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Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions
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 |
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. |
[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. |
[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. |
[4] |
B. Brogliato, Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences, 220, Springer-Verlag, Berlin, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
M. Kunze, Piecewise Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103843. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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), 1409-1418.
doi: 10.1016/j.camwa.2009.12.024. |
[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. |
[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. |
[4] |
B. Brogliato, Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences, 220, Springer-Verlag, Berlin, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
M. Kunze, Piecewise Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103843. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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