Poincaré bifurcation of a three-dimensional system

Consider a three-dimensional system having an invariant surface. By using bifurcation techniques and analyzing the solutions of bifurcation equations, we study the spatial bifurcation phenomena near a family of periodic orbits and a center in the invariant surface respectively. New formula of Melnikov function is derived and sufficient conditions for the existence of periodic orbits are obtained. An application of our results to a modified van der Pol–Duffing electronic circuit is given.