Application of the Feigenbaum-Sharkovskii-Magnitskii theory to the analysis of Hamiltonian systems

Abstract

The modern classical theory of Hamiltonian conservative systems reduces the analysis of dynamics of such systems to the problem of their integrability, i.e., to the construction of a canonical transformation reducing the system to the action–angle variables, in which, as is assumed, the motion occurs on the surface of an n-dimensional torus and is periodic or quasiperiodic. Any nonintegrable nonlinear Hamiltonian system is treated as a perturbation of an integrable system, and the analysis of its dynamics can be reduced to finding out whether the tori of the unperturbed system are destroyed depending on the value of the applied perturbation. In accordance with the Kolmogorov–Arnold–Moser (KAM) theory, it is assumed that the destruction of some tori of the unperturbed system leads to the generation of chaotic dynamics in the perturbed system [1–3]. In the recent years, a harmonious theory of the dynamic and space-time chaos in nonlinear dissipative systems of differential equations was constructed (the Feigenbaum–Sharkovskii–Magnitskii theory, or FSM theory [4]); this theory covers ordinary differential equations, partial differential equations, and equations with delayed argument [5–7]. In this theory, chaos in dissipative systems develops not by an instant destruction of some regular state of the system under an arbitrarily small perturbation but rather by an infinite cascade of bifurcations of generation of periodic or quasiperiodic solutions of increasing complexity. The aim of the present paper is to show, on the basis of numerical simulation, that it is this scenario of passage to chaos that is realized, in complete correspondence with the FSH theorem, at least in three Hamiltonian systems with one and a half degrees of freedom, i.e., in three two-dimensional conservative nonautonomous systems with periodic coefficients, namely, the conservative generalized Mathieu equation, the conservative Duffing–Holmes equation, and the modified conservative system of Crockett equations.