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

Abstract

We introduce a versatile class of prototype dynamical systems for the study of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling bifurcations and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype system consist of a $2d$-dimensional dynamical system with friction forces $f(V(\mathbf{x}))$ functionally dependent exclusively on the mechanical potential $V(\mathbf{x})$, which is typically characterized, here, by a finite number of local minima. We present examples for $d=1,2$ and simple polynomial friction forces $f(V)$, where the zeros of $f(V)$ regulate the relative importance of energy uptake and dissipation respectively, serving as bifurcation parameters. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcation are observed when energy uptake gains progressively in importance.

References in corpus (1)

• Optimization hardness as transient chaos in an analog approach to constraint satisfaction

Cited by in corpus (6)

• Chaos in time delay systems, an educational review
• How to test for partially predictable chaos
• The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles
• Kick control: using the attracting states arising within the sensorimotor loop of self-organized robots as motor primitives
• When the goal is to generate a series of activities: A self-organized simulated robot arm
• Complex activity patterns generated by short-term synaptic plasticity