March  2021, 26(3): 1565-1577. doi: 10.3934/dcdsb.2020173

The Poincaré bifurcation of a SD oscillator

1. 

School of Mathematics, Soochow University, 215006, Suzhou, China

2. 

School of Mathematics (Zhuhai), Sun Yat-sen University, 519082, Zhuhai, China

* Corresponding author

Received  November 2019 Revised  January 2020 Published  March 2021 Early access  May 2020

A van der Pol damped SD oscillator, which was proposed by Ruilan Tian, Qingjie Cao and Shaopu Yang (2010, Nonlinear Dynamics, 59, 19-27), is studied. By improving the criterion function of determining the lowest upper bound of the number of zeros of Abelian Integrals, we show that the number of zeros of Abelian integrals of this SD oscillator is two which is sharp.

Citation: Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
References:
[1]

J. Awrejcewicz and M. M. Holicke, Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-type Methods, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 60, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709103.

[2]

Q. Cao, M. Wiercigroch, E. E. Pavlovskaia, C. Grebogi and J. M. T. Thompson, Archetypal oscillator for smooth and discontinuous dynamics, Phys. Rev. E (3), 74 (2006), 5pp. doi: 10.1103/PhysRevE.74.046218.

[3]

Q. CaoM. WiercigrochE. PavlovskaiaJ. Thompson and C. Grebogi, Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 635-652.  doi: 10.1098/rsta.2007.2115.

[4]

Q. CaoM. WiercigrochE. E. PavlovskaiaC. Grebogi and J. M. T. Thompson, The limit case response of the archetypal oscillator for smooth and discontinuous dynamics, Int. J. Non-Lin. Mech., 43 (2008), 462-473.  doi: 10.1016/j.ijnonlinmec.2008.01.003.

[5]

H. Chen and X. Li, Global phase portraits of memristor oscillators, Internat. J. Bifur. Chaos, 24 (2014), 1-31.  doi: 10.1142/S0218127414501521.

[6]

H. Chen, Global analysis on the discontinuous limit case of a smooth oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 21pp. doi: 10.1142/S0218127416500619.

[7]

H. ChenJ. Llibre and Y. Tang, Global dynamics of a SD oscillator, Nonlinear Dyn., 91 (2018), 1755-1777.  doi: 10.1007/s11071-017-3979-y.

[8]

A. ColomboM. di BernardoS. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems, Phys. D, 241 (2012), 1845-1860.  doi: 10.1016/j.physd.2011.09.017.

[9]

A. F. Filippov, Dierential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[10]

E. FreireE. Ponce and J. Ros, Limit cycle bifurcation from center in symmetric piecewise-linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 895-907.  doi: 10.1142/S0218127499000638.

[11]

M. GrauF. Mañosas and J. Villadelpart, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[12]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.

[13]

R. I. Leine and D. H. van Campen, Bifurcation phenomena in non-smooth dynamical systems, Eur. J. Mech. A Solids, 25 (2006), 595-616.  doi: 10.1016/j.euromechsol.2006.04.004.

[14]

C. Li and Z.-F. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Differential Equations, 124 (1996), 407-424.  doi: 10.1006/jdeq.1996.0017.

[15]

C. LiuG. Chen and Z. Sun, New criteria for the monotonicity of the ratio of two Abelian integrals, J. Math. Anal. Appl., 465 (2018), 220-234.  doi: 10.1016/j.jmaa.2018.04.074.

[16]

C. Liu and D. Xiao, The smallest upper bound on the number of zeros of Abelian integrals, J. Differential Equations, in press. doi: 10.1016/j.jde.2020.03.016.

[17]

O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Phys. D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002.

[18]

F. Mañosas and J. Villadelpart, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[19]

R. TianQ. Cao and S. Yang, The codimension-two bifurcation for the recent proposed SD oscillator, Nonlinear Dynam., 59 (2010), 19-27.  doi: 10.1007/s11071-009-9517-9.

show all references

References:
[1]

J. Awrejcewicz and M. M. Holicke, Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-type Methods, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 60, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709103.

[2]

Q. Cao, M. Wiercigroch, E. E. Pavlovskaia, C. Grebogi and J. M. T. Thompson, Archetypal oscillator for smooth and discontinuous dynamics, Phys. Rev. E (3), 74 (2006), 5pp. doi: 10.1103/PhysRevE.74.046218.

[3]

Q. CaoM. WiercigrochE. PavlovskaiaJ. Thompson and C. Grebogi, Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 635-652.  doi: 10.1098/rsta.2007.2115.

[4]

Q. CaoM. WiercigrochE. E. PavlovskaiaC. Grebogi and J. M. T. Thompson, The limit case response of the archetypal oscillator for smooth and discontinuous dynamics, Int. J. Non-Lin. Mech., 43 (2008), 462-473.  doi: 10.1016/j.ijnonlinmec.2008.01.003.

[5]

H. Chen and X. Li, Global phase portraits of memristor oscillators, Internat. J. Bifur. Chaos, 24 (2014), 1-31.  doi: 10.1142/S0218127414501521.

[6]

H. Chen, Global analysis on the discontinuous limit case of a smooth oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 21pp. doi: 10.1142/S0218127416500619.

[7]

H. ChenJ. Llibre and Y. Tang, Global dynamics of a SD oscillator, Nonlinear Dyn., 91 (2018), 1755-1777.  doi: 10.1007/s11071-017-3979-y.

[8]

A. ColomboM. di BernardoS. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems, Phys. D, 241 (2012), 1845-1860.  doi: 10.1016/j.physd.2011.09.017.

[9]

A. F. Filippov, Dierential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[10]

E. FreireE. Ponce and J. Ros, Limit cycle bifurcation from center in symmetric piecewise-linear systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 895-907.  doi: 10.1142/S0218127499000638.

[11]

M. GrauF. Mañosas and J. Villadelpart, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[12]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.

[13]

R. I. Leine and D. H. van Campen, Bifurcation phenomena in non-smooth dynamical systems, Eur. J. Mech. A Solids, 25 (2006), 595-616.  doi: 10.1016/j.euromechsol.2006.04.004.

[14]

C. Li and Z.-F. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Differential Equations, 124 (1996), 407-424.  doi: 10.1006/jdeq.1996.0017.

[15]

C. LiuG. Chen and Z. Sun, New criteria for the monotonicity of the ratio of two Abelian integrals, J. Math. Anal. Appl., 465 (2018), 220-234.  doi: 10.1016/j.jmaa.2018.04.074.

[16]

C. Liu and D. Xiao, The smallest upper bound on the number of zeros of Abelian integrals, J. Differential Equations, in press. doi: 10.1016/j.jde.2020.03.016.

[17]

O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Phys. D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002.

[18]

F. Mañosas and J. Villadelpart, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[19]

R. TianQ. Cao and S. Yang, The codimension-two bifurcation for the recent proposed SD oscillator, Nonlinear Dynam., 59 (2010), 19-27.  doi: 10.1007/s11071-009-9517-9.

Figure 1.  The global phase portraits of system (1.6) for $ 0<a<1 $ and $ \epsilon = 0 $
Figure 2.  The phase portraits of system (2.1)
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