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.
Highlights
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
We have shown that complex cascades of limit cycle bifurcations can be obtained even for two dimensional phase spaces, when the friction function f(V) alternates between regions of energy uptake and dissipation
We have introduced a generic class of potential functions [7], which allows to define, in a relative straightforward manner, mechanical potentials with an arbitrary number of local minima and varying depth
Summary
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 dynamical behavior of the system should be dominated by the prime phenomenon of interest, with the system being otherwise simple enough to allow for straightforward numerical and (at least partial) analytic investigations1–4 Their dynamical behavior can often be understood in terms of general concepts, such as energy balance, symmetry breaking, etc. Though there are a range of alternative construction methods for dynamical systems in the literature (see e.g.13–15), they generally involve abstract concepts, such as implicitly defined manifolds, or mathematical tools accessible only to researchers with an in-depth math training In contrast to these methods, we provide here a mechanistic design procedure, based on the construction of attractors through the interaction of generalized friction forces with potential forces, an intuitive concept especially suitable for interdisciplinary investigations (e.g. in modeling cardiovascular systems or for solving optimization problems17), www.nature.com/scientificreports/
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