American Institute of Mathematical Sciences

Advanced
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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 3.  Dynamical behaviors near $ D $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Dynamical behaviors near $ I_y $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Dynamical behaviors near saddles
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The closed orbit $ \gamma $
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 8.  The changes of unstable and stable manifolds
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Numerical phase portraits with three equilibria
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

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, 2021  doi: 10.3934/dcdsb.2021101
[2]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[3]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[4]

Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089
[5]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[6]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021093
[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[8]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[9]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[10]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381
[11]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039
[12]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[13]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026
[14]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365
[15]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294
[16]

Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007
[17]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015
[18]

Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021035
[19]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023
[20]

Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai, Jie Sun. Variable metric proximal stochastic variance reduced gradient methods for nonconvex nonsmooth optimization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021084

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (17)
  • HTML views (35)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences