The parameterization method for center manifolds

TL;DR

This paper extends the parameterization method to center manifolds at non-hyperbolic fixed points, providing a new proof of their existence and regularity, along with error bounds for approximations.

Contribution

It generalizes the parameterization method to non-hyperbolic cases, enabling simultaneous computation of the center manifold and its conjugate system with error estimates.

Findings

New proof of existence and regularity of center manifolds.

Error bounds for approximations of the center manifold.

Simultaneous computation of the center manifold and conjugate system.

Abstract

In this paper, we present a generalization of the parameterization method, introduced by Cabr\'{e}, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.

Equations

$$\displaystyle\begindc{\commdiag}[50]\obj(20,15)[1]{$X = X_c \oplus X_h$}\obj(60,15)[2]{$X = X_c \oplus X_h$}\par\obj(20,0)[3]{$X_c$}\obj(60,0)[4]{$X_c$}\par\mor{1}{2}{$F{}{} = A + g$}\mor{3}{1}{$K{}{} = \iota+ \left( \begin{subarray}{c}k_c \\ k_h\end{subarray} \right) $}[-1,0]\mor{4}{2}{$K{}{} = \iota+ \left( \begin{subarray}{c}k_c \\ k_h\end{subarray} \right) $}[1,0]\mor{3}{4}{$R = A_c + r$}\par\enddc$$

$$(A+g)\circ K=K\circ (A_c+r)$$

$$\Vert x\Vert =\underset{1\le i\le n}{\max }\left\{\Vert x_i{\Vert }_i\right\},$$

$$\Vert f{\Vert }_n:=\underset{0\le m\le n}{\max }\underset{x\in X}{\sup }\Vert D^{m}f(x)\Vert$$

$$C_b^n(X,Y)$$

$$\Vert A{\Vert }_{\text{op}}:=\underset{\Vert x\Vert =1}{\sup }\Vert Ax\Vert$$

$$\mathcal{L}(X,Y):=\left\{A:X\to Y\mid A\text{ is a linear operator and }\Vert A{\Vert }_{\text{op}}<\infty \right\}$$

$$\Vert A{\Vert }_{\text{op}}:=\underset{\begin{array}{c}\Vert x_i\Vert \le 1\\ 1\le i\le k\end{array}}{\sup }\Vert A(x_1,\dots ,x_k)\Vert .$$

$${\mathcal{L}}^k(X,Y):=\left\{A:X^k\to Y|A\text{ is a k-linear operator and ∥A∥op<∞}\right\}$$

$$D^k[f\circ g](x)=D^{k}f(g(x))(\underset{k\text{ times }Dg(x)}{\underbrace{Dg(x),\dots ,Dg(x)}})+\text{lower derivatives of }f.$$

$$A^{\otimes k}:=\underset{k\text{ times }A}{\underbrace{\left(A,\dots ,A\right)}}\in \mathcal{L}(X^k,Y^k)$$

$$D^k[f\circ g](x)=D^{k}f(g(x)){\left(Dg(x)\right)}^{\otimes k}+\text{lower derivatives of }f,$$

$$\Vert A_c^{-1}{\Vert }_{\text{op}}^{\tilde{n}}\Vert A_s{\Vert }_{\text{op}}<1$$

$$g$$

$$k_c$$

$$(A+g)\circ K=K\circ (A_c+r).$$

$$r$$

$$t$$

$$K=\iota +\left(\begin{array}{c}{k}_c\\ k_u\\ k_s\end{array}\right)$$

$$k_u$$

$$k_s$$

$$k_0\in \left\{h\in C_b^{m+1}(X_c,X_u\oplus X_s)|h(0)=0,Dh(0)=0\text{ and }\Vert h{\Vert }_{m+1}\le M\right\},$$

$$r_0\in \left\{h\in C_b^{m+1}(X_c,X_c)|h(0)=0,Dh(0)=0,\Vert Dh{\Vert }_0\le L_r\text{ and }\Vert h{\Vert }_{m+1}\le M\right\},$$

$${\Vert F\circ \left(\iota +\left(\begin{array}{c}{k}_c\\ k_0\end{array}\right)\right)-\left(\iota +\left(\begin{array}{c}{k}_c\\ k_0\end{array}\right)\right)\circ \left(A_c+r_0\right)\Vert }_m\le \epsilon ,$$

$$\Vert k-k_0{\Vert }_m\le C(M,m)\epsilon$$

$$L_r$$

$$L_t$$

$$L_u$$

$$L_s$$

$$\begin{array}{rl}{L}_c& <1,\\ L_r\Vert A_c^{-1}{\Vert }_{\text{op}}& <1,\end{array}$$

$${\theta }_{\tilde{n},1}$$

$${\theta }_{\tilde{n},2}$$

$${\theta }_{\tilde{n},3}$$

$$L_{-1}:=\Vert A_c^{-1}{\Vert }_{\text{op}}+L_t,$$

$${\theta }_{\tilde{n},1}$$

Full text

1 Introduction

Cabré, Fontich and De la Llave introduced the parameterization method for invariant manifolds of dynamical systems on Banach spaces in [Cabre03, Cabre03-2, Cabre05]. The goal of the parameterization method is to find a parameterization of the (un)stable manifold associated with an equilibrium point of the dynamical system. This parameterization is defined as a conjugacy between a dynamical system on the (un)stable eigenspace and the original dynamical system on the Banach space. To find a unique conjugacy, the dynamical system on the (un)stable eigenspace is fixed (it is given as a polynomial map). Besides providing a new proof of the classical (un)stable manifold theorem, the method is useful for computational existence proofs of for example homoclinic and heteroclinic orbits, see [VandenBerg11, Lessard14, VandenBerg15-2]. The method has also been used for constructing (un)stable manifolds of periodic orbits, see [Castelli18]. Recent computational advances include applications to delay differential equations, see [Groothedde17], and partial differential equations, see [Reinhardt19].

The goal of this paper is to generalize the parameterization method to center manifolds. As explained above, the proof of the classical parameterization method fixes the dynamics on the (un)stable eigenspace. Having explicit dynamics on the (un)stable eigenspace a priori ensures that the parameterization of the manifold is unique. For center manifolds, we generally cannot choose the dynamics on the center eigenspace ourselves. Thus besides solving for the conjugacy, we also need to solve for the dynamics on the center eigenspace. To ensure a unique parameterization, we fix part of the conjugacy instead of fixing the conjugate dynamics.

To state the main theorem of this paper, we introduce some notation. Let $F{}{}:X\to X$ be a $C^{n}$ (for $n\geq 2$) discrete time dynamical system on a Banach space $X$, with non-hyperbolic fixed point [math]. Denote $F{}{}(x)=Ax+g(x)$, with $A$ the Jacobian of $F{}{}$ at [math]. We assume that $A$ has a spectral gap around the unit circle, and write $X=X_{c}\oplus X_{h}$, where $X_{c}$ is the eigenspace associated with the eigenvalues on the unit circle, and $X_{h}$ the eigenspace associated with the eigenvalues away from the unit circle. We write $A=\left(\begin{smallmatrix}A_{c}&0\\ 0&A_{h}\\ \end{smallmatrix}\right)$ with respect to the splitting $X=X_{c}\oplus X_{h}$. Our goal is to find a conjugacy $K{}{}:X_{c}\to X$ and a dynamical system $R:X_{c}\to X_{c}$ such that the diagram

[TABLE]

commutes, where $\iota:X_{c}\to X_{c}\oplus X_{h}$ is the inclusion map. We impose that the conjugacy $K{}{}$ is tangent to $X_{c}$ at [math], i.e. $DK{}{}(0)=\left(\begin{smallmatrix}\operatorname{Id}\\ 0\\ \end{smallmatrix}\right)$, which implies that the linearization of $R$ at [math] is given by $A_{c}$. We also fix part of the conjugacy $K{}{}$ by an explicit choice of the non-linear map $k_{c}:X_{c}\to X_{c}$. Our result is the following.

Theorem** (Parameterization of the center manifold).**

If $g:X\to X$ and $k_{c}:X_{c}\to X_{c}$ are $C^{n}$, sufficiently small in the $C^{1}$ norm and bounded in $C^{2}$ norm, see Section 2 for the explicit bounds, then there exist a $C^{n}$ map $k_{h}:X_{c}\to X_{h}$ and a $C^{n}$ map $r:X_{c}\to X_{c}$ such that

[TABLE]

Furthermore, we have explicit bounds on the $C^{1}$ norms of $k_{h}$ and $r$ in terms of the spectral gap of $A$ and the $C^{1}$ norms of $g$ and $k_{c}$. These bounds are made precise in Section 2.

Despite $k_{h}$ and $r$ being $C^{n}$, we only give explicit bounds on their $C^{1}$ norm. We note that as $g$ is at least $C^{2}$, we may multiply $g$ as usual with a cut-off function $\xi$ such that $x\mapsto Ax+g(x)\xi(x)$ satisfies the conditions of Theorem 2.1. Then we find a local center manifold for $x\mapsto Ax+g(x)$ on the region where $\xi=1$, as well as local conjugate dynamics on the center eigenspace.

Our theorem provides a new proof of the classical center manifold theorem, see [Carr82, Vanderbauwhede89, Schinas92]. The classical proof of the center manifold theorem is based on the variation of constants formula. Our proof is more elementary. It simultaneously finds $k_{h}$ and $r$ as fixed point of an operator on a function space, which is a contraction on an appropriately chosen subset. The freedom to choose $k_{c}\neq 0$ moreover allows us to obtain the dynamics on the center eigenspace directly in a desired normal form.

As we mentioned in the beginning, our work is inspired by the parameterization method for invariant (un)stable manifolds in [Cabre03, Cabre03-2, Cabre05], which we shall refer to as the classical parameterization method. The main difference between the classical and the proposed parameterization method is the treatment of the conjugate dynamics $R$. The classical parameterization method fixes a polynomial map $R:X_{u/s}\to X_{u/s}$ which determines the dynamical system on the (un)stable eigenspace, whereas our method produces a $C^{n}$ map $R:X_{c}\to X_{c}$.

It follows from [Farres10] that one can find Taylor approximations $R_{0}$ and $K{}{}_{0}$ of the maps $R$ and $K{}{}$ up to any finite order. These $R_{0}$ and $K{}{}_{0}$, will then almost satisfy the conjugacy equation, i.e. $F{}{}\circ K{}{}_{0}-K{}{}_{0}\circ R_{0}$ will be of order $\|x\|^{m}$. This will imply that the real solution of the conjugacy equation lies very close to $R_{0}$ and $K{}{}_{0}$. More precisely, we can find rigorous bounds for $R-R_{0}$ and $K{}{}-K{}{}_{0}$. Hence, from the Taylor approximations we can extract very detailed information of the true dynamical behavior on the center manifold.

Another generalization of the classical parameterization method concerning center manifolds can be found in [Baldoma16, Baldoma16-2]. This generalization finds certain submanifolds of the center manifold at parabolic fixed points. A fixed point is called parabolic if the linearization of the dynamical system is the identity. As in the classical parameterization method, the conjugate vector field is polynomial in this case, and the proof in [Baldoma16, Baldoma16-2] follows roughly the same analysis as the proof of the classical parameterization method. A drawback of this result is that it produces invariant manifolds which are only continuous at the origin. Furthermore, it imposes that the linearization of the dynamical system at the fixed point is the identity on the center eigenspace. Our theorem on the other hand produces invariant manifold which are everywhere $C^{n}$ and does not impose that the linearization restricted to the center eigenspace is the identity.

Outline of the paper

Our paper consists of three parts. We start by introducing notation in the following section and by giving a more precise formulation of the main theorem in Section 2. As a corollary of the main theorem, we find an upper bound between the error of an almost solution $\mathcal{M}$ of 1.0.1 and the actual solution $\Lambda$ of 1.0.1 in terms of how well $\mathcal{M}$ solves 1.0.1. Furthermore, we introduce a fixed point problem in Section 3. This fixed point problem will produce the conjugacy and conjugate dynamics of Theorem 2.1. Finally, we will prove that the fixed point problem defines a contraction in $C^{1}$.

In the second part of our paper, which consists of Sections 4, 5 and 6, we will obtain the smoothness results of the main theorem by means of two bootstrapping arguments. The first bootstrapping argument, in Section 4, shows that once we have a $C^{1}$ conjugacy and conjugate dynamics, they are in fact both $C^{2}$. The second argument, in Section 5, shows that we can inductively increase the smoothness from $C^{2}$ to $C^{n}$. Finally, we prove in Section 6 that our center manifold is unique, and we derive the estimate on $\Lambda-\mathcal{M}$.

Future work

Our proposed method is defined for general discrete time dynamical systems. Although it is not proven in this paper, we are convinced that the method can be generalized to continuous time dynamical systems. Furthermore, our method can be used to parameterize different manifolds than only the center manifold, for instance the center-(un)stable or (un)stable manifold. In the latter case, our method should even work if the (un)stable manifold contains infinitely many resonances.

1.1 Notation and conventions

We use the following notation and conventions in this paper.

• –

For a Banach space $X=\bigoplus_{i=1}^{n}X_{i}$, we assume that the norm on $X$ satisfies

[TABLE]

where $x_{i}\in X_{i}$ and $\|\cdot\|_{i}$ the norm on $X_{i}$. If $X$ is equipped with another norm, we can always define an equivalent norm on $X$ which satisfies 1.1.1 and leaves the norm unchanged on $X_{i}$.

• –

For functions $f:X\to Y$ between Banach spaces, we denote with

[TABLE]

the $C^{n}$ norm of $f$ for $n\geq 0$.

For $X$ and $Y$ Banach spaces, we denote with

[TABLE]

the Banach space of all $C^{n}$ bounded functions between $X$ and $Y$.

• –

For a linear operator $A:X\to Y$ between Banach spaces, we denote with

[TABLE]

the operator norm of $A$.

For $X$ and $Y$ Banach spaces, we denote with

[TABLE]

the Banach space of all bounded linear operators between $X$ and $Y$.

