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

Zhaoxia Wang; Hebai Chen

doi:10.3934/dcdsb.2021101

Article Contents

2022, Volume 27, Issue 3: 1549-1589. Doi: 10.3934/dcdsb.2021101

This issue Previous Article Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction Next Article Linear programming estimates for Cesàro and Abel limits of optimal values in optimal control problems

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

Zhaoxia Wang^1, and
Hebai Chen^2, ,

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
^* Corresponding author: Hebai Chen

Received: October 2020

Early access: March 2021

Published: March 2022

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Abstract

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.

Keywords:

Global phase portrait,
nonsmooth van der Pol-Duffing oscillator,
homoclinic bifurcation,
double limit cycle bifurcation,
Hopf bifurcation.

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

Citation:

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Figure 1. The slice $ \mu_3 = {\mu_3}^{(1)}<2\sqrt{2} $ of the bifurcation diagram and corresponding global phase portraits

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Figure 2. The slice $ \mu_3 = {\mu_3}^{(2)}\ge2\sqrt{2} $ of the bifurcation diagram and corresponding global phase portraits

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Figure 3. Two possibilities of connections in $ S_3 $

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Figure 4. An orbit near $ E_0 $

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Figure 5. Dynamical behaviors near $ I_y^+ $ and $ I_y^- $

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Figure 6. Dynamical behaviors near $ D $

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Figure 7. Dynamical behaviors near infinity

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Figure 8. The orbit $ \Upsilon $ passing through $ (x_*, y_*) $

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Figure 9. Hypothetical limit cycles

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Figure 10. Two large limit cycles

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Figure 11. Unstable manifold in the right half plane of the origin

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Figure 12. Existence of the large limit cycle

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Figure 13. $ P $ is not in the region enclosed by $ \Gamma $

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Figure 14. $ P $ is in the region enclosed by $ \Gamma $

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Figure 15. Unstable and stable manifolds in the right half plane

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Figure 16. Numerical phase portraits with one equilibrium when $ \mu_1 = -4 $ and $ \mu_3 = 1 $

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Figure 17. Numerical phase portraits with one equilibrium when $ \mu_1 = 0 $ and $ \mu_3 = 1 $

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Figure 18. Numerical phase portraits with three equilibrium when $ \mu_1 = 4 $ and $ \mu_3 = 1 $

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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<(\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

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Table 2. 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

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