Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2007-01-17
Nonlinear Sciences
Chaotic Dynamics
laTex, 18 pages, 8 figures
Scientific paper
10.1016/j.physleta.2007.06.046
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 $\lambda_L \pm {\rm i} \omega_L$ on one side of the discontinuity and $-\lambda_R \pm {\rm i} \omega_R$ 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 $\Lambda >0$.
Meiss James D.
Simpson David J. W.
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