A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

Abstract

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 statedependent 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 Poincaré 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.

Hopf-like boundary equilibrium bifurcations involving two foci in Filippov systems

“…The main result is a sufficient condition for a unique limit cycle to be created in the BEB, see Theorem 2.2. The bifurcation resembles a Hopf bifurcation, and is one of 20 Hopf-like bifurcations of piecewise-smooth systems listed in [14]. If the condition is not satisfied, three limit cycles may be created, see §5.…”

“…As in the continuous setting [13], with nonlinear terms added to F L and F R hyperbolic equilibria, pseudo-equilibria, and limit cycles should be of a distance from the origin that is asymptotically proportional to |µ|, instead of directly proportional to |µ|. This will be treated carefully in [17].…”

“…The same is true for hyperbolic limit cycles, if they can be expressed as fixed points of a smooth Poincaré map. For the BEB described here, this will be established formally in [17]. Recently there have been many studies of two-dimensional piecewise-linear ODEs, see [18] and references within.…”

Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis

“…Thus we need to produce a twoparameter unfolding in order to predict stable impacting orbits. This non-smooth bifurcation is unlike other bifurcations known in the literature [15,18]. It relies on the specific form of a mechanical system with at least 2 d.f.…”

Cooperation‐Based Modeling of Sustainable Development: An Approach from Filippov’s Systems

“…Regarding the limit cycles in Filippov's systems, it is recommended to review [19]. On the bifurcations of these systems, there are articles from [20][21][22][23][24][25][26], together with more specialized articles such as [27][28][29] for periodic orbits, [30,31] for sliding bifurcations or the Hopf bifurcation compendium of [32]. Other topics that may be of interest are the numerical aspects of the solution of these differential systems [33,34] or stochastic perturbations to periodic orbits with sliding [35,36].…”