On the forward dynamical behavior of nonautonomous systems

Chunqiu Li; Desheng Li; Xuewei Ju

doi:10.3934/dcdsb.2019190

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2020, Volume 25, Issue 1: 473-487. Doi: 10.3934/dcdsb.2019190

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On the forward dynamical behavior of nonautonomous systems

1.
College of Mathematical Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China
2.
School of Mathematics, Tianjin University, Tianjin 300072, China
3.
Department of Mathematics, School of Science, Civil Aviation University of China, Tianjin 300300, China

^* Corresponding author: Xuewei Ju
^* Corresponding author: Xuewei Ju

Received: August 31, 2018

Revised: March 31, 2019

Early access: September 2019

Published: January 2020

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Abstract

This paper is concerned with the forward dynamical behavior of nonautonomous systems. Under some general conditions, it is shown that in an arbitrary small neighborhood of a pullback attractor of a nonautonomous system, there exists a family of sets $ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $ of phase space $ X $, which is forward invariant such that $ \{\mathcal {A}_\varepsilon(p)\}_{p\in P} $ uniformly forward attracts each bounded subset of $ X $. Furthermore, we can also prove that $ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $ forward attracts each bounded set at an exponential rate.

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Mathematics Subject Classification: 35B41, 37B25, 37B55, 37L30.

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