A versatile class of prototype dynamical systems for complex bifurcation cascades of limit cycles.

Abstract

A general class of prototype dynamical systems is introduced, which allows to study the generation of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling and homoclinic bifurcations, and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima. We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V. We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcations are observed when the energy uptake gains progressively in importance.