Bifurcation of big periodic orbits through symmetric homoclinics‎, ‎application to Duffing equation

Document Type : Research Paper

Authors

Department of Applied Mathematics, University of Birjand, Birjand, Iran

Abstract

‎We consider a planar symmetric vector field that undergoes a homoclinic bifurcation‎. ‎In order to study the existence of exterior periodic solutions of the vector field around broken symmetric homoclinic orbits‎, ‎we investigate the existence of fixed points of the exterior Poincare map around these orbits‎. This Poincare map is the result of the combination of flows inside and outside the homoclinic orbits. It shows how ‎a big periodic orbit converts to two small periodic orbits by passing through a double homoclinic structure‎. Finally‎, ‎we use the results to investigate the existence of periodic solutions of the perturbed Duffing equation.

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[1] Antabli M, Boughariou, M, Homoclinic solutions for singular Hamiltonian systems without the strong force condition, Mathematical Analysis and Applications (2018), 1-19, https://doi.org/10.1016/j.jmaa.2018.11.028.
[2] C. Cerda, E. Toon, P. Ubilla, Existence of one homoclinic orbit for second order Hamiltonian systems involving certain hypotheses of monotonicity on the nonlinearities, Non-linear Analysis: Real World Applications vol. 47 (2019), 348{363.
[3] Y. Y. Chen, S. H. Chen, W. Zhao, Constructing explicit homoclinic solution of oscillators: An improvement for perturbation procedure based on nonlinear time transformation, Communications in Nonlinear Science and Numerical Simulation, vol. 48 (2017), 123{139.
[4] C. Chicone, Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, Publication of university of Missouri (2004),1{34.
[5] K. W. Chung, C. L. Chan, Z. Xu, G. M. Mahmoud, A perturbation-incremental method for strongly nonlinear autonomous oscillators with many degrees of freedom, Nonlinear Dynamics, vol. 28, no. 3 (2002), 243{259.
[6] L. S. Dederick, Implicit functions at a boundary point, Annals of mathematics vol.1, no. 4 (1913-1914),170{178.
[7] G. Deng, D. Zhu Codimension-3 bifurcations of a class of homoclinic loop with saddle-point, vol. 69 (2008), 3761{3773.
[8] P. Glendinning, C. Sparrow, Local and Global Behavior near Homoclinic Orbits, J. Statistical Physics vol. 35 (1984), 645{696.
[9] C. H. He, D. Tian, G. M. Moatimid, H. F. Salman, M. H. Zekry, Hybrid rayleigh{van der pol{dung oscillator: Stability analysis and controller, J. of Low Frequency Noise, Vibration and Active Control: Nonlinear Phenomena, vol. 41, no. 1 (2022), 244{268.
[10] L. Kong, Homoclinic solutions for a higher order di erence equation, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.06.033.
[11] A. P. Kuznetsov, J. P. Roman, Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol{Dung oscillators. Broadband synchronization, Physica D: Nonlinear Phenomena, vol. 238, no. 16 (2009), 1499{1506.
[12] L. Li, L. Huang Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems, J. Math. Anal. Appl. vol. 411 (2014), 83{94.
[13] F. Liang, M. Han, X. Zhang, Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems, J. Di erential Equations vol. 255 (2013), 4403{4436.
[14] X. Ma, Y. Yu, L. Wang, Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Dung oscillator, Applied Mathematics and Computation, vol. 411 (2021), 126522.
[15] X. Ma, D. Xia, W. Jiang, M. Liu, Q. Bi, Compound bursting behaviors in a forced Mathieu-van der Pol-Dung system, Chaos, Solitons & Fractals, vol. 147 (2021), 110967.
[16] J. Palis, J. W. D. Melo, Geometric theory of dynamical systems, Springer, New York, 1982.
[17] A. H. Salas, W. Albalawi, S. A. El-Tantawy, L. S. El-Sherif, Some Novel Approaches for Analyzing the Unforced and Forced Dung{Van der Pol Oscillators, J. Mathematics, vol. 2022 (2022), 2174192.
[18] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, 2nd ed, springer, 2000.
[19] Y. Xiong, M. Han, Limit cycle bifurcations near homoclinic and heteroclinic loops via stability-changing of a homoclinic loop, Chaos, Solitons and Fractals, vol. 78 (2015),107{117.