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Negatively Invariant Sets and Entire Solutions

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Abstract

Negatively invariant compact sets of autonomous and nonautonomous dynamical systems on a metric space, the latter formulated in terms of processes, are shown to contain a strictly invariant set and hence entire solutions. For completeness the positively invariant case is also considered. Both discrete and continuous time systems are considered. In the nonautonomous case, the various types of invariant sets are in fact families of subsets of the state space that are mapped onto each other by the process. A simple example shows the usefulness of the result for showing the occurrence of a bifurcation in a nonautonomous system.

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Authors and Affiliations

  1. Institut für Mathematik, Goethe Universität, 60054, Frankfurt am Main, Germany

    Peter E. Kloeden

  2. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Sevilla, Spain

    Pedro Marín-Rubio

Authors
  1. Peter E. Kloeden
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  2. Pedro Marín-Rubio
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Correspondence to Peter E. Kloeden.

Additional information

Dedicated to Russell Johnson on the occasion of his sixtieth birthday.

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Kloeden, P.E., Marín-Rubio, P. Negatively Invariant Sets and Entire Solutions. J Dyn Diff Equat 23, 437–450 (2011). https://doi.org/10.1007/s10884-010-9196-8

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  • DOI: https://doi.org/10.1007/s10884-010-9196-8

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