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Perturbations of weakly hyperbolic invariant sets of two-dimension periodic systems

Abstract

The question of structural stability is one of the most important areas in a present-day theory of differential equations. In this paper, we study small C 1 perturbations of a systems of differential equations. We introduce the concepts of a weakly hyperbolic invariant set K and leaf Y for a system of ordinary differential equations. The Lipschitz condition is not assumed. We show that, if the perturbation is small enough, then there is a continuous mapping h, i.e., KK Y, where K Y is a weakly hyperbolic set of the perturbed equation system. Moreover, we show that h(Y) is a leaf of the perturbed system.

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Correspondence to N. A. Begun.

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Original Russian Text © N.A. Begun, 2015, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2015, No. 1, pp. 23–33.

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Begun, N.A. Perturbations of weakly hyperbolic invariant sets of two-dimension periodic systems. Vestnik St.Petersb. Univ.Math. 48, 1–8 (2015). https://doi.org/10.3103/S1063454115010033

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Key words

  • stability
  • invariant set
  • small perturbations
  • hyperbolic structures
  • weakly hyperbolic set
  • structural stability