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Embedding of global attractors and their dynamics

Authors: Eleonora Pinto de Moura, James C. Robinson and J. J. Sánchez-Gabites
Journal: Proc. Amer. Math. Soc. 139 (2011), 3497-3512
MSC (2010): Primary 37L30, 54H20, 57N60
Published electronically: February 9, 2011
MathSciNet review: 2813382
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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$.

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Additional Information

Eleonora Pinto de Moura
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

James C. Robinson
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

J. J. Sánchez-Gabites
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

Received by editor(s): August 12, 2010
Received by editor(s) in revised form: August 23, 2010
Published electronically: February 9, 2011
Additional Notes: The first author is sponsored by CAPES and would like to thank CAPES for all their support during her Ph.D
The second and third authors are supported by EPSRC Leadership Fellowship EP/G007470/1.
Communicated by: Professor Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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