American Institute of Mathematical Sciences

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

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
The slice $\mu_3 = {\mu_3}^{(0)}$ of the bifurcation diagram and global phase portraits of system (1.3a)
The relative positions of $HL$ and $FN_1$
Dynamical behaviors near $D$
Dynamical behaviors near $I_y$
Dynamical behaviors near saddles
The closed orbit $\gamma$
Annular regions according the positions of $A$ and $B$
The changes of unstable and stable manifolds
Numerical phase portraits with three equilibria
Numerical phase portraits with one closed orbit surrounding a focus
Numerical phase portraits with one closed orbit surrounding an unidirectional node
Numerical phase portraits with one closed orbit surrounding a bidirectional node
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
 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] Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026 [13] 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 [14] Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294 [15] 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 [16] Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015 [17] 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 [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

Article outline

Figures and Tables

[Back to Top]