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Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise

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

 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China 2 School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China

* Corresponding author: Hebai Chen

Received  October 2020 Published  March 2021

We continue to study the nonsmooth van der Pol-Duffing oscillator $\dot{x} = y$, $\dot{y} = a_1x+a_2x^3+b_1y+b_2|x|y$, where $a_i, b_i$ are real and $a_2b_2\neq0$, $i = 1, 2$. Notice that the sum of indices of equilibria is $-1$ for $a_2>0$ and $1$ for $a_2<0$. When $a_2>0$, the nonsmooth van der Pol-Duffing oscillator has been studied completely in the companion paper. Attention goes to the bifurcation diagram and all global phase portraits in the Poincaré disc of the nonsmooth van der Pol-Duffing oscillator for $a_2<0$ in this paper. The bifurcation diagram is more complex, which includes two Hopf bifurcation surfaces, one pitchfork bifurcation surface, one homoclinic bifurcation surface, one double limit cycle bifurcation surface and one bifurcation surface for equilibria at infinity. When $b_2>0$ is fixed, this nonsmooth van der Pol-Duffing oscillator cannot be changed into a near-Hamiltonian system for small $a_1, b_1$. Moreover, the global dynamics of the nonsmooth van der Pol-Duffing oscillator and the van der Pol-Duffing oscillator are different.

