CATASTROPHE THEORETIC CLASSIFICATION OF NONLINEAR OSCILLATORS

Catastrophe theory is employed to classify different types of nonlinear oscillators, basing on the complication of their potentials. By using Thom's catastrophe unfoldings as oscillator potentials, we have introduced more general models to describe the dynamics of nonlinear oscillators, differing from each other by the form of their potential wells and by the possibility of escape. Spreading the investigation in the space of the parameters of the potential function, we have revealed that our examples defined via Thom's catastrophe unfoldings have some type of universal properties in the context of forced oscillations. For oscillators with nonescaping solutions, we have detected such typical bifurcation structures as crossroad areas and spring areas, and have described the universal scenario of their evolution under the forcing amplitude variation. On increasing the potential function degree, the complexity of the charts of the dynamical regimes results from the repetition of the described bifurcation scenario. For oscillators with escaping solutions, such general properties were investigated, as dependence of the charts of the dynamical regimes and the basins on the parameters of the potential function. We have observed that these properties are typical in a broad range of the control parameters.