• –

For a $k$-linear operator $A:X^{k}\to Y$ between Banach spaces, we denote with

[TABLE]

the operator norm of $A$.

For $X$ and $Y$ Banach spaces, we denote with

[TABLE]

the Banach space of all bounded $k$-linear operators between $X^{k}$ and $Y$.

• –

For Banach spaces $X$, $Y$ and $Z$ and $f\in C^{n}_{b}(Y,Z)$ and $g\in C^{n}_{b}(X,Y)$, the $k^{\text{th}}$ derivative of $f\circ g$ is given by

[TABLE]

We can identify the $k^{\text{th}}$ derivative of $f$ at $g(x)$ with a symmetric $k$-linear operator between $Y^{k}$ and $Z$. This motivates us to define the shorthand notation

[TABLE]

Hence we may rewrite 1.1.2 as

[TABLE]

where $D^{k}f(g(x))\in\mathcal{L}^{k}(Y,Z)$.

2 A quantitative formulation of the parameterized center manifold theorem

We will now formulate the quantitative version of the parameterization of the center manifold theorem.

Theorem 2.1** (Parameterization of the center manifold).**

Let $X$ be a Banach space and $F{}{}:X\to X$ a $C^{n}$, $n\geq 2$, discrete dynamical system on $X$ such that [math] is a fixed point of $F{}{}$. Denote $F{}{}=A+g$ with $A:=Df(0)$ and let $k_{c}:X_{c}\to X_{c}$ be chosen. Assume that

There exist closed $A$-invariant subspaces $X_{c}$, $X_{u}$ and $X_{s}$ such that $X=X_{c}\oplus X_{u}\oplus X_{s}$. We write $A=\left(\begin{smallmatrix}A_{c}&0&0\\ 0&A_{u}&0\\ 0&0&A_{s}\end{smallmatrix}\right)$ where we define A_{c}:=A\big{|}_{X_{c}}, and similarly define $A_{u}$ and $A_{s}$. 2. 2.

The linear operators $A_{c}$ and $A_{u}$ are invertible. 3. 3.

The norm on $X$ is such that

[TABLE] 4. 4.

The non-linearities $g$ and $k_{c}$ satisfy

[TABLE]

for $L_{g}$ and $L_{c}$ small enough; see Remark 2.4 for the explicit inequalities that $L_{g}$ and $L_{c}$ should satisfy.

Then there exist a $C^{n}$ conjugacy $K{}{}:X_{c}\to X$ and $C^{n}$ discrete dynamical system $R=A_{c}+r:X_{c}\to X_{c}$ such that

[TABLE]

Furthermore, $K{}{}$ and $R$ have the following properties:

1. A)

The dynamical system $R=A_{c}+r$ is globally invertible, with its inverse given by $T=A_{c}^{-1}+t$, where

[TABLE]

The constants $L_{r}$ and $L_{t}$ depend on $L_{g}$ and $L_{c}$. See Remark 2.4 for their definition. 2. B)

The conjugacy $K{}{}$ is given by

[TABLE]

with $\iota:X_{c}\to X$ the inclusion map and

[TABLE]

The constants $L_{u}$ and $L_{s}$ depend on $L_{g}$ and $L_{c}$. See Remark 2.4 for their definition.

Remark 2.2*.*

If $A$ has a spectral gap around the unit circle, conditions 1 and 2 hold when we take $X_{c}$ to be the center eigenspace, $X_{u}$ to be the unstable eigenspace and $X_{s}$ to be the stable eigenspace. Furthermore, in this case there exists a norm on $X$ such that condition 3 holds. For the construction of this norm, see Proposition A.1 of [Cabre03].

In many practical situations, using for example numerics or Taylor approximations, one may be able to find $K{}{}_{0}$ and $R_{0}$ which almost solve 2.0.1, i.e. for which $F{}{}\circ K{}{}_{0}-K{}{}_{0}\circ R_{0}$ is small. In other instances, the dynamical system $F{}{}:X\to X$ may be close to another dynamical system $F{}{}_{0}:X\to X$, for which we are able to find a $K{}{}_{0}$ and $R_{0}$ satisfying $F{}{}_{0}\circ K{}{}_{0}-K{}{}_{0}\circ R_{0}=0$ exactly. In both cases, we are interested in estimating the error between $K{}{}_{0}$ and the true conjugacy $K{}{}$, as well as the error between $R_{0}$ and the true dynamical system $R$. The following corollary of Theorem 2.1 gives bounds for these errors.

Corollary 2.3**.**

Let $n\geq 2$ and let $F{}{}:X\to X$ and $k_{c}:X_{c}\to X_{c}$ be $C^{n}$ functions which satisfy the conditions of Theorem 2.1. Let $\varepsilon>0$, $m<n$ and $M>0$. Then there exists a constant $\mathcal{C}(M,m)$, which will be introduced in Section 6.3, such that if

[TABLE]

then we have

[TABLE]

for $k=k_{u}\oplus k_{s}:X_{c}\to X_{u}\oplus X_{s}$ and $r:X_{c}\to X_{c}$ from Theorem 2.1.

Remark 2.4*.*

To state what it means in Theorem 2.1 for $L_{g}$ and $L_{c}$ to be small enough, we first introduce explicit formulas for $L_{r}$, $L_{t}$, $L_{u}$ and $L_{s}$:

[TABLE]

For $L_{r}$, $L_{t}$, $L_{u}$, and $L_{s}$ to be positive, their denominators have to be positive. So we will first of all require $L_{g}$ and $L_{c}$ to be so small that

[TABLE]

Besides these conditions on $L_{g}$ and $L_{c}$, the proof of Theorem 2.1 contains multiple fixed points arguments, with contraction constants depending on $L_{g}$ and $L_{c}$. The corresponding contraction constants are, for $0\leq\tilde{n}\leq n$,

[TABLE]

We note that $\theta_{\tilde{n},1}$ does not depend on $\tilde{n}$, but for consistency we choose to keep the index $\tilde{n}$. Furthermore, we define

[TABLE]

which will act as a bound on $\|D(A_{c}+r)^{-1}\|_{0}$. Now, if $L_{g}$ and $L_{c}$ are small, then we have that $\theta_{\tilde{n},i}<1$. In particular, we require $L_{g}$ and $L_{c}$ to be so small that indeed

[TABLE]

We will often refer to this remark by writing “(…) small in the sense of Remark 2.4 for $n=m$.”. By this, we mean that $\theta_{\tilde{n},i}<1$ holds for all $\tilde{n}\leq m$ and $i=1,2,3$.

2.1 Proof scheme of Theorem 2.1

To prove Theorem 2.1 we use four steps:

We define a fixed point operator and show that its fixed points are solutions to the conjugacy equation 2.0.1. 2. 2.

We show that the fixed point operator is a contraction with respect to the $C^{1}$ norm on an appropriate set of $C^{2}$ functions. Therefore, we find a pair of $C^{1}$ functions $(K{}{},r)$ such that equation 2.0.1 holds. 3. 3.

We show that the $C^{1}$ solution $(K{}{},r)$ of equation 2.0.1 is in fact $C^{2}$. 4. 4.

Finally, we use induction to prove that $K{}{}$ and $r$ are $C^{n}$.

We find it easiest to split the proof in the four steps outlined above. We note that in step 2 we find a contraction with respect to the $C^{1}$ norm in a space of $C^{2}$ functions, which is the reason that we assume $n\geq 2$ in Theorem 2.1. Furthermore, going from a $C^{1}$ solution to a $C^{2}$ solution needs different estimates than when going from a $C^{m}$ solution to a $C^{m+1}$ solution for $m\geq 2$, see for instance Lemma 4.3 and Lemma 5.6.

3 A $C^{1}$ center manifold

3.1 A fixed point operator

We start by setting up a fixed point operator. For this, we consider the conjugacy equation

[TABLE]

When we write $g=\left(\begin{smallmatrix}g_{c}\\ g_{u}\\ g_{s}\end{smallmatrix}\right)$ and $K{}{}=\iota+\left(\begin{smallmatrix}k_{c}\\ k_{u}\\ k_{s}\end{smallmatrix}\right)$, then we have component-wise

[TABLE]

In other words, we obtain the three equations

[TABLE]

The first equation is equivalent to

[TABLE]

Secondly, as $A_{u}$ is invertible, the second equation is equivalent to

[TABLE]

Finally, if we assume that $A_{c}+r$ is invertible, the last equation is equivalent to

[TABLE]

We see that the system 3.1.1 is equivalent with

[TABLE]

Thus $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)$ is a fixed point of the operator $\Theta$, defined by

[TABLE]

The following proposition summarizes this derivation.

Proposition 3.1**.**

Let $K{}{}=\iota+\left(\begin{smallmatrix}k_{c}\\ k_{u}\\ k_{s}\end{smallmatrix}\right):X_{c}\to X$ and $A_{c}+r:X_{c}\to X_{c}$. If $A_{c}+r$ is globally invertible, then the following are equivalent:

1. i)

The dynamical system $F{}{}$ is conjugate to $A_{c}+r$ by $K{}{}$. 2. ii)

The function $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right):X_{c}\to X$ satisfies the system 3.1.2. 3. iii)

The function $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right):X_{c}\to X$ is a fixed point of $\Theta$ defined in 3.1.3.

Proof.

This follows from the derivation above. ∎

3.2 $\Theta$ is well-defined

One of the conditions of Proposition 3.1 is that $A_{c}+r$ is globally invertible. Under a mild condition on the derivative of $r$, we show that $A_{c}+r$ is a global diffeomorphism. We will state a more general result, as we need a more general statement in Proposition 6.3. Furthermore, we note that the result also follows from theorem 2.1 of [Plastock74], which states that under some growth condition on $A_{c}+r$ and the inverse of its derivative, $A_{c}+r$ is a global diffeomorphism. However, we will provide an alternative proof, as this proof is exemplary for the structure of the proof of Theorem 2.1.

Lemma 3.2**.**

Let $B\in\mathcal{L}(X_{c},X_{c})$ be invertible.

1. i)

We have that $B+\psi:X_{c}\to X_{c}$ is a global diffeomorphism for all $\psi\in C_{b}^{1}(X_{c},X_{c})$ with $\|D\psi\|_{0}<\|B^{-1}\|_{\textup{op}}^{-1}$. 2. ii)

If $B+\psi:X_{c}\to X_{c}$ is a homeomorphism and $\psi\in C_{b}^{0}(X_{c},X_{c})$, then $(B+\psi)^{-1}=B^{-1}+\varphi$ with $\varphi\in C_{b}^{0}(X_{c},X_{c})$.

Proof.

i) We will first show that $B+\psi$ is globally invertible, and use its inverse to prove that it is in fact a diffeomorphism, which is similar to how we show the smoothness result of Theorem 2.1.

For $\varphi\in C_{b}^{0}(X_{c},X_{c})$, we have $(B+\psi)\circ(B^{-1}+\varphi)=\operatorname{Id}$ if and only if

[TABLE]

Our first goal is to show that $\Psi$ is a contraction on $C^{0}_{b}(X_{c},X_{c})$.

As $\psi$ is $C^{1}$ with bounded derivative, it is Lipschitz with Lipschitz constant equal to the norm of its derivative. Thus for $\varphi,\tilde{\varphi}\in C^{0}_{b}(X_{c},X_{c})$ we have

[TABLE]

Thus with $\|D\psi\|_{0}<\|B^{-1}\|_{\textup{op}}^{-1}$ we have that $\Psi$ is contraction operator. Let $\varphi$ denote the fixed point of $\Psi$.

If $\varphi$ is $C^{1}$, we can differentiate 3.2.1 to $x$, and the derivative of $\varphi$ satisfies

[TABLE]

In other words, if $\varphi$ is $C^{1}$, then $D\varphi$ is a fixed point of the operator

[TABLE]

We will show that $\Phi$ is a contraction, and that its fixed point is indeed the derivative of $\varphi$.

As $\|B^{-1}D\psi(B^{-1}+\varphi)\|_{0}\leq\|B\|_{\textup{op}}\|D\psi\|_{0}<1$, we see that $\Phi$ is contraction, and we denote the fixed point of $\Phi$ by $\mathcal{A}$. All that is left to show is that $\mathcal{A}$ is the derivative of $\varphi$. We have

[TABLE]

where we define $z(s,x,y):=B^{-1}x+sB^{-1}y+s\varphi(x+y)+(1-s)\varphi(x)$. We find, as $\phi$ is the fixed point of $\Psi$,

[TABLE]

Define $\theta=\|B^{-1}\|_{\textup{op}}\int_{0}^{1}\|D\psi(z(s,x,y))\|\text{d}s$, then $\theta\leq\|B^{-1}\|_{\textup{op}}\|D\psi\|_{0}<1$. Hence 3.2.3 implies

[TABLE]

Furthermore, $\|D\psi\|_{0}$ is finite, hence by using the Lebesgue Dominated Convergence Theorem, we can interchange limits and integrals in

[TABLE]

Using continuity of $\|D\psi(z(s,x,y))\|_{\textup{op}}$ in $y$, the limit is [math]. Therefore,

[TABLE]

and we conclude that the right inverse $B^{-1}+\varphi$ of $B+\psi$ is $C^{1}$.