Citation: Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (II): The sum of indices of equilibria is $1$. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021101
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##### References:
 [1] M. Bikdash, B. Balachandran and A. H. Nayfeh, Melnikov analysis for a ship with a general Roll-damping model, Nonlinear Dyn., 6 (1994), 101-124.   Google Scholar [2] H. Chen and X. Chen, Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (II), Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4141-4170.  doi: 10.3934/dcdsb.2018130.  Google Scholar [3] H. Chen, X. Chen and J. Xie, Global phase portrait of a degenerate Bogdanov-Takens system with symmetry, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1273-1293.  doi: 10.3934/dcdsb.2017062.  Google Scholar [4] H. Chen, Y. Tang and D. Xiao, Global dynamics of a quintic Liénard system with $\mathbb{Z}_2$-symmetry I: Saddle case, Nonlinearity, submitted. Google Scholar [5] S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge Univ. Press, New York, 1994.  doi: 10.1017/CBO9780511665639.  Google Scholar [6] J. F. Dalzell, A note on the form of ship roll damping, J. Ship Research, 22 (1978), 178-185.  doi: 10.5957/jsr.1978.22.3.178.  Google Scholar [7] F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039.  doi: 10.1088/0951-7715/3/4/004.  Google Scholar [8] J. K. Hale, Ordinary Differential Equations, Roberte. Kqieger Publishing Company, Huntington, New York, 1980.  Google Scholar [9] A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics. Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science. A, Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995 doi: 10.1002/9783527617548.  Google Scholar [10] L. M. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2002. doi: 10.1007/978-1-4613-0003-8.  Google Scholar [11] Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity, 17 (2004), 1407-1426.  doi: 10.1088/0951-7715/17/4/015.  Google Scholar [12] Z. Wang and H. Chen, Nonsmooth van der Pol-Duffing oscillators: (I) The sum of indices of equilibria is $-1$, Discrete Contin. Dyn. Syst. Ser. B, to appear. Google Scholar [13] Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr. 101, Amer. Math. Soc., Providence, RI, 1992.  Google Scholar
The slice $\mu_3 = {\mu_3}^{(1)}<2\sqrt{2}$ of the bifurcation diagram and corresponding global phase portraits
The slice $\mu_3 = {\mu_3}^{(2)}\ge2\sqrt{2}$ of the bifurcation diagram and corresponding global phase portraits
Two possibilities of connections in $S_3$
An orbit near $E_0$
Dynamical behaviors near $I_y^+$ and $I_y^-$
Dynamical behaviors near $D$
Dynamical behaviors near infinity
The orbit $\Upsilon$ passing through $(x_*, y_*)$
Hypothetical limit cycles
Two large limit cycles
Unstable manifold in the right half plane of the origin
Existence of the large limit cycle
$P$ is not in the region enclosed by $\Gamma$
$P$ is in the region enclosed by $\Gamma$
Unstable and stable manifolds in the right half plane
Numerical phase portraits with one equilibrium when $\mu_1 = -4$ and $\mu_3 = 1$
Numerical phase portraits with one equilibrium when $\mu_1 = 0$ and $\mu_3 = 1$
Numerical phase portraits with three equilibrium when $\mu_1 = 4$ and $\mu_3 = 1$
Properties of $E_0$, $E_l$ and $E_r$
 possibilities of $(\mu_1, \mu_2)$ types and stabilities $\mu_1> 0$, $\mu_2<(\mu_3-2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ stable bidirectional nodes $\mu_1> 0$, $\mu_2 = (\mu_3-2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ stable unidirectional nodes $\mu_1> 0$, $(\mu_3-2\sqrt{2})\sqrt{\mu_1}<\mu_2<\mu_3\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ stable rough foci $\mu_1> 0$, $\mu_2 = \mu_3\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable weak foci $\mu_1> 0$, $\mu_3\sqrt{\mu_1}<\mu_2<(\mu_3+2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable rough foci $\mu_1> 0$, $\mu_2 = (\mu_3+2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable unidirectional nodes $\mu_1> 0$, $\mu_2>(\mu_3+2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable bidirectional nodes
 possibilities of $(\mu_1, \mu_2)$ types and stabilities $\mu_1> 0$, $\mu_2<(\mu_3-2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ stable bidirectional nodes $\mu_1> 0$, $\mu_2 = (\mu_3-2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ stable unidirectional nodes $\mu_1> 0$, $(\mu_3-2\sqrt{2})\sqrt{\mu_1}<\mu_2<\mu_3\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ stable rough foci $\mu_1> 0$, $\mu_2 = \mu_3\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable weak foci $\mu_1> 0$, $\mu_3\sqrt{\mu_1}<\mu_2<(\mu_3+2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable rough foci $\mu_1> 0$, $\mu_2 = (\mu_3+2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable unidirectional nodes $\mu_1> 0$, $\mu_2>(\mu_3+2\sqrt{2})\sqrt{\mu_1}$ $E_0$ saddle; $E_l$, $E_r$ unstable bidirectional nodes
Properties of $E_0$
 possibilities of $(\mu_1, \mu_2)$ types and stabilities $\mu_2<0$ $E_0$ stable degenerate node $\mu_1 = 0$ $\mu_2 = 0$ $E_0$ stable nilpotent focus $\mu_2>0$ $E_0$ unstable degenerate node $\mu_2<-2\sqrt{-\mu_1}$ $E_0$ stable bidirectional node $\mu_2 = -2\sqrt{-\mu_1}$ $E_0$ stable unidirectional node $-2\sqrt{-\mu_1}<\mu_2<0$ $E_0$ stable rough focus $\mu_1< 0$ $\mu_2 = 0$ $E_0$ stable weak focus $0<\mu_2<2\sqrt{-\mu_1}$ $E_0$ unstable rough focus $\mu_2 = 2\sqrt{-\mu_1}$ $E_0$ unstable unidirectional node $\mu_2>2\sqrt{-\mu_1}$ $E_0$ unstable bidirectional node
 possibilities of $(\mu_1, \mu_2)$ types and stabilities $\mu_2<0$ $E_0$ stable degenerate node $\mu_1 = 0$ $\mu_2 = 0$ $E_0$ stable nilpotent focus $\mu_2>0$ $E_0$ unstable degenerate node $\mu_2<-2\sqrt{-\mu_1}$ $E_0$ stable bidirectional node $\mu_2 = -2\sqrt{-\mu_1}$ $E_0$ stable unidirectional node $-2\sqrt{-\mu_1}<\mu_2<0$ $E_0$ stable rough focus $\mu_1< 0$ $\mu_2 = 0$ $E_0$ stable weak focus $0<\mu_2<2\sqrt{-\mu_1}$ $E_0$ unstable rough focus $\mu_2 = 2\sqrt{-\mu_1}$ $E_0$ unstable unidirectional node $\mu_2>2\sqrt{-\mu_1}$ $E_0$ unstable bidirectional node
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