Recherches et rapports sur les mathématiques

Stability under C1 - Perturbation of Dynamical System

Osipenko GS and Korzh TN

The aim is substantiation of a constructive method for verification of structural stability of discrete dynamical systems. The hyperbolicity of the chain recurrent set is a necessary condition for the structural stability. If the spectrum of the differential Df does not contain 0, then the chain recurrent set is hyperbolic and the system is Ω-stable. The spectrum is estimated through the symbolic image of operation of the differential on projective bundle. A diffeomorphism f is shown to be structurally stable if and only if the spectrum of complementary differential Df ˆ does not contain 0 and there is no connection CR+→CR on the projective bundle, where CR+ and CR denote the chain recurrent components for the positive and negative parts of the spectrum. These conditions are verified through the symbolic image of the complementary differential.