Article Contents
Article Contents

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

• * Corresponding author: Hebai Chen
• 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 $\dot{x} = y$, $\dot{y} = a_1x+a_2x^3+b_1y+b_2|x|y$, where $a_i, b_i$ are real and $a_2b_2\neq0$, $i = 1, 2$. The system is an equivariant system. When the sum of indices of equilibria is $-1$, i.e., $a_2>0$, 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 $C^1$, we still obtain that the heteroclinic bifurcation surface is $C^{\infty}$ 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.

Mathematics Subject Classification: 34C07, 34C23, 34C37, 34K18.

 Citation:

• Figure 1.  The slice $\mu_3 = {\mu_3}^{(0)}$ of the bifurcation diagram and global phase portraits of system (1.3a)

Figure 2.  The relative positions of $HL$ and $FN_1$

Figure 3.  Dynamical behaviors near $D$

Figure 4.  Dynamical behaviors near $I_y$

Figure 5.  Dynamical behaviors near saddles

Figure 6.  The closed orbit $\gamma$

Figure 7.  Annular regions according the positions of $A$ and $B$

Figure 8.  The changes of unstable and stable manifolds

Figure 9.  Numerical phase portraits with three equilibria

Figure 10.  Numerical phase portraits with one closed orbit surrounding a focus

Figure 11.  Numerical phase portraits with one closed orbit surrounding an unidirectional node

Figure 12.  Numerical phase portraits with one closed orbit surrounding a bidirectional node

Table 1.  Properties of $E_0$, $E_l$ and $E_r$

 possibilities of $(\mu_1, \mu_2)$ types and stabilities $\mu_1< 0$, $\mu_2<-2\sqrt{-\mu_1}$ $E_l$, $E_r$ saddles; $E_0$ stable bidirectional node $\mu_1< 0$, $\mu_2 = -2\sqrt{-\mu_1}$ $E_l$, $E_r$ saddles; $E_0$ stable unidirectional node $\mu_1< 0$, $-2\sqrt{-\mu_1}<\mu_2<0$ $E_l$, $E_r$ saddles; $E_0$ stable rough focus $\mu_1< 0$, $\mu_2 = 0$ $E_l$, $E_r$ saddles; $E_0$ stable weak focus $\mu_1< 0$, $0<\mu_2<2\sqrt{-\mu_1}$ $E_l$, $E_r$ saddles; $E_0$ unstable rough focus $\mu_1< 0$, $\mu_2 = 2\sqrt{-\mu_1}$ $E_l$, $E_r$ saddles; $E_0$ unstable unidirectional node $\mu_1< 0$, $\mu_2>2\sqrt{-\mu_1}$ $E_l$, $E_r$ saddles; $E_0$ unstable bidirectional node
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