Forward omega limit sets of nonautonomous dynamical systems

Hongyong Cui; Peter E. Kloeden; Meihua Yang

doi:10.3934/dcdss.2020065

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2020, Volume 13, Issue 4: 1103-1114. Doi: 10.3934/dcdss.2020065

This issue Previous Article Variational discretization of thermodynamical simple systems on Lie groups Next Article On Lie algebra actions

Forward omega limit sets of nonautonomous dynamical systems

School of Mathematics and Statistics, and, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Dedicated to Professor Jürgen Scheurle on his 65th birthday

Received: September 2017

Revised: May 2018

Published: April 2020

HC was partially funded by China Postdoctoral Science Foundation 2017M612430. PEK and MY were partially supported by the Chinese NSF grant 11571125

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Abstract

The forward $ \omega $-limit set $ \omega_{\mathcal{B}} $ of a nonautonomous dynamical system $ \varphi $ with a positively invariant absorbing family $ \mathcal{B} $ $ = $ $ \{ B(t), t \in \mathbb{R}\} $ of closed and bounded subsets of a Banach space $ X $ which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if $ \varphi $ is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for a $ \varphi $-invariant family $ \mathcal{A} $ $ = $ $ \left\{A(t), t \in \mathbb{R}\right\} $ in $ \mathcal{B} $ of nonempty compact subsets of $ X $ to be a forward attractor is generalised to this context.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.

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