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August  2017, 37(7): 3601-3623. doi: 10.3934/dcds.2017155

## On a definition of Morse-Smale evolution processes

 1 Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland 2 Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010,05508-090 São Paulo, Brazil 3 Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

* Corresponding author: Carlos Rocha

Received  June 2016 Revised  February 2017 Published  April 2017

Fund Project: This work was partially supported by FCT/Portugal through the project UID/MAT/04459/2013 and FAPESP thematic project 2015/21049-3.

In this paper we consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework. In particular we consider nonautonomous perturbations of autonomous systems. In this case our definition of Morse-Smale evolution process holds for perturbations of Morse-Smale autonomous systems with or without periodic orbits. We establish that small nonautonomous perturbations of autonomous Morse-Smale evolution processes derived from certain nonautonomous differential equations are Morse-Smale evolution processes. We apply our results to examples of scalar parabolic semilinear differential equations generating evolution processes and possessing periodic orbits.

Citation: Radosław Czaja, Waldyr M. Oliva, Carlos Rocha. On a definition of Morse-Smale evolution processes. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3601-3623. doi: 10.3934/dcds.2017155
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##### References:
Phase portrait of the autonomous equation (10) for $\delta>0$ .
Recurrent behavior for $\xi=(t, z(t))$ . Here $t_+-\bar t>\overline T$ and $\bar{t}-t_->\overline T$ .
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