Bi-center problem and Hopf cyclicity of a Cubic Liénard system
In this paper we investigate the bi-center problem and the total Hopf cyclicity of two center-foci for the general cubic Lienard system which has three distinct equilibria and is equivalent to the general Lienard equation with cubic damping and restoring force. The location of these three equilibria is arbitrary, specially without any kind of symmetry. We find the necessary and sufficient condition for the existence of bi-centers and prove that there is no case of a unique center. On the Hopf cyclicity we prove that there are totally \begin{document}$ 9 $\end{document} possible styles of small amplitude limit cycles surrounding these two center-foci and \begin{document}$ 6 $\end{document} styles of them can occur, from which the total Hopf cyclicity is no more than \begin{document}$ 4 $\end{document} and no less than \begin{document}$ 2 $\end{document} .