2019/05/03 by David J. Simpson, Simpson, David J. W. · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #34A36 #37G15 #37G35 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1905.01329
openalex publication_date 2019/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.