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) +/- i omega(L) on one side of the discontinuity and -lambda(R) +/- i omega(R) on the other, with lambda(L),lambda(R) > 0, and the quantity Lambda = lambda(L)/omega(L) - lambda(R)/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 Lambda < 0 and subcritical if Lambda > 0. (c) 2007 Elsevier B.V. All rights reserved.