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

The Poincaré bifurcation of a SD oscillator

Yangjian Sun 1,2, and  Changjian Liu 1,2,,

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.

Keywords: Poincaré bifurcation, SD oscillator, Abelian integral, limit cycle, Hamiltonian system.
Mathematics Subject Classification: 34C23, 34C25.
Citation: Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

Q. Cao, M. Wiercigroch, E. Pavlovskaia, J. 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.  Google Scholar

[4]

Q. Cao, M. Wiercigroch, E. E. Pavlovskaia, C. 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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

A. Colombo, M. di Bernardo, S. 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.  Google Scholar

[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.  Google Scholar

[10]

E. Freire, E. 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.  Google Scholar

[11]

M. Grau, F. 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.  Google Scholar

[12]

Y. A. Kuznetsov, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Liu, G. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Tian, Q. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Q. Cao, M. Wiercigroch, E. Pavlovskaia, J. 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.  Google Scholar

[4]

Q. Cao, M. Wiercigroch, E. E. Pavlovskaia, C. 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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

A. Colombo, M. di Bernardo, S. 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.  Google Scholar

[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.  Google Scholar

[10]

E. Freire, E. 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.  Google Scholar

[11]

M. Grau, F. 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.  Google Scholar

[12]

Y. A. Kuznetsov, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Liu, G. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Tian, Q. 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.  Google Scholar

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