# American Institute of Mathematical Sciences

May  2016, 36(5): 2803-2825. doi: 10.3934/dcds.2016.36.2803

## Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China 2 Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240

Received  December 2014 Revised  August 2015 Published  October 2015

This paper deals with the maximum number of limit cycles, which can be bifurcated from periodic orbits of planar piecewise smooth Hamiltonian systems, which are located in a neighborhood of a generalized homoclinic loop with a nilpotent saddle on a switch line. First we present asymptotic expressions of the Melnikov functions near the loop. Then using these expressions we study the number of limit cycles which are bifurcated from the periodic orbits near the homoclinic loop under small perturbations. Finally we provide two concrete examples showing applications of our main results.
Citation: Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
##### References:
 [1] R. Asheghi and H. Zangeneh, Bifurcation of limit cycles for a quintic Hamiltonian system with a double cuspidal loop, Comput. Math. Appl., 59 (2010), 1409-1418. doi: 10.1016/j.camwa.2009.12.024.  Google Scholar [2] A. Atabaigi, H. Zangeneh and R. Kazemi, Limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a Hamiltonian system, Nonlinear Anal., 75 (2012), 1945-1958. doi: 10.1016/j.na.2011.09.044.  Google Scholar [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. Available from: http://zh.bookzz.org/book/1174967/8d438a.  Google Scholar [4] B. Brogliato, Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences, 220, Springer-Verlag, Berlin, 1996.  Google Scholar [5] C.A. Buzzi, T. de Carvalho and M.A. Teixeira, Birth of limit cycles bifurcating from a nonsmooth center, J. Math. Pures Appl., 102 (2014), 36-47. doi: 10.1016/j.matpur.2013.10.013.  Google Scholar [6] C. A. Buzzi, J. C. R. Medrado and M. A. Teixeira, Generic bifurcation of refracted systems, Adv. Math., 234 (2013), 653-666. doi: 10.1016/j.aim.2012.11.008.  Google Scholar [7] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936. doi: 10.3934/dcds.2013.33.3915.  Google Scholar [8] F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four II. cuspidal loop, J. Differential Equations, 175 (2001), 209-243. doi: 10.1006/jdeq.2000.3978.  Google Scholar [9] A.F. Filippov, Differential Equations with Discontinuous Right hand Sides, Kluwer Academic, Netherlands, 1988. Available from: http://zh.bookzz.org/book/1049223/dbd89b. doi: 10.1007/978-94-015-7793-9.  Google Scholar [10] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002.  Google Scholar [11] M. Han, J. Yang and D. Xiao, Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle, Internat. J. Bifur. Chaos, 22 (2012), 1250189, 33pp. doi: 10.1142/S0218127412501891.  Google Scholar [12] M. Han, H. Zang and J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differential Equations, 246 (2009), 129-163. doi: 10.1016/j.jde.2008.06.039.  Google Scholar [13] A. Jacquemard and M. A. Teixeira, Periodic solutions of a class of non-autonomous second order differential equations with discontinuous right-hand side, Phys. D, 241 (2012), 2003-2009. doi: 10.1016/j.physd.2011.05.011.  