Abstract
In this work, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three zones separated by two switching manifolds \(x=-\alpha \) and \(x=\beta \). We suppose that the dynamic in each zone is governed by a smooth system. When a trajectory reaches the switching manifolds, then reset maps describing impacting rules on the switching manifolds will be applied instantaneously before the trajectory enters into the other zone. We also assume that the unperturbed system is a piecewise-defined continuous Hamiltonian system and possesses a pair of heteroclinic orbits transversally crossing the switching manifolds. Then, we study the persistence of the heteroclinic orbits under a non-autonomous periodic perturbation and the reset maps. In order to obtain this objective, we derive a Melnikov-type function by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds in this system. Finally, we employ the obtained Melnikov-type function to study the persistence of a heteroclinic cycle and complicated dynamics near the heteroclinic cycle for a concrete planar piecewise-smooth system.
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References
Brogliato, B.: Nonsmooth Mechanics. Springer, London (1999)
Bernardo, M.D., Kowalczyk, P., Nordmark, A.B.: Sliding bifurcations: a novel mechanism for the sudden onset of chaos in dry friction oscillators. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 2935–2948 (2003)
Banerjee, S., Verghese, G.: Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations. Chaos and Nonlinear Control. Wiley-IEEE Press, New York (2001)
Garcia, M., Chatterjee, A., Ruina, A., Coleman, M.: The simplest walking model:stability, complexity and scaling ASME. J. Biomech. Eng. 120, 281–288 (1998)
Bernardo, M.D., Garofalo, L., Vasca, F.: Bifurcations in piecewise-smooth feedback systems. Int. J. Control 75, 1243–1259 (2002)
Kunze, M.: Non-smooth Dynamical Systems. Springer, Berlin (2000)
Awrejcewicz, J., Holicke, M.M.: Smooth and Nonsmooth High Dimensional Chaos and Melnikov-Type Method. World Scientific, Singapore (2007)
Bernardo, M.D., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Application. Springer, London (2008)
Leine, R.I., Van Campen, D.H., Van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000)
Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 1826–1844 (2012)
Melnikov, V.K.: On the stability of the center for time periodic perturbations. Tans. Moscow Math. Soc. 12, 1–57 (1963)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)
Wiggins, S.: Global Bifurcations and Chaos-Analytical Methods. Springer, New York (1988)
Kukučka, P.: Melnikov method for discontinuous planar systems. Nonlinear Anal. 66, 2698–2719 (2007)
Battelli, F., Fečkan, M.: Homoclinic trajectories in discontinuous systems. J. Dyn. Differ. Equ. 20, 337–376 (2008)
Battelli, F., Fečkan, M.: Bifurcation and chaos near sliding homoclinics. J. Differ. Equ. 248, 2227–2262 (2010)
Battelli, F., Fečkan, M.: Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems. Phys. D 241, 1962–1975 (2012)
Li, S.B., Zhang, W., Hao, Y.X.: Melnikov-type method for a class of discontinuous planar systems and applications. Int. J. Bifurc. Chaos 24(1450022), 1–18 (2014)
Du, Z., Zhang, W.: Melnikov method for homoclinic bifurcations in nonlinear impact oscillators. Comput. Math. Appl. 50, 445–458 (2005)
Granados, A., Hogan, S.J., Seara, T.M.: The Melnikov method and subharmonic orbits in a piecewise-smooth system. SIAM J. Appl. Dyn. Syst. 11, 801–830 (2012)
Carmona, V., Fernández-García, S., Freire, E., Torres, F.: Melnikov theory for a class of planar hybrid systems. Phys. D 248, 44–54 (2013)
Granados, A., Hogan, S.J., Seara, T.M.: The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks. Phys. D 269, 1–20 (2014)
Li, S.B., Shen, C., Zhang, W., Hao, Y.X.: Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping. Nonlinear Dyn. 79, 2395–2406 (2015)
Gao, J., Du, Z.: Homoclinic bifurcation in a quasiperiodically excited impact inverted pendulum. Nonlinear Dyn. 79, 445–458 (2015)
Castro, J., Alvarez, J.: Melnikov-type chaos of planar systems with two discontinuities. Int. J. Bifurc. Chaos 25, 1550027 (2015)
Tian, R.L., Zhou, Y.F., Zhang, B.L., Yang, X.W.: Chaotic threshold for a class of impulsive differential system. Nonlinear Dyn. 79, 445–458 (2015)
Li, S.B., Ma, W.S., Zhang, W., Hao, Y.X.: Melnikov method for a three-zonal planar hybrid piecewise-smooth system and application. Int. J. Bifurc. Chaos 26(1650014), 1–13 (2016)
Li, S.B., Ma, W.S., Zhang, W., Hao, Y.X.: Melnikov method for a class of planar hybrid piecewise-smooth systems. Int. J. Bifurc. Chaos 26(1650030), 1–12 (2016)
Bertozzi, A.L.: Heteroclinic orbits and chaotic dynamics in planar fluid flow. SIAM J. Math. Anal. 19, 1271–1294 (1988)
Acknowledgments
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11472298, 11290152, 11427801, 11272063, 11472056, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), the Natural Science Foundation of Tianjin City through Grant No. 13JCQNJC04400 and the Fundamental Research Funds for the Central Universities through Grant No. 3122013k005.
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Li, S., Shen, C., Zhang, W. et al. The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application. Nonlinear Dyn 85, 1091–1104 (2016). https://doi.org/10.1007/s11071-016-2746-9
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DOI: https://doi.org/10.1007/s11071-016-2746-9