Compound motion of conservative systems under periodic perturbation

Abstract

A model is described in which chaos occurs as a motion of an autonomous two-frequency system. The boundaries for the domain of stochastic motions caused by bifurcational and chaotic processes are estimated Keywords: chaos, autonomous two-frequency system, orbital loss of stability, bifurcation Introduction. Physically, the complex behavior of a finite-dimensional system is due to the instability of individual motions. The instability of trajectories located within a bounded domain of the phase space leads to "mixing" and, as a consequence, stochastic motion. The approach used in this case is to set up variational equations with the coefficients of the characteristic equation dependent on a partial solution. The following types of systems are best studied (qualitatively).1. Systems with periodic perturbation. This is the widest class of problems [7,8]. A qualitative analysis shows that orbital loss of stability may occur and cause chaos in some cases [3,5].2. Two-frequency systems described by a system of fourth-order equations. Such systems are also within the limits of applicability of the approach [9].3. Lorentz-type systems. These are systems whose dimension is no less than three and that generate saddle cycles. In many cases, a chain of bifurcations leads from determinacy to chaos. Each such transition is accompanied by loss of stability of a simple manifold and birth of a new, more complicated stable manifold. Internal parametric fluctuations may affect the mechanism of compound motions and cause unexpected qualitative changes in the behavior of the system. Orbital stability analysis is the key aspect in the understanding of the mechanism of complex nonlinear dynamic phenomena.Here we conduct a qualitative analysis of systems of nonlinear conservative oscillators with periodic forcing. An important characteristic of the dynamic behavior of systems is Lyapunov characteristic exponents (LCEs) (see [1,2] and the references therein). We will estimate the initial perturbations for certain types of motion, including chaos.