# American Institute of Mathematical Sciences

August  2020, 13(8): 2303-2326. doi: 10.3934/dcdss.2020092

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

 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  May 2018 Revised  August 2018

Fund Project: 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

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.

Citation: Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2303-2326. doi: 10.3934/dcdss.2020092
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