We still need to show that $B^{-1}+\varphi$ is the inverse of $B+\psi$, or equivalently show that $B+\psi$ is injective. Let $x,y\in X_{c}$ be such that $Bx+\psi(x)=By+\psi(y)$, then

[TABLE]

As $\|B^{-1}\|_{\textup{op}}^{-1}-\|D\psi\|_{0}>0$, we must have $\|x-y\|=0$, and thus $x=y$. With this, we have shown that $B+\psi$ is injective, and thus invertible.

ii) We want to show that for all bounded continuous $\psi:X_{c}\to X_{c}$ such that $B+\psi$ is a homeomorphism, $\varphi:=B^{-1}-(B+\psi)^{-1}$ is uniformly bounded. We have

[TABLE]

From the above lemma it follows that if $r:X_{c}\to X_{c}$ is $C^{1}$ and its derivative $Dr$ is uniformly bounded by $\|A_{c}^{-1}\|_{\textup{op}}^{-1}$, then there exists a $C^{1}$ bounded function $t:X_{c}\to X_{c}$ such that $(A_{c}+r)^{-1}=A_{c}^{-1}+t$. Furthermore, from equation 3.2.2 it follows that $\|Dt\|_{0}$ satisfies

[TABLE]

In particular, we claim in Theorem 2.1 that the dynamical system $A_{c}+r$ will be invertible with inverse $A_{c}^{-1}+t$. Furthermore, we claim that we have the bounds $\|Dr\|_{0}\leq L_{r}$ and $\|Dt\|_{0}\leq L_{t}$, with $L_{r}$ and $L_{t}$ defined in 2.0.2a and 2.0.2b respectively. Since $L_{r}\leq\|A_{c}^{-1}\|_{\textup{op}}^{-1}$, it follows from the lemma that if $\|Dr\|_{0}\leq L_{r}$, we have that $A_{c}+r$ is invertible, and from the definition of $L_{t}$ that $\|Dt\|_{0}\leq L_{t}$. In particular, $\Theta$ is well-defined if we assume $\|Dr\|_{0}\leq L_{r}$ and the desired properties of $R=A_{c}+r$ follow from the above discussion.

3.3 A first invariant set for $\Theta$

We want to find an invariant set for $\Theta$ by putting additional bounds on $\|Dr\|_{0}$, $\|Dk_{u}\|_{0}$ and $\|Dk_{s}\|_{0}$. To this end, we define

[TABLE]

Theorem 3.3**.**

Assume that $L_{g}$ and $L_{c}$ are small in the sense of Remark 2.4 for $n=2$. Then $\Theta$ is well-defined on $\Gamma_{0}$. Furthermore, $\Gamma_{0}$ is non-empty, invariant under $\Theta$, and closed.

Proof.

We first note that by Remark 2.4 it holds that $0\in\Gamma_{0}$ as $L_{r}$, $L_{u}$ and $L_{s}$ are all positive. Furthermore, from Remark 2.4 it follows that $L_{r}<\|A_{c}^{-1}\|_{\textup{op}}^{-1}$. By Section 3.2, we thus have that $A_{c}+r$ is a global diffeomorphism for $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$, hence $\Theta$ is well defined on $\Gamma_{0}$. Finally, it follows directly from the definition of $\Gamma_{0}$ that $\Gamma_{0}$ is closed. All that remains to prove is that $\Gamma_{0}$ is invariant under $\Theta$.

Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$. Then we must prove that $\Theta(\Lambda)=\left(\begin{smallmatrix}\Theta_{1}(\Lambda)\\ \Theta_{2}(\Lambda)\\ \Theta_{3}(\Lambda)\end{smallmatrix}\right)\in\Gamma_{0}$. We will prove the resulting conditions on $\Theta(\Lambda)$ component-wise. We start by showing $\Theta(\Lambda)(0)=0$. We have

[TABLE]

With $\Lambda\in\Gamma_{0}$, we have $r(0)=0$, thus $(A_{c}+r)(0)=0$. So we get

[TABLE]

By assumption 4 of Theorem 2.1 we have $k_{c}(0)=0$. Together with $k_{u}(0)=k_{s}(0)=0$ we find $K{}{}(0)=\iota(0)+\left(\begin{smallmatrix}k_{c}\\ k_{u}\\ k_{s}\end{smallmatrix}\right)(0)=0$. Therefore

[TABLE]

Finally, again by assumption 4 of Theorem 2.1, we have $g_{c}(0)=0$ and we conclude that

[TABLE]

Similarly, $\Theta_{2}(\Lambda)(0)=0$ and $\Theta_{3}(\Lambda)(0)=0$. For the latter equality, we note that $A_{c}+r$ is a diffeomorphism and $(A_{c}+r)(0)=0$, thus $(A_{c}+r)^{-1}(0)=0$. So we find that $\Theta(\Lambda)(0)=0$.

Likewise, we show that $D[\Theta(\Lambda)](0)=0$ component-wise. For example, we have

[TABLE]

By assumption 4 of Theorem 2.1 we have $Dg_{c}(0)=0$ and $Dk_{c}(0)=0$. So we find $D[\Theta_{1}(\Lambda)](0)=0$. Similarly, we find $D[\Theta_{2}(\Lambda)](0)=0$ and $D[\Theta_{3}(\Lambda)](0)=0$. So we see that $D[\Theta(\Lambda)](0)=0$.

For the bounds on the derivatives of the components of $\Theta(\Lambda)$, we use the shorthand notation $R=A_{c}+r$. For the first component we have

[TABLE]

We have by assumption 4 of Theorem 2.1

[TABLE]

Furthermore, we may estimate

[TABLE]

Now note that $1+\|Dk_{c}\|_{0}\leq 1+L_{c}$, $\|Dk_{u}\|_{0}\leq L_{u}$ and $\|Dk_{s}\|_{0}\leq L_{s}$. From the definition of $L_{u}$ in 2.0.2c, we see that $L_{u}\leq 1+L_{c}$ if

[TABLE]

The latter inequality holds since $L_{u}\geq 0$ and it is assumed that $\theta_{1,2}<1$. This holds as $0\leq L_{u}$ and $\theta_{1,2}<1$. Likewise, we can bound $L_{s}\leq 1+L_{c}$ as $\theta_{1,3}<1$. Thus we find

[TABLE]

With this estimate, we get

[TABLE]

Finally, we have

[TABLE]

By combining 3.3.2, LABEL:, 3.3.4, LABEL: and 3.3.5 we obtain from 3.3.1 that

[TABLE]

The inequalities $\|D[\Theta_{2}(\Lambda)]\|_{0}\leq L_{u}$ and $\|D[\Theta_{3}(\Lambda)]\|_{0}\leq L_{s}$ follow from similar calculations.

Thus we see that $\Theta$ leaves $\Gamma_{0}$ invariant. ∎

3.4 A second invariant set for $\Theta$

We would now like to prove that $\Theta:\Gamma_{0}\to\Gamma_{0}$ is a contraction with respect to the $C^{1}$ norm, i.e. we must show that $\|\Theta(\Lambda)-\Theta(\tilde{\Lambda})\|_{1}\leq c\|\Lambda-\tilde{\Lambda}\|_{1}$ for all $\Lambda,\tilde{\Lambda}\in\Gamma_{0}$ and some $c<1$. We will need to restrict to a subset of $\Gamma_{0}$ to obtain such a bound. To motivate our choice of the subset, let us look more carefully at the second component of $\Theta$ and focus on the bound on its derivative. For instance, one obtains an estimate of the form

[TABLE]

If we now estimate the second factor of the first term, we obtain

[TABLE]

We can bound the first term in the right hand side by

[TABLE]

But, to estimate $\|Dk_{u}(A_{c}+r)-Dk_{u}(A_{c}+\tilde{r})\|_{0}$ in terms of $\|r-\tilde{r}\|_{1}$ we would need a uniform bound on the Lipschitz constant of $Dk_{u}$ for $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$.

Therefore, we will restrict $\Theta$ to a subset of $\Gamma_{0}$ consisting of once differentiable functions with uniform bound on the Lipschitz constant of the derivative. In fact, we will restrict $\Theta$ to a subset of twice differentiable functions with uniform bound on the supremum norm of the second derivative. We make this choice rather than working with $C^{1}$ function with a Lipschitz bound on the derivative, because we later want to show that $r$, $k_{u}$ and $k_{s}$ are $C^{n}$. We thus define for $\delta>0$ the space of $C^{2}$ functions

[TABLE]

In Theorem 2.1, we assume that $k_{c}$ and $g$ have bounded second derivative. If we take $\|D^{2}k_{c}\|_{0}\leq\varepsilon$ and $\|D^{2}g\|_{0}\leq\varepsilon$ for some positive $\varepsilon$, then we will be able to construct an explicit $\delta(\varepsilon)$ such that $\Gamma_{1}(\delta(\varepsilon))$ is invariant under $\Theta$.

Proposition 3.4**.**

Let $\varepsilon>0$ and assume that $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$. Furthermore, assume that $L_{g}$ and $L_{c}$ are small in the sense of Remark 2.4 for $n=2$. Then there exists a $\delta(\varepsilon)>0$, which we explicitly define in 3.4.12, such that $\Gamma_{1}(\delta(\varepsilon))$ is invariant under $\Theta$. Furthermore, $\delta(\varepsilon)\downarrow 0$ as $\varepsilon\downarrow 0$.

Proof.

Let $\varepsilon>0$. We want to find $\delta(\varepsilon)>0$ such that $\Gamma_{1}(\delta(\varepsilon))$ is invariant under $\Theta$. Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{1}(\delta)$, and introduce the shorthand notation $R=(A_{c}+r)$ and $T=(A_{c}+r)^{-1}$. We have to show that $\|D^{2}[\Theta_{i}(\Lambda)]\|_{0}\leq\delta$ for $i=1,2,3$.

We estimate the first component by

[TABLE]

We estimate the terms separately. First,

[TABLE]

Second, as $\|D^{2}K{}{}\|_{0}\leq\max\{\|D^{2}k_{c}\|_{0},\|D^{2}k_{u}\|_{0},\|D^{2}k_{s}\|_{0}\}\leq\max\{\varepsilon,\delta\}\leq\varepsilon+\delta$, where we choose the rough estimate of $\varepsilon+\delta$ in order to write $\delta(\varepsilon)$ explicitly in 3.4.12, we find

[TABLE]

Finally, we have

[TABLE]

So inequality 3.4.2 together with estimates 3.4.3, LABEL:, 3.4.4, LABEL: and 3.4.5 gives

[TABLE]

where $C_{1}(\varepsilon):=\left(\|A_{c}\|_{\textup{op}}+\left(1+L_{c}\right)^{2}+L_{g}+(\|A_{c}\|_{\textup{op}}+L_{r})^{2}\right)\varepsilon$.

Likewise, we estimate the second component by

[TABLE]

As before, we estimate the terms separately to find

[TABLE]

Hence, inequality 3.4.7 becomes

[TABLE]

where $C_{2}(\varepsilon):=\|A_{u}^{-1}\|_{\textup{op}}((1+L_{c})^{2}+L_{g})\varepsilon$.

Finally, we estimate the third component by

[TABLE]

Before we estimate the different parts of the right hand side of 3.4.9, we estimate $\|D^{2}T\|_{0}$ by applying the chain rule twice to the right hand side of $0=D^{2}\operatorname{Id}=D^{2}[R\circ T]$.

[TABLE]

We let $DR(T)^{-1}=DT$ act on the left to obtain the upper bound

[TABLE]

We now find the following estimates for the terms in 3.4.9:

[TABLE]

Inequality 3.4.9 thus becomes

[TABLE]

where $C_{3}(\varepsilon):=L_{-1}^{2}((1+L_{c})^{2}+L_{g})\varepsilon$.

If we show that $\delta(\varepsilon)>0$ can be chosen such that

[TABLE]

then we can estimate inequalities 3.4.6, LABEL:, 3.4.8, LABEL: and 3.4.11 by $\delta(\varepsilon)$ which shows that $\Gamma_{1}(\delta(\varepsilon))$ is invariant under $\Theta$. By assumption, we have $\theta_{2,i}<1$ for $i=1,2,3$. Therefore we can define

[TABLE]

This gives for $i=1,2,3$

[TABLE]

Furthermore, we have by construction that $\delta(\varepsilon)\downarrow 0$ when $\varepsilon\downarrow 0$, since $C_{i}(\varepsilon)\downarrow 0$ as $\varepsilon\downarrow 0$ for $i=1,2,3$. ∎

3.5 Estimates for compositions

We already mentioned before Proposition 3.4 that we have to estimate expressions such as $\|D[k_{u}\circ(A_{c}+r)]-D[\tilde{k}_{u}\circ(A_{c}+\tilde{r})]\|_{0}$ in terms of $\|k_{u}-\tilde{k}_{u}\|_{1}$ and $\|r-\tilde{r}\|_{1}$ to show that $\Theta$ is a contraction with respect to the $C^{1}$ norm. Furthermore, we briefly showed part of the steps we would take to achieve the desired estimate. However, to show that $\Theta$ is a contraction with respect to the $C^{1}$ norm, we must also estimate expressions such as $k_{u}\circ(A_{c}+r)-\tilde{k}_{u}\circ(A_{c}+\tilde{r})$ in terms of $\|k_{u}-\tilde{k}_{u}\|_{1}$ and $\|r-\tilde{r}\|_{1}$. We will therefore provide a general result which allows use to bound both $\|k_{u}\circ(A_{c}+r)-\tilde{k}_{u}\circ(A_{c}+\tilde{r})\|_{0}$ and $\|D[k_{u}\circ(A_{c}+r)]-D[\tilde{k}_{u}\circ(A_{c}+\tilde{r})]\|_{0}$, and the corresponding expressions in the first and third component of $\Theta(\Lambda)-\Theta(\tilde{\Lambda})$. As we did in the definition of $\Gamma_{1}$, we prefer to work with twice differentiable functions instead of differentiable functions with Lipschitz first derivative.

Lemma 3.5**.**

Let $X$, $Y$ and $Z$ be Banach spaces.

