Evolution groups and reversibility

Abstract

We give a complete description of the correspondence between the reversibility and equivariance properties of evolution families (so for dynamical systems with continuous time, possibly nonautonomous) and their evolution groups. To the best of our knowledge, no similar result appeared before in the literature for dynamical systems with continuous time, even autonomous. Moreover, based on these results, we describe the faithful correspondence between the reversibility properties of stable and unstable invariant manifolds of evolution families and of their associated evolution groups. Finally, we construct center invariant manifolds for an equation that has a line of equilibria generated by a group of symmetries for which the equation is equivariant.

Appendix A: Nonautonomous center manifold theorem

In this appendix, we establish a nonautonomous center manifold theorem that is required for the proof of Theorem 5.2. The proof is based on a corresponding autonomous center manifold theorem. We shall use the notations of Sect. 3.1. In particular, under suitable conditions, the evolution family $(W(t,s))_{t,s \in {\mathbb {R}}}$ is determined by the solutions of Eq. (3.3).

Theorem A.1

Let $\mathcal {U}$ be an evolution family that is exponentially bounded and partially hyperbolic. Moreover, let $f :{\mathbb {R}}\times X \rightarrow X$ be a continuous function, such that for some constant $\mu >0$, we have

$$\begin{aligned} \Vert f(t,x)-f(t,y)\Vert \le \mu \Vert x -y \Vert \end{aligned}$$

for all $t \in {\mathbb {R}}$ and $x,y \in X$. If $\mu$ is sufficiently small, then there exist $r>0$ and a continuous function

$$\begin{aligned} \phi :\big \{(t,v) \in {\mathbb {R}}\times X: v \in G_t \big \} \rightarrow X, \end{aligned}$$

such that

• 1. letting $\phi _t =\phi (t,\cdot )$, we have $\phi _t(0)=0$, $\phi _t (G_t) \subset E_t \oplus F_t$ and

  $$\begin{aligned} \Vert \phi _t(u)-\phi _t(v) \Vert \le \Vert u -v \Vert \end{aligned}$$

  for all $t \in {\mathbb {R}}$ and $u,v \in G_t$;

• 2. letting

  $$\begin{aligned} \mathcal {V}=\bigl \{(t,u+\phi _t(u)): (t,u) \in {\mathbb {R}}\times (G_t \cap B(0,r)) \bigr \}, \end{aligned}$$

  the following properties hold:

  • (a) if $(s, x) \in \mathcal {V}$ and $W(t,s)x \in B(0,r)$ for all $t \in {\mathbb {R}}$ in some open interval $I \subset {\mathbb {R}}$ containing s, then

    $$\begin{aligned} (t,W(t,s)x) \in \mathcal {V}\quad \text {for all} \ t \in I; \end{aligned}$$

  • (b) if $(s,x) \in \mathcal {V}$ and $W(t,s)x \in B(0,r)$ for all $t \in {\mathbb {R}}$, then

    $$\begin{aligned} (t,W(t,s)x) \in \mathcal {V}\quad \text {for all} \ t \in {\mathbb {R}}. \end{aligned}$$

Proof

With the purpose of constructing center manifolds for the original nonautonomous dynamical system, we first consider a corresponding autonomous problem (for a proof see for example [9]).

Lemma A.2

Let A be a partially hyperbolic bounded linear operator (with stable, unstable, and center spaces E, F, and G) and let $g :X \rightarrow X$ be a function, such that for some constant $\mu >0$, we have

$$\begin{aligned} \Vert g(x)-g(y)\Vert \le \mu \Vert x -y \Vert \quad \text {for all} \ x,y \in X. \end{aligned}$$

If $\mu$ is sufficiently small, then there exist a function $\phi :G\rightarrow E\oplus F$ and constants $\alpha ,\beta >0$, such that:

• 1. $\phi (0)=0$ and $\Vert \phi (u)-\phi (v) \Vert \le \Vert u -v \Vert$ for all $u,v \in G$;

• 2. letting $\mathcal {V}=\{u+\phi (u): u \in G \}$, the following properties hold:

  • (a) $\mathcal {V}$ is invariant under the flow $\Psi _t$ determined by the differential equation $x'=Ax+g(x)$;

  • (b) $x\in \mathcal {V}$ if and only if

    $$\begin{aligned} \sup _{t\le 0}(\Vert \Psi _t(x)\Vert e^{\alpha t})<\infty \quad \text {and}\quad \sup _{t\ge 0}(\Vert \Psi _t(x)\Vert e^{-\beta t})<\infty . \end{aligned}$$

    (6.8)

We continue with the proof of the theorem. Consider the autonomous equation obtained from adding the equation $t'=1$ to Eq. (3.3). This equation has one additional central direction that is perturbed by the constant function 1. Hence, one can apply Lemma A.2 to construct a center manifold $\mathcal {V}$. The invariance property (a) in the theorem follows readily from the invariance property in the lemma.

Finally, if $(s,x) \in \mathcal {V}$ and $W(t,s)x \in B(0,r)$ for all $t \in {\mathbb {R}}$, then the solution $t\mapsto (t,W(t,s)x)$ satisfies property (6.8) for some appropriate constants $\alpha ,\beta >0$. Indeed, the first component only grows polynomially and the second component remains bounded. Therefore, the characterization of the set $\mathcal {V}$ in the lemma yields property (b) in the theorem. $\square$