1}^{p}D_{2}^{q}R_{\delta}(t,u(t))\|<\infty.$

Furthermore, consider any $0\leq l\leq k-(p+q)$ and $\sigma>0$. If
$\eta>q\mu+l\sigma$, then the map $u\mapsto\tilde{R}_{\delta}^{(p,q)}$ from
$\operatorname{BC}_{s}^{\sigma}$ into
$\mathcal{L}^{q}(\operatorname{BC}_{s}^{\mu},\operatorname{BC}_{s}^{\eta})$ is
$C^{l}$-smooth with
$D^{l}\tilde{R}_{\delta}^{(p,q)}=\tilde{R}_{\delta}^{(p,q+l)}$.

[Diekmann et al., 1995] Let $p,q\geq 0$ be positive integers with $p+q<k$ and
let $\eta>q\mu+\sigma$ for some $\mu,\sigma>0$. Consider a map $\Phi\in
C^{1}(E_{0},\operatorname{BC}_{s}^{\sigma})$. Then the map
$\tilde{R}_{\delta}^{(p,q)}\circ\Phi:E_{0}\to\mathcal{L}^{q}(\operatorname{BC}_{s}^{\mu},\operatorname{BC}_{s}^{\eta})$
is $C^{1}$-smooth with

$\displaystyle
D(\tilde{R}_{\delta}^{(p,q)}\circ\Phi)(s_{0},y_{0})(v_{1},\dots,v_{q},(s_{1},y_{1}))=\tilde{R}_{\delta}^{(p,q+1)}(\Phi(s_{0},y_{0}))(v_{1},\dots,v_{q},D\Phi(s_{0},y_{0})(s_{1},y_{1})).$

[[22, Lemma XII.6.6 and XII.6.7]] Let $Y_{0},Y,Y_{1}$ and $\Lambda$ be Banach
spaces with continuous embeddings $J_{0}:Y_{0}\hookrightarrow Y$ and
$J:Y\hookrightarrow Y_{1}$. Consider the fixed point problem $y=f(y,\lambda)$
for $f:Y\times\Lambda\to Y$. Suppose that the following conditions hold.

  1. $1.$

The function $g:Y_{0}\times\Lambda\to Y_{1}$ defined by
$g(y_{0},\lambda):=Jf(J_{0}y_{0},\lambda)$ is of the class $C^{1}$ and there
exist mappings $f^{2:}:J_{0}Y_{0}\times\Lambda\to\mathcal{L}(Y)$ and
$f_{1}^{(1)}:J_{0}Y_{0}\times\Lambda\to\mathcal{L}(Y_{1})$ such that
$D_{1}g(y_{0},\lambda)\xi=Jf^{(1)}(J_{0}y_{0},\lambda)J_{0}$ for all
$(y_{0},\lambda,\xi)\in Y_{0}\times\Lambda\times Y_{0}$ and
$Jf^{(1)}(J_{0}y_{0},\lambda)y=f_{1}^{(1)}(J_{0}y_{0},\lambda)Jy$ for all
$(y_{0},\lambda,y)\in Y_{0}\times\Lambda\times Y$.

  2. $2.$

There exists a $\kappa\in[0,1)$ such that for all $\lambda\in\Lambda$ the map
$f(\cdot,\lambda):Y\to Y$ is Lipschitz continuous with Lipschitz constant
$\kappa$, independent of $\lambda$. Furthermore, for any $\lambda\in\Lambda$
the maps $f^{(1)}(\cdot,\lambda)$ and $f_{1}^{(1)}(\cdot,\lambda)$ are
uniformly bounded by $\kappa$.

  3. $3.$

Under the previous condition, the unique fixed point $\Psi:\Lambda\to Y$
satisfies $\Psi(\lambda)=f(\Psi(\lambda),\lambda)$ and can be written as
$\Psi=J_{0}\circ\Phi$ for some continuous $\Phi:\Lambda\to Y_{0}$.

  4. $4.$

The function $f_{0}:Y_{0}\times\Lambda\to Y$ defined by
$f_{0}(y_{0},\lambda)=f(J_{0}y_{0},\lambda)$ has continuous partial derivative
$D_{2}f:Y_{0}\times\Lambda\to\mathcal{L}(\Lambda,Y).$

  5. $5.$

The mapping $Y_{0}\times\Lambda\ni(y,\lambda)\mapsto J\circ
f^{(1)}(J_{0}y,\lambda)\in\mathcal{L}(Y,Y_{1})$ is continuous.

