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 ± i ω L on one side of the discontinuity and − λ R ± 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 .