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Rarefaction waves for the Toda equation via nonlinear steepest descent
Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems
1. | School of Mathematical Sciences, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, China |
2. | Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, Maribor, SI-2000 Maribor, Slovenia |
In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi-homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi-homogeneous systems. Moreover, the center is global and non-isochronous, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi-homogeneous but non-homogeneous systems are obtained. Finally we investigate limit cycle bifurcations of the piecewise quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.
References:
[1] |
A. Algaba, N. Fuentes and C. García,
Center of quasihomogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431.
doi: 10.1016/j.nonrwa.2011.07.056. |
[2] |
A. Algaba, E. Gamero and C. García,
The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 396-420.
doi: 10.1088/0951-7715/22/2/009. |
[3] |
A. A. Andronov, E. A. Leontovitch, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translations, John Wiley and Sons, New York, 1973. |
[4] |
A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. |
[5] |
W. Aziz, J. Llibre and C. Pantazi,
Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250.
doi: 10.1016/j.aim.2013.12.006. |
[6] |
I. S. Berezin and N. P. Zhidkov,
Computing Methods, Volume Ⅱ, Pergamon Press, Oxford, 1965. |
[7] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk,
Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. |
[8] |
M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. Nordmark, G. Tost and P. Piiroinen,
Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.
doi: 10.1137/050625060. |
[9] |
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. |
[10] |
X. Chen, V. Romanovski and W. Zhang,
Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl., 432 (2015), 1058-1076.
doi: 10.1016/j.jmaa.2015.07.036. |
[11] |
F. Dumortier, J. Llibre and J. C. Artés, Qualititive Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. |
[12] |
A. F. Filippov,
Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. |
[13] |
E. Freire, E. Ponce and F. Torres,
Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.
doi: 10.1137/11083928X. |
[14] |
B. García, J. Llibre and J. S. Pérez del Río,
Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204.
doi: 10.1016/j.jde.2013.07.032. |
[15] |
L. Gavrilov, J. Giné and M. Grau,
On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135.
doi: 10.1016/j.jde.2009.02.010. |
[16] |
F. Giannakopoulos and K. Pliete,
Planar system of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632.
doi: 10.1088/0951-7715/14/6/311. |
[17] |
J. Giné, M. Grau and J. Llibre,
Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547.
doi: 10.3934/dcds.2013.33.4531. |
[18] |
J. Giné, M. Grau and J. Llibre,
Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations, 259 (2015), 7135-7160.
doi: 10.1016/j.jde.2015.08.014. |
[19] |
A. Goriely,
Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.
doi: 10.1063/1.531484. |
[20] |
M. Han and L. Sheng,
Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.
|
[21] |
M. Han and W. Zhang,
On Hopf bifurcation in non-smooth planar systems, J. Differential Equation, 248 (2010), 2399-2416.
doi: 10.1016/j.jde.2009.10.002. |
[22] |
Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems,
Adv. Difference Eqns. , (2007), Art ID 98427, 10 pp. |
[23] |
M. Kunze,
Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. |
[24] |
Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani,
One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos, 13 (2003), 2157-2188.
doi: 10.1142/S0218127403007874. |
[25] |
W. Li, J. Llibre, J. Yang and Z. Zhang,
Limit cycles bifurcating from the period annulus of quasi-homegeneous centers, J. Dyn. Diff. Eqns., 21 (2009), 133-152.
doi: 10.1007/s10884-008-9126-1. |
[26] |
F. Liang, M. Han and V. Romanovski,
Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.
doi: 10.1016/j.na.2012.03.022. |
[27] |
H. Liang, J. Huang and Y. Zhao,
Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681.
doi: 10.1007/s11071-014-1541-8. |
[28] |
J. Llibre and E. Ponce,
Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms, 19 (2012), 325-335.
|
[29] |
J. Llibre and X. Zhang,
Polynomial first integrals for quasihomogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[30] |
O. Makarenkov and J. S. W. Lamb,
Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844.
doi: 10.1016/j.physd.2012.08.002. |
[31] |
J. Reyn,
Phase Portraits of Planar Quadratic Systems, Mathematics and Its Applications, 583, Springer, New York, 2007. |
[32] |
Y. Tang, L. Wang and X. Zhang,
Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191.
|
[33] |
L. Wei and X. Zhang,
Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst., 36 (2016), 2803-2825.
|
[34] |
Y. Xiong and M. Han,
Planar quasi-homogeneous polynomial systems with a given weight degree, Discrete Contin. Dyn. Syst., 36 (2016), 4015-4025.
