Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems

Yejuan Wang; Tomás Caraballo

doi:10.3934/dcdss.2020092

Article Contents

2020, Volume 13, Issue 8: 2303-2326. Doi: 10.3934/dcdss.2020092

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Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems

Yejuan Wang^1, and
Tomás Caraballo^2,

1.
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

Received: April 30, 2018

Revised: July 31, 2018

Early access: June 2019

Published: May 2020

This work was supported by NSF of China (Grants No. 41875084, 11571153), and the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-it58 and lzujbky-2018-ot03, and partially supported by Ministerio de Economía y Competitividad (Spain), FEDER (European Community) under grant MTM2015-63723-P, and Consejería de Innovación Ciencia y Empresa de la Junta de Andalucía (Spain) under grant P12-FQM-1492

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Abstract

In this paper, we first prove that the property of being a gradient-like general dynamical system and the existence of a Morse decomposition are equivalent. Next, the stability of gradient-like general dynamical systems is analyzed. In particular, we show that a gradient-like general dynamical system is stable under perturbations, and that Morse sets are upper semi-continuous with respect to perturbations. Moreover, we prove that any solution of perturbed general dynamical systems should be close to some Morse set of the unperturbed gradient-like general dynamical system. We do not assume local compactness for the metric phase space $ X $, unlike previous results in the literature. Finally, we extend the Morse decomposition theory of single-valued nonautonomous dynamical systems to the multi-valued case, without imposing any compactness of the parameter spaces.

Keywords:

Gradient-like general dynamical systems,
Morse decomposition,
nonautonomous multi-valued dynamical systems.

Mathematics Subject Classification: Primary: 35B40, 35B41, 37L30; Secondary: 58C06, 47H20.

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