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Register a new userTwenty Hopf-like bifurcations in piecewise-smooth dynamical systems
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David J.W. Simpson
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For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such `Hopf-like bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincare maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator-prey systems, ocean circulation, and the McKean and Wilson-Cowan neuron models.
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This paper concerns two-dimensional Filippov systems --- ordinary differential equations that are discontinuous on one-dimensional switching manifolds. In the situation that a stable focus transitions to an unstable focus by colliding with a switching manifold as parameters are varied, a simple sufficient condition for a unique local limit cycle to be created is established. If this condition is violated, three nested limit cycles may be created simultaneously. The result is achieved by constructing a Poincare map and generalising analytical arguments that have been employed for continuous systems. Necessary and sufficient conditions for the existence of pseudo-equilibria (equilibria of sliding motion on the switching manifold) are also determined. For simplicity only piecewise-linear systems are considered.
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We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $Sigma$. Within the class of Filippov solutions, if $Sigma$ is attractive, one should expect solution trajectories to slide on $Sigma$. It is well known, however, that the classical Filippov convexification methodology is ambiguous on $Sigma$. The situation is further complicated by the possibility that, regardless of how sliding on $Sigma$ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where $Sigma$ ceases to be attractive. In this work, we attempt to understand what behavior one should expect of a solution trajectory near $Sigma$ when $Sigma$ is attractive, what to expect when $Sigma$ ceases to be attractive (at least, at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature. Through analysis and experiments we will confirm some known facts, and provide some important insight: (i) when $Sigma$ is attractive, a solution trajectory indeed does remain near $Sigma$, viz. sliding on $Sigma$ is an appropriate idealization (of course, in general, one cannot predict which sliding vector field should be selected); (ii) when $Sigma$ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of $Sigma$; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near $Sigma$ as long as $Sigma$ is attractive, and so that it will be leaving (a neighborhood of) $Sigma$ when $Sigma$ looses attractivity. We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near $Sigma$ (or sliding motion on $Sigma$) should have been taking place.
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