1. i)

For $f_{1}\in C_{b}^{0}(Y,Z)$, $g_{1}\in C_{b}^{1}(Y,Z)$, $f_{2},g_{2}\in C_{b}^{0}(X,Y)$ we have the $C^{0}$-estimate

[TABLE] 2. ii)

For $f_{1},g_{1}\in C_{b}^{2}(Y,Z)$, $f_{2},g_{2}\in C_{b}^{1}(X,Y)$ we have the $C^{1}$-estimate

[TABLE]

Proof.

i) We find by the Mean Value Theorem

[TABLE]

ii) We use the chain rule and triangle inequality to find

[TABLE]

We estimate the first term using the submultiplicativity of the supremum norm:

[TABLE]

For the second term, we again use submultiplicativity of the norm and estimate the factor $Df_{1}(f_{2})-Dg_{1}(g_{2})$ with part i) of this lemma:

[TABLE]

When we estimate the third component of $\Theta(\Lambda)-\Theta(\tilde{\Lambda})$ with the previous lemma, we get an estimate involving $\|(A_{c}+r)^{-1}-(A_{c}-\tilde{r})^{-1}\|_{i}$ instead of $\|r-\tilde{r}\|_{i}$ for $i=0,1$. So if we want to use the previous lemma to estimate $\Theta(\Lambda)-\Theta(\tilde{\Lambda})$ by $\|\Lambda-\tilde{\Lambda}\|_{1}$, then we must find an estimate for $\|(A_{c}+r)^{-1}-(A_{c}+\tilde{r})^{-1}\|_{0}$ and $\|D(A_{c}+r)^{-1}-D(A_{c}+\tilde{r})^{-1}\|_{0}$ in terms of $\|r-\tilde{r}\|_{0}$ and $\|Dr-D\tilde{r}\|_{0}$.

Lemma 3.6**.**

Let $r_{1},r_{2}\in C_{b}^{2}(X_{c},X_{c})$ be such that $A_{c}+r_{i}$ is a diffeomorphism with $D(A_{c}+r_{i})^{-1}\in C_{b}^{1}(X_{c},\mathcal{L}(X_{c},X_{c}))$ for $i=1,2$.

1. i)

We have the $C^{0}$-estimate

[TABLE] 2. ii)

We have the $C^{1}$-estimate

[TABLE]

Proof.

i) We denote $R_{i}=A_{c}+r_{i}$ for $i=1,2$. Lemma 3.5i) implies that

[TABLE]

ii) For the $C^{1}$-estimate we denote $T_{i}=DR_{i}^{-1}\in C_{b}^{1}(X_{c},\mathcal{L}(X_{c},X_{c}))$ for $i=1,2$. Let $x\in X_{c}$, then we know that $T_{i}(x)\in\mathcal{L}(X_{c},X_{c})$ is invertible for $i=1,2$ by the Inverse Function Theorem. We find

[TABLE]

Furthermore, the Inverse Function Theorem allows us to rewrite

[TABLE]

Taking the supremum over $x\in X_{c}$ gives

[TABLE]

By using Lemma 3.5i) we find

[TABLE]

Then we use part i) of this lemma applied to $\|R_{1}^{-1}-R_{2}^{-1}\|_{0}$ to obtain

[TABLE]

The result now follows from taking the supremum of $x\in X_{c}$ in inequality 3.5.1 and using the above estimate to bound $\sup_{x\in X_{c}}\|T_{2}(x)^{-1}-T_{1}(x)^{-1}\|_{\textup{op}}$. ∎

To bound $\Theta(\Lambda)-\Theta(\tilde{\Lambda})$ by $\|\Lambda-\tilde{\Lambda}\|_{1}$, we can use Lemma 3.5 for the first two components. For the third component, we can use the same lemma together with Lemma 3.6 for the desired bound. However, while the first and second component consists of a single composition, the third component contains a double composition.

Lemma 3.7**.**

Let $r,\tilde{r}\in C_{b}^{2}(X_{c},X_{c})$ such that $\|Dr\|_{0},\|D\tilde{r}\|_{0}\leq L_{r}$ and assume that $L_{r}\leq\|A_{c}^{-1}\|_{\textup{op}}^{-1}$. Furthermore, let $X$ and $Y$ be Banach spaces with functions $h\in C^{2}_{b}(Y,X)$ and $f_{1},f_{2}\in C^{2}_{b}(X_{c},Y)$.

1. i)

We have the $C^{0}$-estimate

[TABLE] 2. ii)

If $\|D^{2}\tilde{r}\|_{0}\leq\delta(\varepsilon)$ for some $\varepsilon>0$, then we have the $C^{1}$-estimate

[TABLE]

Proof.

i) For the $C^{0}$-estimate, we first use Lemma 3.5i) twice:

[TABLE]

Then we use Lemma 3.6i) to estimate $(A_{c}+r)^{-1}-(A_{c}+\tilde{r})^{-1}$:

[TABLE]

Recall from Remark 2.4 that $L_{r}<\|A_{c}^{-1}\|_{\textup{op}}^{-1}$ implies that $\|D(A_{c}+r)^{-1}\|_{0}\leq L_{-1}$, which proves the first estimate.

ii) To prove the $C^{1}$-estimate, we follow the same steps using Lemma 3.5ii) and Lemma 3.6ii) instead of Lemma 3.5i) and Lemma 3.6i) respectively. We then use the estimates $\|D^{2}\tilde{r}\|_{0}\leq\delta(\varepsilon)$, $\|D^{i}f_{1}-D^{i}f_{2}\|_{0}\leq\|f_{1}-f_{2}\|_{1}$ and $\|D^{i}r-D^{i}\tilde{r}\|_{0}\leq\|r-\tilde{r}\|_{1}$ for $i=1,2$. From this, the desired estimate follows. ∎

Remark 3.8*.*

The assumptions on $r$ and $f_{1}$ can be weakened in part i). We only need the assumptions that $A_{c}+r$ is a homeomorphism and $f_{1}$ is continuous and bounded.

3.6 A contraction

Following our proof scheme for Theorem 2.1, which we described in Section 2.1, we want to show that our fixed point operator $\Theta:\Gamma_{1}(\delta(\varepsilon))\to\Gamma_{1}(\delta(\varepsilon))$, which is defined in 3.1.3, has a $C^{1}$ fixed point. We note that in Theorem 3.9 we will impose an upper bound on the second derivatives of the nonlinearities $k_{c}:X_{c}\to X_{c}$ and $g:X\to X$, whereas we only assume boundedness of the second derivatives in Theorem 2.1. However, we will see in Lemma 6.1 that we can always find a scaling such that the second derivative is sufficiently small.

Theorem 3.9**.**

Assume that $L_{g}$ and $L_{c}$ are small in the sense of Remark 2.4 for $n=2$. There exists an $\varepsilon_{0}>0$ such for all $\varepsilon<\varepsilon_{0}$ it holds that if $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$, then $\Theta:\Gamma_{1}(\delta(\varepsilon))\to\Gamma_{1}(\delta(\varepsilon))$ is a contraction with respect to the $C^{1}$ norm.

Proof.

Let $\varepsilon>0$ and $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$. Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right),\tilde{\Lambda}=\left(\begin{smallmatrix}\tilde{r}\\ \tilde{k}_{u}\\ \tilde{k}_{s}\end{smallmatrix}\right)\in\Gamma_{1}(\delta(\varepsilon))$. We denote $R=A_{c}+r$ and $\tilde{R}=A_{c}+\tilde{r}$.

Our proof that $\Theta$ is a $C^{1}$-contraction is divided in three steps.

1. A)

We prove that $\Theta$ is a contraction with respect to the $C^{0}$ norm, independent of $\varepsilon$. 2. B)

We show the existence of a constant $\theta_{1}(\varepsilon)$ such that

[TABLE] 3. C)

We show that $\varepsilon>0$ can be chosen so that $\theta_{1}(\varepsilon)<1$, thus proving that $\Theta$ is a contraction with respect to the $C^{1}$ norm.

Step A) We want to find $\theta_{0}<1$ such that

[TABLE]

Recall from equation 3.1.3 that

[TABLE]

We will find the contraction constant component-wise, i.e we will show that

[TABLE]

with $\theta_{0,i}$ given explicitly in equation 2.0.3a, 2.0.3b and 2.0.3c for $i=1,2,3$.

$r$-component: We start with

[TABLE]

By using Lemma 3.5i) we find the estimate

[TABLE]

Here we recall that $\|Dg_{c}\|_{0}\leq L_{g}$, which follows from assumption 4 of Theorem 2.1. Likewise, we estimate

[TABLE]

Thus inequality 3.6.1 together with estimates 3.6.2 and 3.6.3 gives

[TABLE]

$k_{u}$-component: Similarly, we have

[TABLE]

We again use Lemma 3.5i), which gives

[TABLE]

Here we used that $\|D\tilde{k}_{u}\|_{0}\leq L_{u}$, which follows from the fact that $\Gamma_{1}(\delta(\varepsilon))\subset\Gamma_{0}$, with the latter space defined before Theorem 3.3. Thus inequality 3.6.5 becomes

[TABLE]

$k_{s}$-component: Let $T=\left(A_{c}+r\right)^{-1}$ and $\tilde{T}=(A_{c}+\tilde{r})^{-1}$, then we have

[TABLE]

We use Lemma 3.7i), where the condition $L_{r}<\|A_{c}^{-1}\|_{\textup{op}}^{-1}$ is satisfied by Remark 2.4, to obtain

[TABLE]

We used $\|D\tilde{K{}{}}\|_{0}\leq 1+L_{c}$, see 3.3.3, in the last estimate. Thus inequality 3.6.7 becomes

[TABLE]

Contraction constant: We can now estimate $\|\Theta(\Lambda)-\Theta(\tilde{\Lambda})\|_{0}$ with inequalities 3.6.4, LABEL:, 3.6.8, LABEL: and 3.6.6. We obtain

[TABLE]

Here we define $\theta_{0}:=\max\left\{\theta_{0,1},\theta_{0,2},\theta_{0,3}\right\}$. Since it is assumed that Remark 2.4 holds for $n=2$, we have $\theta_{0,i}<1$ and thus $\theta_{0}<1$. This implies that $\Theta$ is a contraction with respect to the $C^{0}$ norm.

Step B) Analogous to step A), we want to prove the component-wise inequality

[TABLE]

with $\theta_{1,i}$ defined in 2.0.3a, 2.0.3b and 2.0.3c and $C_{1,i}$ defined below in the proof. We note that $\Lambda,\tilde{\Lambda}\in\Gamma_{1}(\delta(\varepsilon))$, so we have $\|D^{2}r\|_{0},\|D^{2}k_{u}\|_{0},\|D^{2}k_{s}\|_{0}\leq\delta(\varepsilon)$.

$r$-component: We start with

[TABLE]

We infer from Lemma 3.5ii) that

[TABLE]

where we have used 3.3.3. Likewise, we find the estimate

[TABLE]

Thus inequality 3.6.10 together with estimates 3.6.11 and 3.6.12 gives

[TABLE]

where we define $C_{1,1}(\varepsilon):=\left(1+L_{c}+\|A_{c}\|_{\textup{op}}+L_{r}\right)\varepsilon$.

$k_{u}$-component: Similarly, we have

[TABLE]

Using Lemma 3.5ii) we get

[TABLE]

Thus inequality 3.6.14 becomes

[TABLE]

where we define $C_{1,2}(\varepsilon):=\|A_{u}^{-1}\|_{\textup{op}}\left(L_{g}(1+L_{c})\varepsilon+(\|A_{c}\|_{\textup{op}}+L_{r})\delta(\varepsilon)\right)$.

$k_{s}$-component: Recall that $T=(A_{c}+r)^{-1}$ and $\tilde{T}=(A_{c}+\tilde{r})^{-1}$, then we have

[TABLE]

We will estimate both terms with Lemma 3.7ii). For the first term, we note that $\|DA_{s}\|_{0}=\|A_{s}\|_{\textup{op}}$ and $\|D^{2}A_{s}\|_{0}=0$, which gives us

[TABLE]

where we grouped the terms with and without a factor $\delta(\varepsilon)$. The second term in 3.6.16 involves the first and second derivative of $\tilde{K{}{}}$. We estimate the first derivate again with $1+L_{c}$ and we estimate the second derivative with $\|D^{2}\tilde{K{}{}}\|_{0}=\max\{\|D^{2}k_{c}\|_{0},\|D^{2}\tilde{k}_{u}\|_{0},\|D^{2}\tilde{k}_{s}\|_{0}\}\leq\max\{\varepsilon,\delta(\varepsilon)\}=:\gamma(\varepsilon)$. Hence we obtain

[TABLE]

Here we again grouped the terms with and without $\varepsilon$. We see that 3.6.17 and 3.6.19 together are $\theta_{1,3}\|\Lambda-\tilde{\Lambda}\|_{1}$. Likewise, we can estimate 3.6.18 and 3.6.20 together by $\theta_{2,3}\gamma(\varepsilon)\|\Lambda-\tilde{\Lambda}\|_{1}$ as $\delta(\varepsilon)\leq\gamma(\varepsilon)$. Then inequality 3.6.16 reduces to

[TABLE]

where we define $C_{1,3}(\varepsilon):=\theta_{2,3}\gamma(\varepsilon)+L_{-1}(1+L_{c})\varepsilon+L_{-1}^{2}(1+L_{c})^{2}\varepsilon$.

Lipschitz constant: Inequalities 3.6.13, LABEL:, 3.6.15, LABEL: and 3.6.21 give

[TABLE]

Here we define

[TABLE]

Step C) From Remark 2.4 it follows that

[TABLE]

As $\delta(\varepsilon)\downarrow 0$ and thus also $\gamma(\varepsilon)\downarrow 0$ when $\varepsilon\downarrow 0$, see Proposition 3.4, we have

[TABLE]

We infer that

[TABLE]

Hence, we can find an $\varepsilon_{0}>0$ such that $\theta_{1}(\varepsilon)<1$ for all $\varepsilon<\varepsilon_{0}$. Then estimates 3.6.9 and 3.6.22 imply that

[TABLE]

We define the contraction constant $\lambda_{1}:=\max\{\theta_{0},\theta_{1}(\varepsilon)\}$, which is smaller than $1$ for $\varepsilon<\varepsilon_{0}$ by our previous discussion. We conclude that $\Theta:\Gamma_{1}(\delta(\varepsilon))\to\Gamma_{1}(\delta(\varepsilon))$ is a contraction with respect to the $C^{1}$ norm for $\varepsilon<\varepsilon_{0}$. ∎

We can now prove the existence of a $C^{1}$ center manifold under the assumption that the second derivative of $k_{c}$ and $g$ are small enough. As we will see in Lemma 6.1, we can always find a scaling such that these second derivatives will be sufficiently small.