Google Scholar [14] D. John and W. Simpson, Bifurcations in Piecewise-smooth Continuous Systems, World Scientific, Singapore, 2010. Available from: http://zh.bookzz.org/book/1270815/2cadf8. Google Scholar [15] R. Kazemi and H. Zangeneh, Bifurcation of limit cycles in small perturbations of a hyperelliptic Hamiltonian system with two nilpotent saddles, J. Appl. Anal. Comput., 2 (2012), 395-413. Available from: http://jaac-online.com/index.php/jaac/article/view/96.  Google Scholar [16] M. Kunze, Piecewise Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.  Google Scholar [17] F. Liang and M. Han, Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Solitons Fract., 45 (2012), 454-464. doi: 10.1016/j.chaos.2011.09.013.  Google Scholar [18] F. Liang, M. Han and X. Zhang, Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems, J. Differential Equations, 255 (2013), 4403-4436. doi: 10.1016/j.jde.2013.08.013.  Google Scholar [19] X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos, 20 (2010), 1379-1390. doi: 10.1142/S021812741002654X.  Google Scholar [20] J. Llibre, B. D. Lopes and J. R. De Moraes, Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems, Qual. Theory Dyn. Syst., 13 (2014), 129-148. doi: 10.1007/s12346-014-0109-9.  Google Scholar [21] J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.  Google Scholar [22] J. Llibre, M. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012. doi: 10.1016/j.nonrwa.2013.02.004.  Google Scholar [23] J. Llibre, E. Ponce and X. Zhang, Existence of piecewise linear differential systems with exactly $n$ limit cycles for all $n\in \mathbb N$, Nonlinear Anal., 54 (2003), 977-994. doi: 10.1016/S0362-546X(03)00122-6.  Google Scholar [24] J. Llibre, P. R. da Silva and M. A. Teixeira, Study of singularities in nonsmooth dynamical systems via singular perturbation, SIAM J. Appl. Dyn. Syst., 8 (2009), 508-526. doi: 10.1137/080722886.  Google Scholar [25] D. Pi, J. Yu and X. Zhang, On the sliding bifurcation of a class of planar Filippov systems, Internat. J. Bifur. Chaos, 23 (2013), 1350040, 18pp. doi: 10.1142/S0218127413500405.  Google Scholar [26] D. Pi and X. Zhang, The sliding bifurcations in planar piecewise smooth differential systems, J. Dyn. Diff. Equat., 25 (2013), 1001-1026. doi: 10.1007/s10884-013-9327-0.  Google Scholar [27] E. Pratt, A. Léger and X. Zhang, Study of a transition in the qualitative behavior of a simple oscillator with Coulomb friction, Nonlinear Dynam., 74 (2013), 517-531. doi: 10.1007/s11071-013-0985-6.  Google Scholar [28] R. Prohens and A. E. Teruel, Canard trajectories in $3D$ piecewise linear systems, Discrete Contin. Dyn. Syst., 33 (2013), 4595-4611. doi: 10.3934/dcds.2013.33.4595.  Google Scholar [29] M. A. Teixeira and P. R. da Silva, Regularization and singular perturbation techniques for non-smooth systems, Phys. D, 241 (2012), 1948-1955. doi: 10.1016/j.physd.2011.06.022.  Google Scholar [30] J. Wang and D. Xiao, On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle, J. Differential Equations, 250 (2011), 2227-2243. doi: 10.1016/j.jde.2010.11.004.  Google Scholar [31] L. Wei, F. Liang and S. Lu, Limit cycle bifurcations near a generalized homoclinic loop in piecewise smooth systems with a hyperbolic saddle on a switch line, Appl. Math. Comput., 243 (2014), 298-310. doi: 10.1016/j.amc.2014.05.041.  Google Scholar [32] L. Zhao, The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle, Nonlinear Anal., 95 (2014), 374-387. doi: 10.1016/j.na.2013.09.020.  Google Scholar

