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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 $

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 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.

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. BikdashB. 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]

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[3]

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[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. ChenX. 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

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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]

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[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. LinsW. 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. BikdashB. 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. ChenX. 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. LinsW. 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 2.  The relative positions of $ HL $ and $ FN_1 $
Figure 3.  Dynamical behaviors near $ D $
Figure 4.  Dynamical behaviors near $ I_y $
Figure 5.  Dynamical behaviors near saddles
Figure 6.  The closed orbit $ \gamma $
Figure 7.  Annular regions according the positions of $ A $ and $ B $
Figure 8.  The changes of unstable and stable manifolds
Figure 9.  Numerical phase portraits with three equilibria
Figure 10.  Numerical phase portraits with one closed orbit surrounding a focus
Figure 11.  Numerical phase portraits with one closed orbit surrounding an unidirectional node
Figure 12.  Numerical phase portraits with one closed orbit surrounding a bidirectional node
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} $$ E_l $, $ E_r $ saddles; $ E_0 $ unstable bidirectional node
possibilities of $ (\mu_1, \mu_2) $types and stabilities
$ \mu_1< 0 $, $ \mu_2<-2\sqrt{-\mu_1} $$ 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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