Corollary 3.10**.**

Let $\varepsilon>0$ be such that $F{}{}:X\to X$ satisfies the assumptions of Theorems 2.1 and 3.9. Then the conclusion of Theorem 2.1 holds for $K{}{}\in C^{1}(X_{c},X)$ and $r\in C^{1}(X_{c},X_{c})$. In particular, the image of $K{}{}$ is a $C^{1}$ center manifold for $F{}{}$.

Proof.

By assumption, $\varepsilon>0$ is such that $\Theta$ is a contraction. In Theorems 3.9 and 3.1 we proved the existence of a conjugacy $K{}{}$ and conjugate dynamics $A_{c}+r$. Furthermore, from the definition of $\Gamma_{0}$, it follows that $K{}{}$ and $r$ satisfy the properties B) and A) respectively. In particular, it follows that image of $K{}{}$ is invariant under $F{}{}$ and tangent to $X_{c}$ at [math], hence the image of $K{}{}$ is a $C^{1}$ center manifold for $F{}{}$. ∎

4 A $C^{2}$ center manifold

Now that we have a $C^{1}$ conjugacy, the third step in our proof scheme in Section 2.1 is showing that the conjugacy is $C^{2}$. We will prove the equivalent statement that the derivative of the $C^{1}$ conjugacy is also $C^{1}$. For this, we define another fixed point operator acting on $C^{1}$ functions, and show that its fixed point is the derivative of the conjugacy from Corollary 3.10.

4.1 A new fixed point operator

We first note that $\Theta$ is a contraction with respect to the $C^{1}$ norm on $\Gamma_{1}(\delta(\varepsilon))$, a set that is not closed with respect to the $C^{1}$ norm. That means that the fixed point of $\Theta$ lies in the $C^{1}$ closure of $\Gamma_{1}(\delta(\varepsilon))$, which is enclosed by $\Gamma_{0}$.

Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$ denote any fixed point of $\Theta$, i.e. $\Lambda$ consists of three $C^{1}$ functions and we have

[TABLE]

We can therefore take the derivative at both sides of the equation, which gives

[TABLE]

where we define $R:=A_{c}+r$ and $T:=(A_{c}+r)^{-1}$, notation that we will use throughout the rest of the paper. To express $DT=D(A_{c}+R)^{-1}$ in terms of $r$ and $Dr$, we use the Inverse Function Theorem and write

[TABLE]

This motivates us to introduce for $\rho:X_{c}\to\mathcal{L}(X_{c},X_{c})$ the functions

[TABLE]

so that we can write $DT(x)=Q_{Dr}(x)$ and $DR(x)=P_{Dr}(x)$. In view of 4.1.1 we use these functions to introduce the fixed point operator

[TABLE]

where $\kappa=\left(\begin{smallmatrix}\operatorname{Id}+Dk_{c}\\ \kappa_{u}\\ \kappa_{s}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$ is a fixed point of $\Theta$. To summarize, we have the following proposition:

Proposition 4.1**.**

Let $\Lambda\in\Gamma_{0}$ be any fixed point of the operator $\Theta$. Then $D\Lambda$ is a fixed point of the operator $\Theta^{[2]}$ defined in 4.1.3.

Proof.

This follows immediately from the above discussion. ∎

We want to use $\Theta^{[2]}$ to show that $D\Lambda$ is $C^{1}$ instead of only $C^{0}$. To this end, we want to show that $\Theta^{[2]}$ is a contraction in $C^{1}$ on a suitable set of $C^{1}$ functions, and show that its fixed point in this set is $D\Lambda$. We therefore want to restrict $\Theta^{[2]}$ to a space similar to $\Gamma_{1}(\delta(\varepsilon))$. In particular, we want to reflect that $\Theta$ is a fixed point operator acting on functions and $\Theta^{[2]}$ is a fixed point operator acting on derivatives. So where functions in $\Gamma_{1}(\delta(\varepsilon))$ have restrictions on the first and second derivative, we want the same restrictions on the function and its derivative in our new space respectively. Therefore, let $\delta>0$, and define the set

[TABLE]

Proposition 4.2**.**

Let $\varepsilon>0$ and assume that $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$. Furthermore, assume that $L_{g}$ and $L_{c}$ are small in the sense of Remark 2.4 for $n=2$. Then, for $\delta(\varepsilon)>0$ from Proposition 3.4, the set $\Gamma_{2}(\delta(\varepsilon))$ is invariant under $\Theta^{[2]}$.

Proof.

The proof follows from similar estimates as in Theorem 3.3 for the bounds on $\rho$, $\kappa_{u}$ and $\kappa_{s}$ as well as for $\mathcal{M}(0)=0$. The bound on the second derivative follows from the same estimates as in Proposition 3.4. We will illustrate this for the derivative of the second component, i.e. we will show that $D\Theta^{[2]}_{1}(\mathcal{M})$ is bounded by $\delta(\varepsilon)$ for $\mathcal{M}\in\Gamma_{2}(\delta(\varepsilon))$.

We start as we did in Proposition 3.4 with

[TABLE]

We again estimate the terms separately:

[TABLE]

where we used $D\kappa\leq\varepsilon+\delta(\varepsilon)$ as we did in 3.4.4. All together, we find

[TABLE]

Here we used the definition of $C_{1}(\varepsilon)$ just below 3.4.8, and the last inequality follows from the definition of $\delta(\varepsilon)$ in 3.4.12. The other estimates are similar. ∎

4.2 Estimates for products

In Section 3.5 we gave some preliminary results for Theorem 3.9 in Lemmas 3.5 and 3.7. We want to derive similar results for derivatives instead of functions in Lemmas 4.3 and 4.4 respectively. The results below will be framed in a slightly more general setting, so that we can use them in the next section as well.

Lemma 4.3**.**

Let $X$, $Y$ and $Z$ be Banach spaces, $m\in\mathbb{N}$ and $h\in C^{1}_{b}(X,Y)$.

1. i)

For $f_{1},g_{1}\in C^{0}_{b}(Y,\mathcal{L}(Y,Z))$, $f_{2},g_{2}\in C^{0}_{b}(X,\mathcal{L}^{m}(X,Y))$ we have the $C^{0}$-estimate

[TABLE] 2. ii)

For $f_{1},g_{1}\in C^{1}_{b}(Y,\mathcal{L}(Y,Z))$, $f_{2},g_{2}\in C^{1}_{b}(X,\mathcal{L}^{m}(X,Y))$ we have the $C^{1}$-estimate

[TABLE]

Proof.

i) The $C^{0}$-estimate follows from the triangle inequality and submultiplicativity of the norm.

[TABLE]

ii) For the $C^{1}$-estimate we use the product rule and triangle inequality to find

[TABLE]

where we introduce $F_{1}=f_{1}\circ h$ and $G_{1}=g_{1}\circ h$. We then estimate

[TABLE]

For those estimates we have used that $DF_{1}=Df_{1}(h)Dh$, and thus $DF_{1}$ is bounded by $\|Df_{1}\|_{0}\|Dh\|_{0}$ and likewise we have bounded $DF_{1}-DG_{1}$ by $\|Df_{1}-Dg_{2}\|_{0}\|Dh\|_{0}$. For the last estimate, we have used that $F_{1}$ is bounded by $\|f_{1}\|_{0}$ and $F_{1}-G_{1}$ is bounded by $\|f_{1}-g_{1}\|_{0}$. We obtain the desired estimate by adding the two estimates together ∎

Lemma 4.4**.**

Let $\rho,\tilde{\rho}\in C_{b}^{1}(X_{c},\mathcal{L}(X_{c},X_{c}))$ be such that $\|\rho\|_{0},\|\tilde{\rho}\|_{0}\leq L_{r}$ and $\|D\rho\|_{0},\|D\tilde{\rho}\|_{0}\leq\delta(\varepsilon)$ for some $\varepsilon>0$. Furthermore, let $X$ and $Y$ be Banach spaces. Let $h\in C^{2}_{b}(Y,\mathcal{L}(Y,X))$ and $f_{1},f_{2}\in C^{2}_{b}(X_{c},\mathcal{L}(X_{c},Y))$. Furthermore, assume that $L_{r}<\|A_{c}^{-1}\|_{\textup{op}}^{-1}$ and recall the definition of $Q_{\rho}$ in 4.1.2.

1. i)

We have the $C^{0}$-estimate

[TABLE] 2. ii)

We have the $C^{1}$-estimate

[TABLE]

Proof.

i) For the $C^{0}$-estimate, we note that $h(y)$ is a linear operator for all $y\in Y$ and we use submultiplicativity of the norm and Lemma 4.3i)

[TABLE]

All that is left to do is to show that $Q_{\rho}-Q_{\tilde{\rho}}$ is bounded by $L_{-1}^{2}\|\rho-\tilde{\rho}\|_{0}$ and that $Q_{\tilde{\rho}}$ is bounded $L_{-1}$. For the latter bound, we use similar calculations as performed at the end of the proof of Lemma 3.2i). Namely, fix $x\in X_{c}$, denote $y=T(x)$ and $\tau=Q_{\tilde{\rho}}(x)-A_{c}^{-1}$, then we have $(A_{c}^{-1}+\tau)(A_{c}+\tilde{\rho}(y))=Q_{\tilde{\rho}}(x)P_{\tilde{\rho}}(y)=\operatorname{Id}$. We can rewrite this as $\tau=-A_{c}^{-1}\tilde{\rho}(y)A_{c}^{-1}-\tau\tilde{\rho}(y)A_{c}^{-1}$. This implies that the norm of $\tau$ is bounded by $\|A_{c}^{-1}\|_{\textup{op}}^{2}\|\tilde{\rho}(y)\|_{\textup{op}}/\left(1-\|A_{c}^{-1}\|_{\textup{op}}\|\tilde{\rho(y)}\|_{\textup{op}}\right)\leq L_{t}$, as $\|\tilde{\rho}(y)\|_{\textup{op}}\leq L_{r}$, where $L_{r}$ and $L_{t}$ are defined in 2.0.2a and 2.0.2b respectively. Therefore, we have the desired bound

[TABLE]

The bound on $Q_{\rho}-Q_{\tilde{\rho}}$ now follows from submultiplicativity and

[TABLE]

ii) For the $C^{1}$-estimate, we start by with the product rule and triangle inequality to find

[TABLE]

The first two terms of the right hand side are estimated using similar arguments as those used in part i), that is

[TABLE]

which are precisely the first two terms of the right hand side of our desired $C^{1}$-estimate. The last term can be estimated using Lemma 4.3ii):

[TABLE]

We will estimate the four terms separately. With the estimates of $Q_{\tilde{\rho}}$ and $Q_{\rho}-Q_{\tilde{\rho}}$ from the proof of part i), and given that $\|DT\|_{0}\leq L_{-1}$, we find

[TABLE]

which are, up to the factor $\|h\|_{0}$, the third and fourth term of the right hand side of our desired $C^{1}$-estimate. Finally, we have to find an upper bound for $DQ_{\tilde{\rho}}$ and $DQ_{\rho}-DQ_{\tilde{\rho}}$ in terms of $L_{-1}$ and $\|\rho-\tilde{\rho}\|_{0}$ to estimate the final two terms in 4.2.1. The product rule gives us, since $Q_{\tilde{\rho}}=\left(P_{\tilde{\rho}}\circ T\right)^{-1}$,

[TABLE]

We isolate $DQ_{\tilde{\rho}}(P_{\tilde{\rho}}\circ T)$ and multiply from the right with $(P_{\tilde{\rho}}\circ T)^{-1}=Q_{\tilde{\rho}}$:

[TABLE]

Furthermore, we note that $P_{\tilde{\rho}}=A_{c}+\tilde{\rho}$, hence $DP_{\tilde{\rho}}=D\tilde{\rho}$, which is bounded by $\delta(\varepsilon)$. We also saw that $\|Q_{\tilde{\rho}}\|_{0}\leq L_{-1}$ in the proof of part i). Hence we find with the triangle inequality

[TABLE]

For the last inequality we used that $P_{\tilde{\rho}}=A_{c}+\tilde{\rho}$, and thus $DP_{\tilde{\rho}}=D\tilde{\rho}$, which is bounded by $\delta(\varepsilon)$. Furthermore, we used that $Q_{\tilde{\rho}}$ is bounded by $L_{-1}$ and $Q_{\rho}-Q_{\tilde{\rho}}$ is bounded by $L_{-1}^{2}\|\rho-\tilde{\rho}\|_{0}$, as shown in the proof of part i). Hence the third factor of 4.2.1 is bounded by the fifth and sixth term appearing in the right hand side of our desired $C^{1}$-estimate. Finally, we estimate the last term of 4.2.1, where we use 4.2.2 to bound $DQ_{\tilde{\rho}}$:

[TABLE]

This is precisely the final term appearing in the asserted estimate. ∎

4.3 A new contraction

With the previous two lemmas, we will show that $\Theta^{[2]}$ is a contraction on $\Gamma_{2}(\delta(\varepsilon))$ for $\varepsilon>0$ small enough. We note again that we will later show that we can always scale our functions to satisfy the bound on the second derivative.

Proposition 4.5**.**

Assume that $L_{g}$ and $L_{c}$ are small in the sense of Remark 2.4 for $n=2$. There exists an $\varepsilon_{0}>0$ such for all $\varepsilon<\varepsilon_{0}$ it holds that if $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$, then $\Theta^{[2]}:\Gamma_{2}(\delta(\varepsilon))\to\Gamma_{2}(\delta(\varepsilon))$ is a contraction with respect to the $C^{1}$ norm.

Proof.

Let $\varepsilon>0$ and $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$. Let $\mathcal{M}=\left(\begin{smallmatrix}\rho\\ \kappa_{u}\\ \kappa_{s}\end{smallmatrix}\right),\tilde{\mathcal{M}}=\left(\begin{smallmatrix}\tilde{\rho}\\ \tilde{\kappa}_{u}\\ \tilde{\kappa}_{s}\end{smallmatrix}\right)\in\Gamma_{2}(\delta(\varepsilon))$. To show that $\Theta^{[2]}$ is a $C^{1}$ contraction, we will use the same steps as we used in the proof of Theorem 3.9.