doi: 10.3934/dcds.2016.36.4015. |
[35] |
J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Preprint. |
[36] |
Y. Zou, T. Kupper and W. J. Beyn,
Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Science, 16 (2006), 159-177.
doi: 10.1007/s00332-005-0606-8. |
show all references
References:
[1] |
A. Algaba, N. Fuentes and C. García,
Center of quasihomogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431.
doi: 10.1016/j.nonrwa.2011.07.056. |
[2] |
A. Algaba, E. Gamero and C. García,
The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 396-420.
doi: 10.1088/0951-7715/22/2/009. |
[3] |
A. A. Andronov, E. A. Leontovitch, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translations, John Wiley and Sons, New York, 1973. |
[4] |
A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. |
[5] |
W. Aziz, J. Llibre and C. Pantazi,
Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250.
doi: 10.1016/j.aim.2013.12.006. |
[6] |
I. S. Berezin and N. P. Zhidkov,
Computing Methods, Volume Ⅱ, Pergamon Press, Oxford, 1965. |
[7] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk,
Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. |
[8] |
M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. Nordmark, G. Tost and P. Piiroinen,
Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.
doi: 10.1137/050625060. |
[9] |
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. |
[10] |
X. Chen, V. Romanovski and W. Zhang,
Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl., 432 (2015), 1058-1076.
doi: 10.1016/j.jmaa.2015.07.036. |
[11] |
F. Dumortier, J. Llibre and J. C. Artés, Qualititive Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. |
[12] |
A. F. Filippov,
Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. |
[13] |
E. Freire, E. Ponce and F. Torres,
Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.
doi: 10.1137/11083928X. |
[14] |
B. García, J. Llibre and J. S. Pérez del Río,
Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204.
doi: 10.1016/j.jde.2013.07.032. |
[15] |
L. Gavrilov, J. Giné and M. Grau,
On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135.
doi: 10.1016/j.jde.2009.02.010. |
[16] |
F. Giannakopoulos and K. Pliete,
Planar system of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632.
doi: 10.1088/0951-7715/14/6/311. |
[17] |
J. Giné, M. Grau and J. Llibre,
Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547.
doi: 10.3934/dcds.2013.33.4531. |
[18] |
J. Giné, M. Grau and J. Llibre,
Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations, 259 (2015), 7135-7160.
doi: 10.1016/j.jde.2015.08.014. |
[19] |
A. Goriely,
Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.
doi: 10.1063/1.531484. |
[20] |
M. Han and L. Sheng,
Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.
|
[21] |
M. Han and W. Zhang,
On Hopf bifurcation in non-smooth planar systems, J. Differential Equation, 248 (2010), 2399-2416.
doi: 10.1016/j.jde.2009.10.002. |
[22] |
Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems,
Adv. Difference Eqns. , (2007), Art ID 98427, 10 pp. |
[23] |
M. Kunze,
Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. |
[24] |
Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani,
One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos, 13 (2003), 2157-2188.
doi: 10.1142/S0218127403007874. |
[25] |
W. Li, J. Llibre, J. Yang and Z. Zhang,
Limit cycles bifurcating from the period annulus of quasi-homegeneous centers, J. Dyn. Diff. Eqns., 21 (2009), 133-152.
doi: 10.1007/s10884-008-9126-1. |
[26] |
F. Liang, M. Han and V. Romanovski,
Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.
doi: 10.1016/j.na.2012.03.022. |
[27] |
H. Liang, J. Huang and Y. Zhao,
Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681.
doi: 10.1007/s11071-014-1541-8. |
[28] |
J. Llibre and E. Ponce,
Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms, 19 (2012), 325-335.
|
[29] |
J. Llibre and X. Zhang,
Polynomial first integrals for quasihomogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[30] |
O. Makarenkov and J. S. W. Lamb,
Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844.
doi: 10.1016/j.physd.2012.08.002. |
[31] |
J. Reyn,
Phase Portraits of Planar Quadratic Systems, Mathematics and Its Applications, 583, Springer, New York, 2007. |
[32] |
Y. Tang, L. Wang and X. Zhang,
Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191.
|
[33] |
L. Wei and X. Zhang,
Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst., 36 (2016), 2803-2825.
|
[34] |
Y. Xiong and M. Han,
Planar quasi-homogeneous polynomial systems with a given weight degree, Discrete Contin. Dyn. Syst., 36 (2016), 4015-4025.
doi: 10.3934/dcds.2016.36.4015. |
[35] |
J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Preprint. |
[36] |
Y. Zou, T. Kupper and W. J. Beyn,
Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Science, 16 (2006), 159-177.
doi: 10.1007/s00332-005-0606-8. |




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Figure 3 | Parameter conditions |
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