Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows |
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Authors: | D.J.W. Simpson J.D. Meiss |
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Affiliation: | Department of Applied Mathematics, University of Colorado Boulder, CO 80309-0526, USA |
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Abstract: | An equilibrium of a planar, piecewise-C1, 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 λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR 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. |
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Keywords: | 02.30.Oz 05.45.-a |
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