Andronov Hopf bifurcations in planar, piecewisesmooth, continuous flows
Abstract
An equilibrium of a planar, piecewiseC^{}, 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 bordercrossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λ_{}±iω_{} on one side of the discontinuity and λ_{}±iω_{} on the other, with λ_{},λ_{}>0, and the quantity Λ=λ_{}/ω_{}λ_{}/ω_{} 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.
 Publication:

Physics Letters A
 Pub Date:
 November 2007
 DOI:
 10.1016/j.physleta.2007.06.046
 arXiv:
 arXiv:nlin/0701036
 Bibcode:
 2007PhLA..371..213S
 Keywords:

 02.30.Oz;
 05.45.a;
 Bifurcation theory;
 Nonlinear dynamics and chaos;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 laTex, 18 pages, 8 figures