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##### References:
 [1] R. Asheghi and H. Zangeneh, Bifurcation of limit cycles for a quintic Hamiltonian system with a double cuspidal loop, Comput. Math. Appl., 59 (2010), 1409-1418. doi: 10.1016/j.camwa.2009.12.024.  Google Scholar [2] A. Atabaigi, H. Zangeneh and R. Kazemi, Limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a Hamiltonian system, Nonlinear Anal., 75 (2012), 1945-1958. doi: 10.1016/j.na.2011.09.044.  Google Scholar [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. Available from: http://zh.bookzz.org/book/1174967/8d438a.  Google Scholar [4] B. Brogliato, Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences, 220, Springer-Verlag, Berlin, 1996.  Google Scholar [5] C.A. Buzzi, T. de Carvalho and M.A. Teixeira, Birth of limit cycles bifurcating from a nonsmooth center, J. Math. Pures Appl., 102 (2014), 36-47. doi: 10.1016/j.matpur.2013.10.013.  Google Scholar [6] C. A. Buzzi, J. C. R. Medrado and M. A. Teixeira, Generic bifurcation of refracted systems, Adv. Math., 234 (2013), 653-666. doi: 10.1016/j.aim.2012.11.008.  Google Scholar [7] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936. doi: 10.3934/dcds.2013.33.3915.  Google Scholar [8] F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four II. cuspidal loop, J. Differential Equations, 175 (2001), 209-243. doi: 10.1006/jdeq.2000.3978.  Google Scholar [9] A.F. Filippov, Differential Equations with Discontinuous Right hand Sides, Kluwer Academic, Netherlands, 1988. Available from: http://zh.bookzz.org/book/1049223/dbd89b. doi: 10.1007/978-94-015-7793-9.  Google Scholar [10] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002.  Google Scholar [11] M. Han, J. Yang and D. Xiao, Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle, Internat. J. Bifur. Chaos, 22 (2012), 1250189, 33pp. doi: 10.1142/S0218127412501891.  Google Scholar [12] M. Han, H. Zang and J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differential Equations, 246 (2009), 129-163. doi: 10.1016/j.jde.2008.06.039.  Google Scholar [13] A. Jacquemard and M. A. Teixeira, Periodic solutions of a class of non-autonomous second order differential equations with discontinuous right-hand side, Phys. D, 241 (2012), 2003-2009. doi: 10.1016/j.physd.2011.05.011.  Google Scholar [14] D. John and W. Simpson, Bifurcations in Piecewise-smooth Continuous Systems, World Scientific, Singapore, 2010. Available from: http://zh.bookzz.org/book/1270815/2cadf8. Google Scholar [15] R. Kazemi and H. Zangeneh, Bifurcation of limit cycles in small perturbations of a hyperelliptic Hamiltonian system with two nilpotent saddles, J. Appl. Anal. Comput., 2 (2012), 395-413. Available from: http://jaac-online.com/index.php/jaac/article/view/96.  Google Scholar [16] M. Kunze, Piecewise Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.  Google Scholar [17] F. Liang and M. Han, Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Solitons Fract., 45 (2012), 454-464. doi: 10.1016/j.chaos.2011.09.013.  Google Scholar [18] F. Liang, M. Han and X. Zhang, Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems, J. Differential Equations, 255 (2013), 4403-4436. doi: 10.1016/j.jde.2013.08.013.  Google Scholar [19] X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos, 20 (2010), 1379-1390. doi: 10.1142/S021812741002654X.  Google Scholar [20] J. Llibre, B. D. Lopes and J. R. De Moraes, Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems, Qual. Theory Dyn. Syst., 13 (2014), 129-148. doi: 10.1007/s12346-014-0109-9.  Google Scholar [21] J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.  Google Scholar [22] J. Llibre, M. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012. doi: 10.1016/j.nonrwa.2013.02.004.  Google Scholar [23] J. Llibre, E. Ponce and X. Zhang, Existence of piecewise linear differential systems with exactly $n$ limit cycles for all $n\in \mathbb N$, Nonlinear Anal., 54 (2003), 977-994. doi: 10.1016/S0362-546X(03)00122-6.  Google Scholar [24] J. Llibre, P. R. da Silva and M. A. Teixeira, Study of singularities in nonsmooth dynamical systems via singular perturbation, SIAM J. Appl. Dyn. Syst., 8 (2009), 508-526. doi: 10.1137/080722886.  Google Scholar [25] D. Pi, J. Yu and X. Zhang, On the sliding bifurcation of a class of planar Filippov systems, Internat. J. Bifur. Chaos, 23 (2013), 1350040, 18pp. doi: 10.1142/S0218127413500405.  Google Scholar [26] D. Pi and X. Zhang, The sliding bifurcations in planar piecewise smooth differential systems, J. Dyn. Diff. Equat., 25 (2013), 1001-1026. doi: 10.1007/s10884-013-9327-0.  Google Scholar [27] E. Pratt, A. Léger and X. Zhang, Study of a transition in the qualitative behavior of a simple oscillator with Coulomb friction, Nonlinear Dynam., 74 (2013), 517-531. doi: 10.1007/s11071-013-0985-6.  Google Scholar [28] R. Prohens and A. E. Teruel, Canard trajectories in $3D$ piecewise linear systems, Discrete Contin. Dyn. Syst., 33 (2013), 4595-4611. doi: 10.3934/dcds.2013.33.4595.  Google Scholar [29] M. A. Teixeira and P. R. da Silva, Regularization and singular perturbation techniques for non-smooth systems, Phys. D, 241 (2012), 1948-1955. doi: 10.1016/j.physd.2011.06.022.  Google Scholar [30] J. Wang and D. Xiao, On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle, J. Differential Equations, 250 (2011), 2227-2243. doi: 10.1016/j.jde.2010.11.004.  Google Scholar [31] L. Wei, F. Liang and S. Lu, Limit cycle bifurcations near a generalized homoclinic loop in piecewise smooth systems with a hyperbolic saddle on a switch line, Appl. Math. Comput., 243 (2014), 298-310. doi: 10.1016/j.amc.2014.05.041.  Google Scholar [32] L. Zhao, The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle, Nonlinear Anal., 95 (2014), 374-387. doi: 10.1016/j.na.2013.09.020.  Google Scholar
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