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 - A, 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.  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.  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, (2008).   Google Scholar

[4]

B. Brogliato, Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences, 220,, Springer-Verlag, (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.  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.  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.  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.  doi: 10.1006/jdeq.2000.3978.  Google Scholar

[9]

A.F. Filippov, Differential Equations with Discontinuous Right hand Sides,, Kluwer Academic, (1988).  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.  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).  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.  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.  doi: 10.1016/j.physd.2011.05.011.  Google Scholar

[14]

D. John and W. Simpson, Bifurcations in Piecewise-smooth Continuous Systems,, World Scientific, (2010).   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.   Google Scholar

[16]

M. Kunze, Piecewise Smooth Dynamical Systems,, Springer-Verlag, (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.  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.  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.  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.  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.  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.  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.  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.  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).  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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.na.2013.09.020.  Google Scholar

show all references

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.  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.  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, (2008).   Google Scholar

[4]

B. Brogliato, Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences, 220,, Springer-Verlag, (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.  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.  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.  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.  doi: 10.1006/jdeq.2000.3978.  Google Scholar

[9]

A.F. Filippov, Differential Equations with Discontinuous Right hand Sides,, Kluwer Academic, (1988).  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.  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).  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.  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.  doi: 10.1016/j.physd.2011.05.011.  Google Scholar

[14]

D. John and W. Simpson, Bifurcations in Piecewise-smooth Continuous Systems,, World Scientific, (2010).   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.   Google Scholar

[16]

M. Kunze, Piecewise Smooth Dynamical Systems,, Springer-Verlag, (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.  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.  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.  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.  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.  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.  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.  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.  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).  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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.na.2013.09.020.  Google Scholar

[1]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[2]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[3]

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

[4]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[5]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[6]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[7]

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

[8]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[9]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[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]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[12]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[13]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[14]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[15]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[16]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[17]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[18]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[19]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[20]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]