A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $

Abstract

<p style='text-indent:20px;'>The paper deals with the bifurcation diagram and all global phase portraits in the Poincaré disc of a nonsmooth van der Pol-Duffing oscillator with the form <inline-formula><tex-math id="M2">\begin{document}$ \dot{x} = y $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \dot{y} = a_1x+a_2x^3+b_1y+b_2|x|y $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ a_i, b_i $\end{document}</tex-math></inline-formula> are real and <inline-formula><tex-math id="M5">\begin{document}$ a_2b_2\neq0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ i = 1, 2 $\end{document}</tex-math></inline-formula>. The system is an equivariant system. When the sum of indices of equilibria is <inline-formula><tex-math id="M7">\begin{document}$ -1 $\end{document}</tex-math></inline-formula>, i.e., <inline-formula><tex-math id="M8">\begin{document}$ a_2&gt;0 $\end{document}</tex-math></inline-formula>, it is proven that the bifurcation diagram includes one Hopf bifurcation surface, one pitchfork bifurcation surface and one heteroclinic bifurcation surface. Although the vector field is only <inline-formula><tex-math id="M9">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>, we still obtain that the heteroclinic bifurcation surface is <inline-formula><tex-math id="M10">\begin{document}$ C^{\infty} $\end{document}</tex-math></inline-formula> and a generalized Hopf bifurcation occurs. Moreover, we also find that the heteroclinic bifurcation surface and the focus-node surface have exactly one intersection curve.</p>