Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

We investigate a generalized Hopf bifurcation emerged from a corner located at the origin which is the intersection of n$n$ discontinuity boundaries in planar piecewise smooth dynamical systems with the Jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues. We obtain a novel result that the generalized Hopf bifurcation can occur even when the Jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues. According to the eigenvalues of the Jacobian matrices and the number of smooth subsystems, we provide a general method and prove some generalized Hopf bifurcation theorems by studying the associated Poincaré map.