Embedding of global attractors and their dynamics

Abstract:

Suppose that $\mathcal A$ is the global attractor associated with a dissipative dynamical system on a Hilbert space $H$.

If the set $\mathcal A-\mathcal A$ has finite Assouad dimension $d$, then for any $m>d$ there are linear homeomorphisms $L:\mathcal {A}\rightarrow \mathbb {R}^{m+1}$ such that $L{\mathcal A}$ is a cellular subset of $\mathbb {R}^{m+1}$ and $L^{-1}$ is log-Lipschitz (i.e. Lipschitz to within logarithmic corrections). We give a relatively simple proof that a compact subset $X$ of $\mathbb {R}^k$ is the global attractor of some smooth ordinary differential equation on $\mathbb {R}^k$ if and only if it is cellular, and hence we obtain a dynamical system on $\mathbb {R}^k$ for which $L{\mathcal A}$ is the global attractor. However, $L\mathcal {A}$ consists entirely of stationary points.

In order for the dynamics on $L\mathcal A$ to reproduce those on $L\mathcal A$ we need to make an additional assumption, namely that the dynamics restricted to $\mathcal A$ are generated by a log-Lipschitz continuous vector field (this appears overly restrictive when $H$ is infinite-dimensional, but is clearly satisfied when the initial dynamical system is generated by a Lipschitz ordinary differential equation on $\mathbb {R}^N$). Given this we can construct an ordinary differential equation in some $\mathbb {R}^k$ (where $k$ is determined by $d$ and $\alpha$) that has unique solutions and reproduces the dynamics on $\mathcal A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $\mathcal X$ arbitrarily close to $L\mathcal A$.