1. A)

We prove that $\Theta^{[2]}$ is a contraction with respect to the $C^{0}$ norm, independent of $\varepsilon$, 2. B)

We show the existence of a constant $\theta_{2}(\varepsilon)$ such that

[TABLE] 3. C)

We show that $\varepsilon>0$ can be chosen so that $\theta_{2}(\varepsilon)<1$, thus proving that $\Theta^{[2]}$ is a contraction with respect to the $C^{1}$ norm.

Step A) We recall that $\theta_{1}(\varepsilon)$, defined in 3.6.23, has the property $\theta_{1}(0)<1$. We want to show that

[TABLE]

Recall from equation 4.1.3 that

[TABLE]

which was derived by taking the derivative of $\Theta\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)$. Therefore, we will use similar arguments as in step B) of the proof of Theorem 3.9 to show that

[TABLE]

for $\theta_{1,i}$ given explicitly in equation 2.0.3a, 2.0.3b and 2.0.3c for $i=1,2,3$.

$\rho$-component: We have

[TABLE]

The first term is estimated by Lemma 4.3i):

[TABLE]

Here we recall that $\|Dg_{c}\|_{0}\leq L_{g}$, which follows from assumption 4 of Theorem 2.1. Likewise, we estimate

[TABLE]

Thus inequality 4.3.1 together with estimates 4.3.2 and 4.3.3 gives

[TABLE]

$\kappa_{u}$-component: Similarly, we have

[TABLE]

We again use Lemma 4.3i), which gives

[TABLE]

Here we used that $\kappa_{u}$ is bounded by $L_{u}$ and $\tilde{\rho}$ by $L_{r}$. Thus inequality 4.3.5 becomes

[TABLE]

$\kappa_{s}$-component: Let us denote $G_{s}=Dg_{s}\circ K{}{}\circ T$. Then

[TABLE]

We will estimate both terms with Lemma 4.4i). We note that $\kappa$ is bounded by $1+L_{c}$, and hence we obtain

[TABLE]

Thus inequality 4.3.7 becomes

[TABLE]

Contraction constant: We can now estimate $\|\Theta^{[2]}(\mathcal{M})-\Theta^{[2]}(\tilde{\mathcal{M}})\|_{0}$ with inequalities 4.3.4, 4.3.6 and 4.3.8. We have

[TABLE]

where the last equality follows from the definition of $\theta_{1}$ in 3.6.23. Since $\theta_{1}(0)<1$, we conclude that $\Theta^{[2]}$ is a contraction with respect to the $C^{0}$ norm.

Step B) Analogous to step B) of the proof of Theorem 3.9, we want to prove the component-wise inequality

[TABLE]

with $\theta_{2,i}$ defined in 2.0.3a, 2.0.3b and 2.0.3c and $C_{2,i}$ will be defined during the proof. We note that $\mathcal{M},\tilde{\mathcal{M}}\in\Gamma_{2}(\delta(\varepsilon))$, hence $D\rho$, $D\kappa_{u}$ and $D\kappa_{s}$ are bounded by $\delta(\varepsilon)$.

$\rho$-component: We start with

[TABLE]

By applying Lemma 4.3ii) we find that

[TABLE]

where we recall that $DK{}{}$ is bounded by $1+L_{c}$, see 3.3.3. Likewise, we find the estimate

[TABLE]

Thus inequality 4.3.10 together with the above estimates gives

[TABLE]

where we define $C_{2,1}(\varepsilon):=\left(1+L_{c}+\|A_{c}\|_{\textup{op}}+L_{r}\right)\varepsilon$.

$\kappa_{u}$-component: We have

[TABLE]

By Lemma 4.3ii) we get

[TABLE]

Thus inequality 4.3.12 becomes, as $\rho-\tilde{\rho}$ and $\kappa-\tilde{\kappa}$ are bounded by $\|\mathcal{M}-\tilde{\mathcal{M}}\|_{1}$,

[TABLE]

where we define $C_{2,2}(\varepsilon):=\|A_{u}^{-1}\|_{\textup{op}}((2+L_{c})\varepsilon+(\|A_{c}\|_{\textup{op}}+L_{r})\delta(\varepsilon))$.

$\kappa_{s}$-component: Let us again denote $G_{s}=Dg_{s}\circ K{}{}\circ T$, then we have

[TABLE]

We will estimate both terms with Lemma 4.4ii). For the first term, we note that $A_{s}$ is the constant operator $x\mapsto A_{s}$, hence $\|A_{s}\|_{0}=\|A_{s}\|_{\textup{op}}$ and $DA_{s}=0$, which gives us

[TABLE]

The second term in the right hand side of 4.3.14 involves $\kappa$, which is bounded by $1+L_{c}$, its derivative, which is bounded by

[TABLE]

and the derivative of $G_{s}$. The derivative of $G_{s}$ is bounded by $\|D^{2}g_{s}\|_{0}\|DK{}{}\|_{0}\|DT\|_{0}$, which in turn is bounded by $L_{-1}(1+L_{c})\varepsilon$. We obtain

[TABLE]

We see that 4.3.15 and 4.3.17 together are $\theta_{2,3}\|\mathcal{M}-\tilde{\mathcal{M}}\|_{1}$. Likewise, we can estimate 4.3.16 and 4.3.18 together by $2\theta_{3,3}\gamma(\varepsilon)\|\mathcal{M}-\tilde{\mathcal{M}}\|_{1}$ as $\delta(\varepsilon)\leq\gamma(\varepsilon)$.Then inequality 4.3.14 becomes

[TABLE]

where we define $C_{2,3}(\varepsilon):=2\theta_{3,3}\gamma(\varepsilon)+L_{-1}^{2}(1+L_{c})\varepsilon+L_{-1}^{3}(1+L_{c})^{2}\varepsilon$.

Lipschitz constant: Inequalities 4.3.11, 4.3.13 and 4.3.19 imply

[TABLE]

Here we define $\theta_{2}(\varepsilon):=\max\left\{\theta_{2,1},\theta_{2,2},\theta_{2,3}\right\}+\max\{C_{2,1}(\varepsilon),C_{2,2}(\varepsilon),C_{2,3}(\varepsilon)\}$.

Step C) From Remark 2.4 it follows that

[TABLE]

As $\delta(\varepsilon)\downarrow 0$ when $\varepsilon\downarrow 0$, see Proposition 3.4, we have

[TABLE]

For the limit of $\varepsilon$ to [math] of $\theta_{2}(\varepsilon)$ we find

[TABLE]

Therefore, we can find an $\varepsilon_{0}>0$ such that $\theta_{2}(\varepsilon)<1$ for all $\varepsilon<\varepsilon_{0}$. Then estimates 4.3.9 and 4.3.20 give

[TABLE]

We define the contraction constant $\lambda_{2}:=\max\{\theta_{1}(0),\theta_{2}(\varepsilon)\}$, which is smaller than $1$ by the above discussion. Thus we see that $\Theta^{[2]}:\Gamma_{2}(\delta(\varepsilon))\to\Gamma_{2}(\delta(\varepsilon))$ is a contraction with respect to the $C^{1}$ norm for all $\varepsilon<\varepsilon_{0}$. ∎

Corollary 4.6**.**

Let $\varepsilon>0$ such that $F{}{}:X\to X$ satisfies the conditions of Theorems 2.1, 3.9 and 4.5. Then the image of $K{}{}$ is a $C^{2}$ center manifold for $F{}{}$.

Proof.

By assumption, $\varepsilon>0$ is such that both $\Theta$ and $\Theta^{[2]}$ are contractions. Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$ be a fixed point of $\Theta$. Then Corollary 3.10 implies that $K{}{}=\iota+\left(\begin{smallmatrix}k_{c}\\ k_{u}\\ k_{s}\end{smallmatrix}\right)$ parameterizes a $C^{1}$ center manifold for $F{}{}$. Thus if we prove that $\Lambda$ is $C^{2}$, or, equivalent, that $D\Lambda$ is $C^{1}$, we are done.

We will show that $D\Lambda$ is in fact the fixed point of $\Theta^{[2]}$. Let

[TABLE]

then $D\Lambda\in\Gamma_{3/2}$ and $\Gamma_{2}(\delta(\varepsilon))\subset\Gamma_{3/2}$. Furthermore, from step A) of the proof of the previous proposition, it follows that $\Theta^{[2]}:\Gamma_{3/2}\to\Gamma_{3/2}$ is a contraction with respect to the $C^{0}$ norm. Therefore, $D\Lambda$ is the unique fixed point of $\Theta^{[2]}$ in $\Gamma_{3/2}$. However, $\Theta^{[2]}:\Gamma_{2}(\delta(\varepsilon))\to\Gamma_{2}(\delta(\varepsilon))$ is also a contraction with respect to the $C^{1}$ norm for $\varepsilon$ sufficiently small. Let $\mathcal{M}\in\Gamma_{2}(\delta(\varepsilon))$ be the fixed point of $\Theta^{[2]}$. Then $\mathcal{M}\in\Gamma_{3/2}$ is a fixed point of $\Theta^{[2]}$, which means that $D\Lambda=\mathcal{M}\in\Gamma_{2}(\delta(\varepsilon))$. We conclude that $D\Lambda$ is $C^{1}$. ∎

We have now shown in two steps that there exists a $C^{2}$ center manifold. In particular, we used the existence of a $C^{1}$ center manifold to obtain a $C^{2}$ center manifold in the second step. Furthermore, to obtain the $C^{1}$ center manifold, we explicitly used that our dynamical system $F$ is at least $C^{2}$. Hence, with the current proof we cannot obtain a center manifold for a $C^{1}$ dynamical system.

We want to remark that if $F{}{}$ is $C^{0+\text{Lip}}$, we could slightly alter the definition of $\Gamma_{0}$ and show that $\Theta$ is a contraction with respect to the $C^{0}$ norm. Then both $K$ and $r$ would have been $C^{0+\text{Lip}}$ and we would obtain a $C^{0+\text{Lip}}$ center manifold and dynamical system.

Furthermore, if $F{}{}$ is $C^{1}$ instead of $C^{2}$, we could adapt our proof to obtain a $C^{1}$ center manifold and dynamical system if $X$ is uniformly convex, e.g. $X=\mathbb{R}^{m}$. In this case, we would also prove that $\Theta$ is a contraction with respect to the $C^{0}$ norm, which would give us $C^{0+\text{Lip}}$ functions $K{}{}$ and $r{}{}$. Using the results from [Clarkson36], we know that both $K{}{}$ and $r$ would be almost everywhere differentiable. We could still define the fixed point operator $\Theta^{[2]}$ and prove that $\Theta^{[2]}$ would be a contraction with respect to the $C^{0}$ norm. Using similar arguments as in the previous corollary, we could then show that $K{}{}$ and $r$ would be everywhere continuously differentiable. Therefore, we would obtain a $C^{1}$ center manifold with $C^{1}$ dynamical system if we start with a $C^{1}$ dynamical system $F:X\to X$ on a uniformly convex space $X$.

5 A $C^{m}$ center manifold

The final step of our proof scheme in Section 2.1 is inductively showing that the conjugacy is $C^{n}$. Similar to what we did in the $C^{2}$ case in the previous section, we will show that if the conjugacy is $C^{m}$, then the $m^{\text{th}}$ derivative will be $C^{1}$. We start again by defining a fixed point operator for the $m^{\text{th}}$ derivative of

[TABLE]

To make the definition of the fixed point operator more insightful, we will use several lemmas before we define it. We start by stating Faà di Bruno’s formula for derivatives of compositions, see for instance [Mennucci17]

Lemma 5.1** (Faà di Bruno’s Formula).**

Let $X$, $Y$ and $Z$ be Banach spaces, and $f_{1}:Y\to Z$ and $f_{2}:X\to Y$ $C^{m}$ functions. Then the $m^{\text{th}}$ derivative of $f_{1}\circ f_{2}$ is given by

[TABLE]

where $P_{m}^{i}$ is the set of ordered partitions of length $i$ of the set $\{1,\dots,m\}$.

Since we want to define a fixed point operator for the $m^{\text{th}}$ derivative of $\Lambda$ using $\Lambda=\Theta(\Lambda)$, we want to isolate the $m^{\text{th}}$ derivatives from Faà di Bruno’s formula, and apply Faà di Bruno’s formula to the composition of three functions.

Lemma 5.2**.**

Let $X$, $Y$ and $Z$ be Banach spaces, and $f_{1}:Y\to Z$ and $f_{2}:X\to Y$ $C^{m}$ functions.

1. i)

The $m^{\text{th}}$ derivative of $f_{1}\circ f_{2}$ is given by

[TABLE]

where we use the shorthand notation $\left(Df_{2}(x)\right)^{\otimes m}:=\underbrace{\left(Df_{2}(x),\dots,Df_{2}(x)\right)}_{m\text{ times}}$ and $\mathcal{P}_{m}(f_{1},f_{2})(x):=\sum_{i=2}^{m-1}\sum_{\pi\in P_{m}^{i}}D^{i}f_{1}(f_{2}(x))\left(D^{\pi(1)}f_{2}(x),\cdots,D^{\pi(i)}f_{2}(x)\right)$. 2. ii)

Let $f_{3}:X\to X$ be another $C^{m}$ function, then we find

[TABLE]

where $z=f_{2}(f_{3}(x))$ and $y=f_{3}(x)$.

Proof.

i) The equality follows from Faà di Bruno’s formula and the fact that there is one ordered partition of $m$ of length 1 and one of length $m$.

ii) The equality follows from applying Faà di Bruno’s formula twice. We first apply it to the composition of $f_{1}\circ f_{2}$ and $f_{3}$, which gives

[TABLE]

Then the first derivative of $f_{1}\circ f_{2}$ is given by $Df_{1}(f_{2})Df_{2}$, and we apply Faà di Bruno’s formula a second time for the $m^{\text{th}}$ derivative of $f_{1}\circ f_{2}$ to find the desired equality. ∎

The third component of $\Theta(\Lambda)$ contains the function $(A_{c}+r)^{-1}$. We want to express the $m^{\text{th}}$ derivative in terms of the $m^{\text{th}}$ derivative of $A_{c}+r$.

