Investigation of the complex dynamics of nonlinear oscillators
by using the scheme of the catastrophe theory

Anna Potapova

Department of Nonlinear processes, Saratov State University, Russia

Abstract: In present work the behavior of the nonlinear oscillators is considered. New classification for oscillators, based on the scheme of the catastrophe theory, is performed. It allows one to describe the different oscillators, which differ from each other in the quantity of the potential wells and in the possibility to escape, by using the oscillator equation with some elementary Thom`s catastrophe at different fixed values of the controlling parameters. The investigation of the oscillators with Thom`s catastrophes is developed in full parameter space. First, the investigation of the dynamics of the systems with escaping olutions is carried out. The basin erosion under the variation of the potential function parameter is shown for the oscillator with fold catastrophe. The escape region and region of the stable periodic or chaotic solutions is estimated for this oscillator and for oscillator with swallowtail catastrophe in the parameter space. To study the dynamics of the oscillators without the escaping solutions the evolution of the dynamical regimes topography under the increasing forcing amplitude is considered. We investigated the case of the oscillator with cusp catastrophe and the oscillators with the potential function of sixth and eighth degree. The effect of the increasing of the degree of the potential function on the system dynamics is studied. It turns out that the topological configurations of "crossroad area" and "spring area" arise for smaller amplitudes in case of the potential function of higher degree and scenario of its appearance repeat itself several times on the regarded range of the forcing amplitude. So the topographies contain a great number of the "crossroad area" and "spring area". The basin transformations were considered for the oscillator with utterfly catastrophe. These transformations take place under the increasing of the forcing amplitude and under the variation of the potential function parameters. The topographies of the dynamical regimes on the plane of the different controlling parameters demonstrate the period-doubling cascade to chaos and "crossroad areas" and "spring areas".