Then the map $J\circ\Psi$ is of the class $C^{1}$ and
$D(J\circ\Psi)(\lambda)=J\circ\mathcal{A}(\lambda)$ for all
$\lambda\in\Lambda$, where $A=\mathcal{A}(\lambda)\in\mathcal{L}(\Lambda,Y)$
is the unique solution of the fixed point equation
$A=f^{(1)}(\Psi(\lambda),\lambda)A+D_{2}f_{0}(\Psi(\lambda),\lambda)$.

For each $l\in\\{1,\dots,k\\}$ and
$\eta\in(l\eta_{-},\eta_{+}]\subset(0,\min\\{-a,b\\})$, the map
$\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-}}:E_{0}\to\operatorname{BC}_{s}^{\eta}$
is $C^{l}$-smooth provided that $\delta>0$ is sufficiently small. To begin, we
choose $\delta>0$ small enough so that (39) holds. We prove the assertion by
induction on $l$. Let $l=k=1$ and $\eta\in(\eta_{-},\eta_{+}]$ be given. We
show that lemma:fixedpointsmooth applies with the Banach spaces
$Y_{0}=Y=\operatorname{BC}_{s}^{\eta_{-}},Y_{1}=\operatorname{BC}_{s}^{\eta}$
and $\Lambda=E_{0}$, and operators

$\displaystyle f(u,s,y_{0})=\mathcal{G}^{\eta_{-}}(u,s,y_{0}),\quad
f^{(1)}(u,s,y_{0})=\mathcal{K}_{s}^{\eta_{-}}\circ\tilde{R}_{\delta}^{(0,1)}(u),\quad
f_{1}^{(1)}=\mathcal{K}_{s}^{\eta}\circ\tilde{R}_{\delta}^{(0,1)}(u),$

with embeddings $J=\mathcal{J}_{s}^{\eta,\eta_{-}}$ and $J_{0}$ denotes the
identity map. In the context of lemma:fixedpointsmooth, the map $g$ is given
by $\mathcal{G}^{\eta}$ due to the linearity of the embedding $J$. Because
$(s,y_{0})\mapsto U(\cdot,s)y_{0},s\mapsto\mathcal{K}_{s}^{\eta}$ and
$u\mapsto\tilde{R}_{\delta}(u)$ are $C^{1}$-smooth (sec:Floquet, prop:ketas
and lemma:smoothness3), the map $g$ is $C^{1}$-smooth and one can easily
verify the additional equalities. The second condition follows from (39) and
the fact that the Lipschitz constant is independent of $s\in\mathbb{R}$ due to
prop:ketas. The third condition follows from the fact that $\Psi$ is given by
$\hat{u}^{\eta_{-}}$ and therefore well-defined due to thm:contraction. The
mentioned results show that the fourth condition is satisfied. It follows from
prop:ketas and lemma:smoothness3 that the fifth condition is satisfied as
well. Hence, we conclude that the map
$\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-}}$ is of the class
$C^{1}$ and that
$D(\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-}})=\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-},(1)}\in\mathcal{L}(E_{0},\operatorname{BC}_{s}^{\eta})$,
where $\hat{u}^{\eta_{-},(1)}(s,y_{0})$ is the unique solution of

$\displaystyle
w^{(1)}=\mathcal{K}_{s}^{\eta_{-}}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))w^{(1)}+U_{s}^{\eta_{-}}=:F_{\eta_{-}}^{(1)}(w^{(1)},s,y_{0})$

in the space $\mathcal{L}(E_{0},\operatorname{BC}_{s}^{\eta_{-}})$. Here
$F_{\eta_{-}}^{(1)}:\mathcal{L}(E_{0},\operatorname{BC}_{s}^{\eta_{-}})\times
E_{0}\to\mathcal{L}(E_{0},\operatorname{BC}_{s}^{\eta_{-}})$ and notice that
$F_{\eta_{-}}^{(1)}(\cdot,s,y_{0})$ is a uniform contraction
(lemma:smoothness3), which proves the uniqueness of the fixed point.

To specify the induction hypothesis, consider any integer $1\leq l<k$ and
suppose that for all $1\leq q\leq l$ and all $\eta\in(q\eta_{-},\eta_{+}]$
that the map $\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-}}$ is
$C^{q}$-smooth with
$D^{q}(\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-}})=\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-},(q)}\in\mathcal{L}^{q}(E_{0},\operatorname{BC}_{s}^{q\eta})$,
where $\hat{u}^{\eta_{-},(q)}$ is the unique solution of

$\displaystyle
w^{(l)}=\mathcal{K}_{s}^{l\eta_{-}}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))w^{(l)}+H_{\eta_{-}}^{(l)}(s,y_{0})=:F_{l\eta_{-}}^{(l)}(w^{(l)},s,y_{0})$

in the space $\mathcal{L}^{q}(E_{0},\operatorname{BC}_{s}^{q\eta_{-}})$. Here
$H_{\eta_{-}}^{(1)}(s,y_{0})=U_{s}^{\eta_{-}}y_{0}$ and for
$\nu\in[\eta_{-},\eta_{+}]$ and $l\geq 2$ we have that
$H_{\nu}^{(l)}(s,y_{0})$ is a finite sum of terms of the form