Lemma 5.3**.**

Let $R=A_{c}+r:X_{c}\to X_{c}$ be an invertible $C^{m}$ function with inverse $T$. Then

[TABLE]

Proof.

We apply Faà di Bruno’s formula to $R\circ T$, and notice that its $m^{\text{th}}$ derivative is [math] as $R\circ T=\operatorname{Id}$. Thus we find

[TABLE]

The asserted identity follows by multiplying both sides by the inverse of $DR(T)$, which is $DT$. ∎

Remark 5.4*.*

This lemma is another reason why we start the inductive step after we have proven $C^{2}$ smoothness, because the first derivative of $\operatorname{Id}$ does not vanish.

5.1 Another fixed point operator

We can now take the $m^{\text{th}}$ derivative of the fixed point identity $\Lambda=\Theta(\Lambda)$, where $\Theta(\Lambda)$ is defined in 3.1.2. We will do this component-wise. For the first component we find, by Lemma 5.2i),

[TABLE]

where

[TABLE]

For the second component, we find with Lemma 5.2i)

[TABLE]

where

[TABLE]

Finally, for the third component we find from Lemma 5.2ii) and 5.3, with $T=(A_{c}+r)^{-1}$,

[TABLE]

where we define

[TABLE]

and

[TABLE]

Hence, we introduce the fixed point operator, where we use $\kappa=\left(\begin{smallmatrix}D^{m}k_{c}\\ \kappa_{u}\\ \kappa_{s}\end{smallmatrix}\right)$,

[TABLE]

We note that if $\Lambda\in C^{m}_{b}(X_{c},X)$, then the functions $f_{m,1}$, $f_{m,2}$ and $f_{m,3}$ are bounded, since we assume that $g\in C^{m}_{b}(X,X)$. Hence, if $\Lambda\in C^{m}_{b}(X_{c},X)$, then we have that $\Theta^{[m+1]}:\Gamma_{m+1}\to\Gamma_{m+1}$.

Proposition 5.5**.**

Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{0}$ be a $C^{m}$ fixed point of $\Theta$ for $2\leq m<n$. If $\Lambda\in C^{m}_{b}(X_{c},X)$, then $D^{m}\Lambda$ is a fixed point of $\Theta^{[m+1]}:\Gamma_{m+1}\to\Gamma_{m+1}$.

Proof.

This follows from the above discussion. ∎

5.2 Another contraction

We note that we can apply the estimates of Lemma 4.3 to terms such as $Dg_{c}(K{}{})\kappa-Dg_{c}(K{}{})\tilde{\kappa}$. However, we do not have estimates for terms such as $(\kappa_{u}\circ R)(DR)^{\otimes m}-(\tilde{\kappa}_{u}\circ R)(DR)^{\otimes m}$ with $\kappa_{u},\tilde{\kappa}_{u}\in C^{1}_{b}(X_{c},\mathcal{L}^{m}(X_{c},X_{c}))$. Therefore, we start with some preliminary results about the differences of products.

Lemma 5.6**.**

Let $X$, $Y$ and $Z$ be Banach spaces, $m\geq 2$ and $h\in C^{1}_{b}(X,Y)$.

1. i)

Let $f,g\in C^{0}_{b}(Y,\mathcal{L}^{m}(Y,Z))$. For $\tilde{h}\in C^{0}_{b}(X,\mathcal{L}(X,Y))$ we have the $C^{0}$-estimate

[TABLE] 2. ii)

Let $f,g\in C^{1}_{b}(X,\mathcal{L}^{m}(Y,Z))$. For $\tilde{h}\in C^{1}_{b}(X,\mathcal{L}(X,Y))$ we have the $C^{1}$-estimate

[TABLE]

Proof.

i) Let $x\in X$ and recall that the norm of the $k$-linear operator $(f-g)(h(x))$ is given by

[TABLE]

from which the desired $C^{0}$-estimate follows.

iii) For the $C^{1}$-estimate we use the product rule and triangle inequality to find

[TABLE]

We note that $D[\tilde{h}^{\otimes m}]=\sum_{i=0}^{m-1}\left(\tilde{h}^{\otimes i},D\tilde{h},\tilde{h}^{\otimes m-1-i}\right)$, hence we find

[TABLE]

from which the desired estimate follows. ∎

Proposition 5.7**.**

Let $m\geq 2$ and assume that $L_{g}$ and $L_{c}$ are small in the sense of Remark 2.4 for $n=m$. There exists an $\varepsilon_{0}>0$ such for all $\varepsilon<\varepsilon_{0}$ it holds that if $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$ and the fixed point $\Lambda\in\Gamma_{1}(\delta(\varepsilon))$ of $\Theta$ lies in $C^{m}_{b}(X_{c},X)$, then $\Theta^{[m+1]}:\Gamma_{m+1}\to\Gamma_{m+1}$ is a contraction with respect to the $C^{1}$ norm.

Proof.

Let $\varepsilon>0$ and $\|D^{2}g\|_{0},\|D^{2}k_{c}\|_{0}\leq\varepsilon$. Let $\mathcal{M}=\left(\begin{smallmatrix}\rho\\ \kappa_{u}\\ \kappa_{s}\end{smallmatrix}\right),\tilde{\mathcal{M}}=\left(\begin{smallmatrix}\tilde{\rho}\\ \tilde{\kappa}_{u}\\ \tilde{\kappa}_{s}\end{smallmatrix}\right)\in\Gamma_{\tilde{m}}$, where we denote $\tilde{m}=m+1$.

To show that $\Theta^{[\tilde{m}]}$ is a $C^{1}$ contraction, we use the same steps as we used in the proofs of Theorems 3.9 and 4.5.

1. A)

We prove that $\Theta^{[\tilde{m}]}$ is a contraction with respect to the $C^{0}$ norm, independent of $\varepsilon$, 2. B)

We show the existence of a constant $\theta_{\tilde{m}}(\varepsilon)$ such that

[TABLE] 3. C)

We show that $\varepsilon>0$ can be chosen so that $\theta_{\tilde{m}}(\varepsilon)<1$, thus showing that $\Theta^{[\tilde{m}]}$ is a contraction with respect to the $C^{1}$ norm.

Step A) We want to find $\theta_{m}<1$ such that

[TABLE]

As we did in the proofs of Theorems 3.9 and 4.5, we will find the contraction constant component-wise, i.e. we will show that

[TABLE]

for $\theta_{m,i}$ given explicitly in equation 2.0.3a, 2.0.3b and 2.0.3c and $i=1,2,3$. Furthermore, we note that we will directly apply Lemma 4.3ii) and Lemma 5.6ii) to obtain the estimates 5.2.1, LABEL:, 5.2.2, LABEL: and 5.2.4

$\rho$-component: We recall that $\|Dg_{c}\|_{0}\leq L_{g}$ and $\|Dk_{c}\|_{0}\leq L_{c}$, which follows from assumption 4 of Theorem 2.1. We estimate

[TABLE]

$\kappa_{u}$-component: Similarly, we have the estimates $\|Dg_{u}\|_{0}\leq L_{g}$, $\|(DR)\|_{0}\leq\|A_{c}\|_{\textup{op}}+L_{r}$ and $\|k_{u}\|_{0}\leq L_{u}$, hence we find

[TABLE]

$\kappa_{s}$-component: Finally, we have the bounds $\|DT\|_{0}\leq L_{-1}$, $\|Dk_{s}\|_{0}\leq L_{s}$, $\|Dg_{s}\|_{0}\leq L_{g}$, $\|DK{}{}\|_{0}\leq 1+L_{c}$ and

[TABLE]

We infer that

[TABLE]

Contraction constant: We can now estimate $\|\Theta^{[\tilde{m}]}(\mathcal{M})-\Theta^{[\tilde{m}]}(\tilde{\mathcal{M}})\|_{0}$ with inequalities 5.2.1, 5.2.2 and 5.2.4. We have

[TABLE]

Here we define $\theta_{m}:=\max\left\{\theta_{m,1},\theta_{m,2},\theta_{m,3}\right\}$. We have assumed that Remark 2.4 holds for $n=m$. Therefore, we have $\theta_{m,i}<1$ and thus $\theta_{m}<1$. We conclude that $\Theta^{[\tilde{m}]}$ is a contraction with respect to the $C^{0}$ norm.

Step B) Analogous to step A), we want to prove the component-wise inequality

[TABLE]

with $\theta_{\tilde{m},i}$ defined in 2.0.3a, 2.0.3b and 2.0.3c and $C_{\tilde{m},i}$ will be defined during the proof. We note that $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in\Gamma_{1}(\delta(\varepsilon))$, hence $\|D^{2}r\|_{0},\|D^{2}k_{u}\|_{0},\|D^{2}k_{s}\|_{0}\leq\delta=\delta(\varepsilon)$.

$\rho$-component: Using the same estimates as we did for the $\rho$-component in step B) of the proof of Proposition 4.5, we find

[TABLE]

where we define $C_{\tilde{m},1}(\varepsilon):=\left(1+L_{c}+\|A_{c}\|_{\textup{op}}+L_{r}\right)\varepsilon$.

$\kappa_{u}$-component: We have

[TABLE]

By applying Lemma 4.3ii) and Lemma 5.6ii) we find the estimates

[TABLE]

By combining these with 5.2.7 we estimate

[TABLE]

where we define

[TABLE]

$\kappa_{s}$-component: We have

[TABLE]

Before we apply Lemma 4.3ii), we start by deriving an upper bound for $Dh_{m}(\rho)-Dh_{m}(\tilde{\rho})$.

[TABLE]

From Lemma 5.3 we see that $D^{2}T$ is bounded by $\|DTD^{2}r(T)\left(DT\right)^{\otimes 2}\|_{0}$ since $\mathcal{P}_{2}=0$. Using the estimates $\|DT\|_{0}\leq L_{-1}$ and $\|D^{2}T\|_{0}\leq L_{-1}^{3}\delta(\varepsilon)$, see 3.4.10 for the latter bound, we find

[TABLE]

We can now apply Lemma 4.3ii) and estimate 5.2.3 for the bound on $h_{m}(\rho)-h_{m}(\tilde{\rho})$ to get

[TABLE]

By applying Lemma 5.6ii) we find

[TABLE]

For the third term of the right hand side of 5.2.9, we take $x\mapsto Dg_{s}(K{}{}(x))DK{}{}(x)$ for the functions $f_{1}$ and $g_{1}$ in Lemma 4.3ii) and use $T$ as $h$. Recall that $\gamma(\varepsilon)=\max\{\varepsilon,\delta(\varepsilon)\}$, so that

[TABLE]

Hence we find the estimate, using Lemma 4.3ii) and 5.2.10 and 5.2.3,

[TABLE]

For the final term, we will apply Lemma 5.6, which gives

[TABLE]

We first recall that 5.2.11 and 5.2.12 should be multiplied by $\|A_{s}\|_{\textup{op}}$. We will collect the terms without $\varepsilon$ in 5.2.11, 5.2.12 and 5.2.14, which sum up to $\theta_{\tilde{m},3}$. Then we collect the terms containing $\delta(\varepsilon)$ and $\gamma(\varepsilon)$ in 5.2.11, LABEL:, 5.2.12, LABEL:, 5.2.13, LABEL: and 5.2.15. We estimate $\delta(\varepsilon)$ by $\gamma(\varepsilon)$, so that those terms together are bounded by $\tilde{m}\theta_{\tilde{m}+1,3}\gamma(\varepsilon)$. Together with terms containing $\varepsilon$, we get

[TABLE]

Finally, we conclude that 5.2.9 reduces to

[TABLE]

Lipschitz constant: Inequalities 5.2.6, 5.2.8 and 5.2.16 give

[TABLE]

Here we define $\theta_{\tilde{m}}(\varepsilon):=\max\left\{\theta_{\tilde{m},1},\theta_{\tilde{m},2},\theta_{\tilde{m},3}\right\}+\max\{C_{\tilde{m},1}(\varepsilon),C_{\tilde{m},2}(\varepsilon),C_{\tilde{m},3}(\varepsilon)\}$.

Step C) From Remark 2.4 it follows that

[TABLE]

As $\delta(\varepsilon)\downarrow 0$ when $\varepsilon\downarrow 0$, see Proposition 3.4, we have

[TABLE]

Therefore, we can find an $\varepsilon_{0}>0$ such that $\theta_{\tilde{m}}(\varepsilon)<1$ for all $\varepsilon<\varepsilon_{0}$. Then estimates 5.2.5 and 5.2.17 give

[TABLE]

Here we define the contraction constant $\lambda_{\tilde{m}}:=\max\{\theta_{m},\theta_{\tilde{m}}(\varepsilon)\}$ which is smaller than $1$ by our previous discussion. We conclude that $\Theta^{[\tilde{m}]}:\Gamma_{\tilde{m}}\to\Gamma_{\tilde{m}}$ is a contraction with respect to the $C^{1}$ norm for all $\varepsilon<\varepsilon_{0}$. ∎

Corollary 5.8**.**

Let $\varepsilon>0$ be such that $F{}{}:X\to X$ satisfies the conditions of Theorems 2.1, 3.9, 4.5 and 5.7. Then the image of $K{}{}$ is a $C^{n}$ center manifold for $F{}{}$.

Proof.

The proof follows by induction on the smoothness of the conjugacy and the conjugate dynamics, with similar arguments as we had in the proof of Proposition 4.5. ∎

6 Existence and uniqueness of the center manifold

6.1 Proof of Theorem 2.1

From Corollary 5.8 it follows that there exists an $\varepsilon>0$ such that if $\|D^{2}g\|_{0}$ and $\|D^{2}k_{c}\|_{0}$ are smaller than $\varepsilon$, then there exists a $C^{n}$ center manifold for $F{}{}:X\to X$. However, in Theorem 2.1, the only assumption on the second derivative of $k_{c}$ and $g$ is that they are both bounded. We will use a simple scaling argument to show that we can bound $D^{2}k_{c}$ and $D^{2}g$ by $\varepsilon$ without affecting the assumed bound on $Dk_{c}$ and $Dg$ from Theorem 2.1.

