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  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 & 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. 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.  Google Scholar

[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.  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. ChenJ. 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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. ChenJ. 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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

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)
[1]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[2]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[3]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[4]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[5]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[6]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013

[7]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[8]

Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020375

[9]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[10]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[11]

Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394

[12]

Yujuan Li, Huaifu Wang, Peipei Zhou, Guoshuang Zhang. Some properties of the cycle decomposition of WG-NLFSR. Advances in Mathematics of Communications, 2021, 15 (1) : 155-165. doi: 10.3934/amc.2020050

[13]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[14]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[15]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[16]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109

[17]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[18]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[19]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[20]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]