Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations

Xuewei Ju; Desheng Li; Jinqiao Duan

doi:10.3934/dcdsb.2019011

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2019, Volume 24, Issue 3: 1175-1197. Doi: 10.3934/dcdsb.2019011

This issue Previous Article A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere Next Article On relation between attractors for single and multivalued semiflows for a certain class of PDEs

Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations

1.
Department of Mathematics, School of Science, Civil Aviation University of China, Tianjin 300300, China
2.
School of Mathematics, Tianjin University, Tianjin 300072, China
3.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

^* Corresponding author
^* Corresponding author

Received: May 2017

Revised: December 2017

Early access: January 2019

Published: January 2019

Supported by NNSF of China under the grant 11471240 and the Foundation Research Funds for the Central Universities under the grant 3122014K008

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Abstract

We consider the nonautonomous perturbation $ x_t+Ax = f(x)+\varepsilon h(t) $ of a gradient-like system $ x_t+Ax = f(x) $ in a Banach space $ X $, where $ A $ is a sectorial operator with compact resolvent. Assume the non-perturbed system $ x_t+Ax = f(x) $ has an attractor $ {\mathscr A} $. Then it can be shown that the perturbed one has a pullback attractor $ {\mathscr A} _\varepsilon $ near $ {\mathscr A} $. If all the equilibria of the non-perturbed system in $ {\mathscr A} $ are hyperbolic, we also infer from [4,6] that $ {\mathscr A} _\varepsilon $ inherits the natural Morse structure of $ {\mathscr A} $. In this present work, we introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of $ {\mathscr A} _\varepsilon $ and the asymptotically synchronizing behavior of the perturbed system. Based on the above result we further prove that the sections of $ {\mathscr A} _\varepsilon $ depend on time symbol continuously in the sense of Hausdorff distance. Consequently, one concludes that $ {\mathscr A} _\varepsilon $ is a forward attractor of the perturbed nonautonomous system. It will also be shown that the perturbed system exhibits completely a global forward synchronizing behavior with the external force.

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Mathematics Subject Classification: 37B35, 37B55, 37E10, 34D06, 34D45, 34D06.

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