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Embedding of global attractors and their dynamics. (English) Zbl 1250.37032

Summary: 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 \( L\mathcal A\).

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
54H20 Topological dynamics (MSC2010)
57N60 Cellularity in topological manifolds
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References:

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