Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows

Abstract

An equilibrium of a planar, piecewise-$C^1$, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to $λ_L \pm {\rm i} ω_L$ on one side of the discontinuity and $-λ_R \pm {\rm i} ω_R$ on the other, with $λ_L, λ_R >0$, and the quantity $ Λ= λ_L / ω_L -λ_R / ω_R $ is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if $Λ< 0$ and subcritical if $Λ>0$.