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Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems. (English) Zbl 1256.34028
Imagine a one-parameter family of piecewise smooth planar systems of ordinary differential equations for which a “corner” that is formed by the intersection of finitely many curves that bound the domains of smoothness is a singularity for all values of the parameter. One can then consider the possibility of the creation of a limit cycle from the singularity as the parameter is varied, i.e., a “generalized Hopf bifurcation”. The authors examine this situation and prove that even when the eigenvalues of the linear parts of all the individual smooth systems that meet at the corner are real (and nonzero and distinct) a generalized Hopf bifurcation can occur. They provide a numerical example illustrating the phenomenon. They also examine the situation when the eigenvalues of some of the linear parts involved are as above, but others are complex conjugates. In each case treated, the corner has a finite number $n$ of boundary rays emanating from it, determining $n$ sectors, on each of which the system is at least ${C}^{3}$ and has a linear part that depends analytically on the parameter $\lambda \in ℝ$.
##### MSC:
 34C23 Bifurcation (ODE) 34A36 Discontinuous equations