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Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension
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 |
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.
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. |










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