Lemma 6.1**.**

Let $\varepsilon>0$ and define $h^{\varepsilon}(x):=\varepsilon^{-1}h(\varepsilon x)$ for $h\in C^{2}_{b}(X,Y)$.

1. i)

We have $h^{\varepsilon}(0)=0$ if $h(0)=0$, $Dh^{\varepsilon}(0)=Dh(0)$ and $\|Dh^{\varepsilon}\|_{0}=\|Dh\|_{0}$. 2. ii)

We have $\|D^{2}h^{\varepsilon}\|_{0}=\varepsilon\|D^{2}h\|_{0}$. 3. iii)

For $h_{1}\in C_{b}^{2}(Y,Z)$ and $h_{2}\in C_{b}^{2}(X,Y)$ we have $h_{1}^{\varepsilon}\circ h_{2}^{\varepsilon}=(h_{1}\circ h_{2})^{\varepsilon}$.

Proof.

These results follow from straightforward computations. ∎

We can now prove Theorem 2.1.

Proof of Theorem 2.1.

Let $k_{c}:X_{c}\to X$ be chosen such that

[TABLE]

Then it follows from Lemma 6.1i) that for all $\varepsilon>0$ we have

[TABLE]

Likewise, if $F{}{}=A+g$ satisfies the conditions of Theorem 2.1, we have for all $\varepsilon>0$ that $F{}{}^{\varepsilon}=A+g^{\varepsilon}$ with

[TABLE]

Hence $F{}{}^{\varepsilon}$ also satisfies the conditions of Theorem 2.1.

By Lemma 6.1ii) we can apply Corollary 5.8 to $F{}{}^{\varepsilon}$ and $k_{c}^{\varepsilon}$ for $\varepsilon$ sufficiently small. We fix $\varepsilon>0$ sufficiently small. Corollary 5.8 then provides a $K{}{}:X_{c}\to X$ which conjugates $f^{\varepsilon}:X\to X$ with $A_{c}+r:X_{c}\to X_{c}$. Then we find from Lemma 6.1iii) that

[TABLE]

and, again by Lemma 6.1iii),

[TABLE]

Thus we see that $K{}{}^{1/\varepsilon}:X_{c}\to X$ conjugates $F{}{}:X\to X$ with $A_{c}+r^{1/\varepsilon}:X_{c}\to X_{c}$. Furthermore, by combining Lemma 6.1i) with Corollary 3.10, it follows that $r^{1/\varepsilon}$ satisfies property A) and $k_{u}^{1/\varepsilon}$ and $k_{s}^{1/\varepsilon}$ satisfy property B). ∎

6.2 Uniqueness of the center manifold

Since we found our conjugacy $K{}{}$ and dynamical system $A_{c}+r$ with a contraction argument, we have uniqueness of $\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)$ in $\Gamma_{0}$. However, we can improve the uniqueness of $k_{u}$ and $k_{s}$ to arbitrary bounded functions, and the uniqueness of $r$ to arbitrary bounded functions such that $A_{c}+r$ is invertible.

Lemma 6.2**.**

Let $F{}{}:X\to X$ and $k_{c}:X_{c}\to X_{c}$ satisfy the conditions of Theorem 2.1. Then

[TABLE]

has a unique solution for $k_{u}\in C^{0}_{b}(X_{c},X_{u})$, $k_{s}\in C^{0}_{b}(X_{c},X_{s})$ and $r\in C^{0}_{b}(X_{c},X_{c})$ with the property that $A_{c}+r$ is a homeomorphism.

Proof.

Let $\Lambda=\left(\begin{smallmatrix}r\\ k_{u}\\ k_{s}\end{smallmatrix}\right)\in C_{b}^{n}(X_{c},X)$ be obtained from Theorem 2.1 and let $\mathcal{M}=\left(\begin{smallmatrix}\tilde{r}\\ \tilde{k}_{u}\\ \tilde{k}_{s}\end{smallmatrix}\right)\in C^{0}_{b}(X_{c},X)$ be such that 6.2.1 holds. Then we know from Proposition 3.1 that $\mathcal{M}=\Theta(\mathcal{M})$ and $\Lambda=\Theta(\Lambda)$. We can mimic step A) of the proof of Theorem 3.9, since Lemma 3.5i) and Lemma 3.7i) can both be applied to the components of $\Theta(\mathcal{M})-\Theta(\Lambda)$, even though $\mathcal{M}$ is merely $C^{0}$, see Remark 3.8. Hence we obtain

[TABLE]

Since $\theta_{0}<1$, we find $\mathcal{M}=\Lambda$. ∎

We conclude from Lemma 6.2 that, given $k_{c}:X_{c}\to X_{c}$, the conjugacy between the center dynamical system $R:X_{c}\to X_{c}$ and the original dynamical system $F{}{}:X\to X$ is unique in the space of continuous functions with bounded unstable and stable components. Additionally, we want to show that the center manifold is unique independent of our choice of $k_{c}:X_{c}\to X_{c}$. That is, we want to prove that the image of the conjugacy does not depend on a given $k_{c}:X_{c}\to X_{c}$.

Proposition 6.3**.**

Let $F{}{}:X\to X$ and $k_{c},\tilde{k}_{c}:X_{c}\to X_{c}$ satisfy the conditions of Theorem 2.1. Then the image of $K{}{}=\iota+\left(\begin{smallmatrix}k_{c}\\ k_{u}\\ k_{s}\end{smallmatrix}\right)$ and $\tilde{K{}{}}=\iota+\left(\begin{smallmatrix}\tilde{k}_{c}\\ \tilde{k}_{u}\\ \tilde{k}_{s}\end{smallmatrix}\right)$ are the same, for $K{}{},\tilde{K{}{}}$ the (unique) conjugacy obtained from Theorem 2.1.

Proof.

From Lemma 3.2i) it follows that $\operatorname{Id}+k_{c}$ is invertible, and from Lemma 3.2ii) we see that its inverse is given by $\operatorname{Id}+\psi$ for some bounded function $\psi:X_{c}\to X_{c}$. In particular, we can write

[TABLE]

where $\varphi=\psi\circ(\operatorname{Id}+\tilde{k}_{c})+\tilde{k}_{c}$ is a bounded function. Likewise, $\operatorname{Id}+\tilde{k}_{c}$ is invertible, and thus $\operatorname{Id}+\varphi=(\operatorname{Id}+k_{c})^{-1}\circ(\operatorname{Id}+\tilde{k}_{c})$ is invertible, as it is the composition of two invertible functions. By Lemma 3.2i) its inverse is given by $\operatorname{Id}+\phi$ for some bounded function $\phi$. We infer that

[TABLE]

where we define $\Phi=r\circ(\operatorname{Id}+\varphi)+\phi\circ(A_{c}+r)\circ(\operatorname{Id}+\varphi)$. By property A) of Theorem 2.1, we have that $A_{c}+r$ is invertible, and thus $A_{c}+\Phi$ is the composition of three invertible functions, hence invertible itself. Therefore, we use Lemma 6.2 to conclude $\tilde{k}_{u}=k_{u}\circ(\operatorname{Id}+\varphi)$ and $\tilde{k}_{s}=k_{s}\circ(\operatorname{Id}+\varphi)$. Since $\operatorname{Id}+\tilde{k}_{c}=(\operatorname{Id}+k_{c})\circ(\operatorname{Id}+\varphi)$, we see that $\tilde{K{}{}}=K{}{}\circ(\operatorname{Id}+\varphi)$. As $\operatorname{Id}+\varphi$ is invertible, we conclude that

[TABLE]

6.3 Proof of Corollary 2.3

Finally, we want to show that if we have found approximations of the center manifold and the center dynamics, i.e., we have found an approximate conjugacy $K{}{}_{0}$ and approximate dynamical system $R_{0}$ such that

[TABLE]

then the center manifold and dynamical system lie close to these approximations.

Proof of Corollary 2.3.

Let $m<n$ and let $k_{0}=\left(\begin{smallmatrix}k_{u,0}\\ k_{s,0}\\ \end{smallmatrix}\right):X_{c}\to X_{u}\oplus X_{s}$ and $r_{0}:X_{c}\to X_{c}$. We use $\mathcal{M}$ to denote $\left(\begin{smallmatrix}r_{0}\\ k_{u,0}\\ k_{s,0}\end{smallmatrix}\right)$ and $\mathcal{F}=F\circ\left(\begin{smallmatrix}k_{c}\\ k_{u,0}\\ k_{s,0}\end{smallmatrix}\right)-\left(\begin{smallmatrix}k_{c}\\ k_{u,0}\\ k_{s,0}\end{smallmatrix}\right)\circ(A_{c}+r_{0})$. We assume that $k_{0}$ and $r_{0}$ satisfy the conditions of Corollary 2.3, that is, there exists constants $M>0$ and $\varepsilon>0$ such that

[TABLE]

Our proof consists of the following steps:

We prove that if $\mathcal{F}$ is small in $C^{m}$, then $\mathcal{M}-\Theta(\mathcal{M})$ is small in $C^{m}$. 2. 2.

We prove that if $\mathcal{M}-\Theta(\mathcal{M})$ is small in $C^{0}$, then $\mathcal{M}-\Lambda$ is small in $C^{0}$, where $\Lambda$ is the fixed point of $\Theta$. 3. 3.

Using induction, we prove that if $\mathcal{M}-\Theta(\mathcal{M})$ is small in $C^{m}$, then $D^{m}\mathcal{M}-D^{m}\Lambda$ is small in $C^{0}$.

For the first step, we want an explicit estimate for $\mathcal{M}-\Theta(\mathcal{M})$. By definition of $\Theta$, see 3.1.3, we have

[TABLE]

We can clearly estimate the $C^{m}$ norm of the first two components by $\|\mathcal{F}\|_{m}\leq\varepsilon$ and $\|A_{u}^{-1}\|_{\textup{op}}\|\mathcal{F}\|_{m}\leq\|A_{u}^{-1}\|_{\textup{op}}\varepsilon$ respectively. For the third component, we use Faá di Bruno’s formula and Lemma 5.3 to obtain an estimate. We find

[TABLE]

where we used that the derivatives of $r_{0}$ are bounded by $M$ and $D(A_{c}+r_{0})^{-1}$ is bounded by $L_{-1}$ as $Dr_{0}$ is bounded by $L_{r}$. Hence we obtain

[TABLE]

Here $\mathcal{C}_{2}(M,m):=\max\{1,\|A_{u}^{-1}\|_{\textup{op}},\mathcal{C}_{1}(M,m)\}$. Hence we have shown that if $\mathcal{F}$ is small, then $\mathcal{M}$ is an almost fixed point of $\Theta$.

To prove the second step, we use that $\Theta$ is a contraction in the $C^{0}$ norm with contraction constant $\theta_{0}$, see the proof of Lemma 6.2, thus we have

[TABLE]

By rewriting, we obtain the estimate

[TABLE]

Combining 6.3.2 and 6.3.1 proves Corollary 2.3 for $m=0$, that is

[TABLE]

To prove step 3, we will use induction. So let us assume that $\|\Lambda-\mathcal{M}\|_{m-1}\leq\mathcal{C}(M,m-1)\varepsilon$. By construction of the contraction $\Theta^{[m+1]}$, its fixed point is $D^{m}\Lambda$. Thus similarly to 6.3.2, we have

[TABLE]

From 6.3.1, we know that $\|D^{m}\left(\mathcal{M}-\Theta(\mathcal{M})\right)\|_{0}\leq\mathcal{C}_{1}(M,m)\varepsilon$. Hence we estimate 6.3.3 with the triangle inequality and the estimate on $D^{m}\left(\mathcal{M}-\Theta(\mathcal{M})\right)$ to obtain

[TABLE]

Therefore, it remains to find a bound on $D^{m}\Theta(\mathcal{M})-\Theta^{[m+1]}(D^{m}\mathcal{M})$. Recall from 3.1.2 the definition of $\Theta$. By Faá di Bruno’s formula, there exists a function $\mathcal{G}$ such that for all $\mathcal{T}:X_{c}\to X$

[TABLE]

By construction, $\mathcal{G}(\mathcal{T},\dots,D^{m}\mathcal{T})(x)$ is a linear combination of products of various derivatives of $\mathcal{T}$, evaluated at either $x$ or $\mathcal{T}(x)$. In particular, we have for $\mathcal{M}:X_{c}\to X$

[TABLE]

Subtracting 6.3.7 from 6.3.6 we thus obtain

[TABLE]

where $\mathcal{N}(s)=(1-s)\left(\mathcal{M},\dots,D^{m-1}\mathcal{M}\right)+s\left(\Lambda,\dots,D^{m-1}\Lambda\right)$.

To calculate the partial derivative of $\mathcal{G}$ in the direction of its first input, we use the chain rule and obtain an expression depending on $\mathcal{G}$, $\mathcal{N}(s)$, $D\mathcal{N}(s)$, $D^{m}\mathcal{M}$ and $D^{m+1}\mathcal{M}$. The other partial derivatives of $\mathcal{G}$ are partial derivatives of polynomials, hence only depend on $\mathcal{G}$, $\mathcal{N}(s)$ and $D^{m}\mathcal{M}$. In particular, $D\mathcal{G}$ is continuous and evaluated on the compact set $\left\{(\mathcal{N}(s),D^{m}\mathcal{M})\mid s\in[0,1]\right\}$ in 6.3.8 – note that $\Lambda,\mathcal{M}\in C_{b}^{m+1}(X_{c},X)$. Thus $\|D\mathcal{G}(\mathcal{N}(s),D^{m}\mathcal{M})\|_{\textup{op}}\leq\mathcal{C}_{3}(M,m)$ for all $s\in[0,1]$. Therefore, we can estimate 6.3.8 by

[TABLE]

Using 6.3.4 and our induction hypothesis, we thus conclude that

[TABLE]