$\displaystyle\mathcal{K}_{s}^{l\nu}\circ\tilde{R}_{\delta}^{(0,q)}(\hat{u}^{\eta_{-}}(s,y_{0}))(\hat{u}^{\eta_{-},(r_{1})}(s,y_{0}),\dots,\hat{u}^{\eta_{-},(r_{q})}(s,y_{0})),$

with $2\leq q\leq l$ and $1\leq r_{i}<l$ for $i=1,\dots,q$ such that
$r_{1}+\dots+r_{q}=l$. Here
$F_{l\eta}^{(l)}:\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{l\eta})\times
E_{0}\to\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{l\eta})$ is a uniform
contraction (lemma:smoothness3) for any $\eta\in[\eta_{-},\eta_{+}]$, which
guarantees the uniqueness of the fixed point.

For the induction step, fix some $\eta\in((l+1)\eta_{-},\eta_{+}]$ and choose
$\sigma,\mu>0$ such that $\eta_{-}<\sigma<(l+1)\sigma<\mu<\eta$. We show that
lemma:fixedpointsmooth applies with the Banach spaces
$Y_{0}=\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{l\sigma}),Y=\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\mu}),Y_{1}=\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\eta})$
and $\Lambda=E_{0}$, and operators

$\displaystyle f(u,s,y_{0})$
$\displaystyle=\mathcal{K}_{s}^{\mu}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))u+H_{\mu/l}^{(l)}(s,y_{0}),$
$\displaystyle f^{(1)}(u,s,y_{0})$
$\displaystyle=\mathcal{K}_{s}^{\mu}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))\in\mathcal{L}(\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\mu})),$
$\displaystyle f_{1}^{(1)}(u,s,y_{0})$
$\displaystyle=\mathcal{K}_{s}^{\eta}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))\in\mathcal{L}(\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\eta})).$

To verify the first condition, we have to check that
$g:\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{l\sigma})\times
E_{0}\to\mathcal{L}(E_{0},\operatorname{BC}_{s}^{\eta})$ given by

$\displaystyle g(u,s,y_{0})$
$\displaystyle=\mathcal{K}_{s}^{\eta}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))u+\mathcal{J}_{s}^{\eta,\mu}\circ
H_{\mu/l}^{(l)}(s,y_{0})$

is $C^{1}$-smooth, where now
$\mathcal{J}_{s}^{\eta,\mu}:\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\mu})\hookrightarrow\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\eta})$
is the continuous embedding. Clearly, $g$ is $C^{1}$-smooth in the first
variable since it is linear. For the second variable, notice that the map
$(s,y_{0})\mapsto\mathcal{K}_{s}^{\eta}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))u$
is $C^{1}$-smooth due to lemma: smoothness4 with $\mu>(l+1)\sigma$ and the
$C^{1}$-smoothness of
$(s,y_{0})\mapsto\mathcal{J}_{s}^{\sigma,\eta_{-}}\hat{u}^{\eta_{-}}(s,y_{0})$
for any $\sigma\geq\eta_{-}$. For the $C^{1}$-smoothness of the map
$H_{\mu/l}^{(l)}$, we get differentiability from lemma: smoothness4 and so we
have that the derivative of this map is a finite sum of terms of the form

$\displaystyle\mathcal{K}_{s}^{\mu}$
$\displaystyle\circ\tilde{R}_{\delta}^{(0,q+1)}(\hat{u}^{\eta_{-}}(s,y_{0}))(\hat{u}^{\eta_{-},(r_{1})}(s,y_{0}),\dots,\hat{u}^{\eta_{-},(r_{q})}(s,y_{0}))$
$\displaystyle+\sum_{j=1}^{q}\mathcal{K}_{s}^{\mu}\circ\tilde{R}_{\delta}^{(0,q)}(\hat{u}^{\eta_{-}}(s,y_{0}))(\hat{u}^{\eta_{-},(r_{1})}(s,y_{0}),\dots,\hat{u}^{\eta_{-},(r_{j}+1)}(s,y_{0}),\dots,\hat{u}^{\eta_{-},(r_{q})}(s,y_{0}))$

and each $\hat{u}^{\eta_{-},(r_{j})}(s,y_{0})$ is a map from $E_{0}$ into
$\operatorname{BC}_{s}^{j\sigma}$ for $j=1,\dots,q$. An application of
lemma:smoothness3 with $\mu>(l+1)\sigma$ ensures the continuity of
$DH_{\mu/l}^{(l)}(s,y_{0})$ and consequently that of
$\mathcal{J}_{s}^{\eta,\mu}DH_{\mu/l}^{(l)}(s,y_{0})$. The remaining
calculations from the first condition are then easily checked, and condition
four can be proven similarly. The Lipschitz condition and boundedness for the
second condition follows by the choice of $\delta>0$ chosen at the beginning
of the proof and the contractivity of $H_{\mu/l}^{(l)}$ described above. To
prove the third condition, observe that one can write

