On three types of dynamics and the notion of attractor

S. V. Gonchenko ; D. V. Turaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 133-157

Résumé

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor-repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.