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

Zhaoxia Wang; Hebai Chen

doi:10.3934/dcdsb.2021096

Article Contents

2022, Volume 27, Issue 3: 1421-1446. Doi: 10.3934/dcdsb.2021096

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A nonsmooth van der Pol-Duffing oscillator (I): 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: August 2020

Revised: January 2021

Early access: March 2021

Published: March 2022

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Abstract

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.

Keywords:

Equivariant bifurcation,
nonsmooth van der Pol-Duffing oscillator,
global phase portrait,
limit cycle,
$ 2 $-saddle loop.

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

Citation:

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

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

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Figure 3. Dynamical behaviors near $ D $

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Figure 4. Dynamical behaviors near $ I_y $

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Figure 5. Dynamical behaviors near saddles

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Figure 6. The closed orbit $ \gamma $

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

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Figure 8. The changes of unstable and stable manifolds

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Figure 9. Numerical phase portraits with three equilibria

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

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

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

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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<-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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