$\mathcal{K}_{s}^{\eta}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))=\mathcal{J}_{s}^{\eta,\mu}\mathcal{K}_{s}^{\mu}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))$

and applying lemma:smoothness3 together with the $C^{1}$-smoothness of
$\hat{u}^{\eta_{-}}$ to obtain the continuity of
$(s,y_{0})\mapsto\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-}}(s,y_{0}))$.
This also proves the fifth condition, and so we conclude that
$\hat{u}^{\eta_{-}}:E_{0}\mapsto\mathcal{L}^{l}(E_{0},\operatorname{BC}_{s}^{\eta})$
is of the class $C^{1}$ with derivative
$\hat{u}^{\eta_{-},(l+1)}=D\hat{u}^{\eta_{-},(l)}\in\mathcal{L}^{l+1}(E_{0},\operatorname{BC}_{s}^{\eta})$
that is the unique solution of

$w^{(l+1)}=\mathcal{K}_{s}^{\mu}\circ\tilde{R}_{\delta}^{(0,1)}(\hat{u}^{\eta_{-},(l+1)})w^{(l+1)}+H_{\mu/(l+1)}^{(l+1)}(s,y_{0}),$

where

$\displaystyle
H_{\mu/(l+1)}^{(l+1)}(s,y_{0})=\mathcal{K}_{s}^{\mu}\circ\tilde{R}_{\delta}^{(0,2)}(\hat{u}^{\eta_{-}}(s,y_{0}))(\hat{u}^{\eta_{-},(l)}(s,y_{0}),\hat{u}^{\eta_{-},(1)}(s,y_{0}))+DH_{\mu/l}^{(l)}(s,y_{0}).$

A similar argument as in the proof of the $l=k=1$ case shows that the unique
fixed point $\hat{u}^{\eta_{-},(l+1)}$ is also contained in
$\mathcal{L}^{l+1}(E_{0},\operatorname{BC}_{s}^{(l+1)\eta_{-}})$. Hence, the
map $\mathcal{J}_{s}^{\eta,\eta_{-}}\circ\hat{u}^{\eta_{-}}$ is of the class
$C^{l+1}$ provided that $\eta\in((l+1)\eta_{-},\eta_{+}]$ and $\delta>0$ is
sufficiently small. The map $\mathcal{C}:E_{0}\to\mathbb{R}^{n}$ from (12) is
$C^{k}$-smooth. Let $\eta\in[\eta_{-},\eta_{+}]\subset(0,\min\\{-a,b\\})$ such
that $k\eta_{-}<\eta_{+}$. Let $\operatorname{ev}_{s}$ denote the bounded
linear evolution operator (at time $s$) defined in the proof of prop:bundle.
Recall that
$\mathcal{C}(s,y_{0})=\hat{u}^{\eta}(s,y_{0})(s)=\operatorname{ev}_{s}(\hat{u}^{\eta}(s,y_{0})),$
and so
$\mathcal{C}(s,y_{0})=\operatorname{ev}_{s}(\mathcal{J}_{s}^{\eta,\eta_{-}}\hat{u}^{\eta_{-}}(s,y_{0}))$.
The result follows now from thm:smoothnesscmt.

To study in prop:bundle the tangent bundle of the center manifold, we have to
use the partial derivative of the map $\mathcal{C}$ in the second component.
The following result shows that such (higher order) partial derivatives are
uniformly Lipschitz continuous. For each $l\in\\{0,\dots,k\\}$, there exists a
constant $L(l)>0$ such that

$\|D_{2}^{l}\mathcal{C}(s,y_{0})-D_{2}^{l}\mathcal{C}(s,z_{0})\|\leq
L(l)\|y_{0}-z_{0}\|$

for all $(s,y_{0}),(s,z_{0})\in E_{0}$. For $l=0$, the result is already
proven in lemma:lipschitzCMT. Now let $l\in\\{1,\dots,k\\}$. Then, from the
proof of thm:smoothnesscmt we see that $\hat{u}^{\eta_{-},(l)}$ is the unique
solution of a fixed point problem, where the right hand-side is a contraction
with a Lipschitz constant $L(l)$ independent of $s$. Using the same strategy
as the proof of lemma:lipschitzCMT, we obtain the desired result.

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