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

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 ${\lambda }_L\pm \mathrm{i}{\omega }_L$ on one side of the discontinuity and $-{\lambda }_R\pm \mathrm{i}{\omega }_R$ on the other, with ${\lambda }_L,{\lambda }_R>0$, and the quantity $\mathrm{\Lambda }={\lambda }_L\mathrm{/}{\omega }_L-{\lambda }_R\mathrm{/}{\omega }_R$ 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 $\mathrm{\Lambda }<0$ and subcritical if $\Lambda

0$.