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March  2022, 27(3): 1421-1446. doi: 10.3934/dcdsb.2021096

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

Received  August 2020 Revised  January 2021 Published  March 2022 Early access  March 2021

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: Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1421-1446. doi: 10.3934/dcdsb.2021096
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.  doi: 10.1007/BF00045435.

[2]

H. Chen and X. Chen, Dynamical analysis of a cubic Liénard system with global parameters, Nonlinearity, 28 (2015), 3535-3562.  doi: 10.1088/0951-7715/28/10/3535.

[3]

H. Chen and X. Chen, Dynamical analysis of a cubic Liénard system with global parameters: (II), Nonlinearity, 29 (2016), 1978-1826.  doi: 10.1088/0951-7715/29/6/1798.

[4]

H. Chen and X. Chen, Dynamical analysis of a cubic Liénard system with global parameters: (III), Nonlinearity, 33 (2020), 1443-1465.  doi: 10.1088/1361-6544/ab5e29.

[5]

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.

[6]

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.

[7]

H. Chen, Y. Tang and D. Xiao, Global dynamics of a quintic Liénard system with $\mathbb{Z}_2$-symmetry I: Saddle case, Nonlinearity, submitted.

[8]

X. Chen and H. Chen, Complete bifurcation diagram and global phase portraits of Liénard differential equations of degree four, J. Math. Anal. Appl., 485 (2020), 123802, 12 pages. doi: 10.1016/j.jmaa.2019.123802.

[9]

S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Reprint of the 1994 original. Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511665639.

[10]

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.

[11]

G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Arch. Ration. Mech. An., 97 (1987), 321-352.  doi: 10.1007/BF00280410.

[12]

F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations, Nonlinearity, 9 (1996), 1489-1500.  doi: 10.1088/0951-7715/9/6/006.

[13]

F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59.  doi: 10.1006/jdeq.1997.3291.

[14]

F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039.  doi: 10.1088/0951-7715/3/4/004.

[15]

M. R. Haddara and P. Bennett, A study of the angle dependence of roll damping moment, Ocean Engng., 16 (1989), 411-427.  doi: 10.1016/0029-8018(89)90016-4.

[16]

J. K. Hale, Ordinary Differential Equations, Roberte. Kqieger Publishing, Company, Huntington, New York, 1980.

[17]

C. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162.  doi: 10.1016/j.jde.2011.11.002.

[18]

A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 335-357.  doi: 10.1007/BFb0085364.

[19]

A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics, Wiley Series in Nonlinear Science. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. doi: 10.1002/9783527617548.

[20]

G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon, New York, 1964. doi: 10.1016/C2013-0-05338-8.

[21]

S. Smale, Dynamics retrospective: Great problems, attempts that failed. Nonlinear science: the next decade, Physica D, 51 (1991), 267-273.  doi: 10.1016/0167-2789(91)90238-5.

[22]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.  doi: 10.1007/BF03025291.

[23]

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.

[24]

L. Yang and X. Zeng, An upper bound for the amplitude of limit cycles in Liénard systems with symmetry, J. Differential Equations, 258 (2015), 2701-2710. 

[25]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr. 101, Amer. Math. Soc., Providence, RI, 1992.

show all references

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.  doi: 10.1007/BF00045435.

[2]

H. Chen and X. Chen, Dynamical analysis of a cubic Liénard system with global parameters, Nonlinearity, 28 (2015), 3535-3562.  doi: 10.1088/0951-7715/28/10/3535.

[3]

H. Chen and X. Chen, Dynamical analysis of a cubic Liénard system with global parameters: (II), Nonlinearity, 29 (2016), 1978-1826.  doi: 10.1088/0951-7715/29/6/1798.

[4]

H. Chen and X. Chen, Dynamical analysis of a cubic Liénard system with global parameters: (III), Nonlinearity, 33 (2020), 1443-1465.  doi: 10.1088/1361-6544/ab5e29.

[5]

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.

[6]

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.

[7]

H. Chen, Y. Tang and D. Xiao, Global dynamics of a quintic Liénard system with $\mathbb{Z}_2$-symmetry I: Saddle case, Nonlinearity, submitted.

[8]

X. Chen and H. Chen, Complete bifurcation diagram and global phase portraits of Liénard differential equations of degree four, J. Math. Anal. Appl., 485 (2020), 123802, 12 pages. doi: 10.1016/j.jmaa.2019.123802.

[9]

S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Reprint of the 1994 original. Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511665639.

[10]

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.

[11]

G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Arch. Ration. Mech. An., 97 (1987), 321-352.  doi: 10.1007/BF00280410.

[12]

F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations, Nonlinearity, 9 (1996), 1489-1500.  doi: 10.1088/0951-7715/9/6/006.

[13]

F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59.  doi: 10.1006/jdeq.1997.3291.

[14]

F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039.  doi: 10.1088/0951-7715/3/4/004.

[15]

M. R. Haddara and P. Bennett, A study of the angle dependence of roll damping moment, Ocean Engng., 16 (1989), 411-427.  doi: 10.1016/0029-8018(89)90016-4.

[16]

J. K. Hale, Ordinary Differential Equations, Roberte. Kqieger Publishing, Company, Huntington, New York, 1980.

[17]

C. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162.  doi: 10.1016/j.jde.2011.11.002.

[18]

A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 335-357.  doi: 10.1007/BFb0085364.

[19]

A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics, Wiley Series in Nonlinear Science. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. doi: 10.1002/9783527617548.

[20]

G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon, New York, 1964. doi: 10.1016/C2013-0-05338-8.

[21]

S. Smale, Dynamics retrospective: Great problems, attempts that failed. Nonlinear science: the next decade, Physica D, 51 (1991), 267-273.  doi: 10.1016/0167-2789(91)90238-5.

[22]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.  doi: 10.1007/BF03025291.

[23]

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.

[24]

L. Yang and X. Zeng, An upper bound for the amplitude of limit cycles in Liénard systems with symmetry, J. Differential Equations, 258 (2015), 2701-2710. 

[25]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr. 101, Amer. Math. Soc., Providence, RI, 1992.

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} $$ E_l $, $ E_r $ saddles; $ E_0 $ unstable bidirectional node
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possibilities of $ (\mu_1, \mu_2) $types and stabilities
$ \mu_1< 0 $, $ \mu_2<-2\sqrt{-\mu_1} $$ 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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