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- W3138413721 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_2neq0 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=M6>begin{document}$ 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>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>" @default.
- W3138413721 created "2021-03-29" @default.
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- W3138413721 date "2022-01-01" @default.
- W3138413721 modified "2023-10-14" @default.
- W3138413721 title "A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $" @default.
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- W3138413721 doi "https://doi.org/10.3934/dcdsb.2021096" @default.
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