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On a definition of Morse-Smale evolution processes

  • * Corresponding author: Carlos Rocha

    * Corresponding author: Carlos Rocha

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

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  • 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.

    Mathematics Subject Classification: Primary: 37B55, 35B41; Secondary: 35B10, 37B20, 37B35.


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  • Figure 1.  Phase portrait of the autonomous equation (10) for $\delta>0$ .

    Figure 2.  Recurrent behavior for $\xi=(t, z(t))$ . Here $t_+-\bar t>\overline T$ and $\bar{t}-t_->\overline T$ .

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