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December  2019, 24(12): 6495-6509. doi: 10.3934/dcdsb.2019150

Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems

a. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

b. 

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350108, China

c. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

d. 

School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

Received  December 2018 Revised  January 2019 Published  July 2019

In this paper we investigate the center problem for the discontinuous piecewise smooth quasi–homogeneous but non–homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi–homogeneous cubic and quartic polynomial differential systems.

Citation: Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150
References:
[1]

A. AlgabaE. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.  doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[2]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.  Google Scholar

[3]

W. AzizJ. 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.  Google Scholar

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008.  Google Scholar

[5]

M. di BernardoC. J. BuddA. R. ChampneysP. KowalczykA. NordmarkG. Tost and P. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.  doi: 10.1137/050625060.  Google Scholar

[6]

H. ChenS. DuanY. Tang and J. Xie, Global dynamics of a mechanical system with fry friction, J. Differential Equations, 265 (2018), 5490-5519.  doi: 10.1016/j.jde.2018.06.013.  Google Scholar

[7]

H. Chen and Y. Tang, At most two limit cycles in a piecewise linear differential system with three zones and asymmetry, Physica D, 386/387 (2019), 23-30.  doi: 10.1016/j.physd.2018.08.004.  Google Scholar

[8]

H. Dulac, Détermination et integration d'une certaine classe d'équations différentielle ayant par point singulier un centre, Bull. Sci. Math., Sér. (2), 32 (1908), 230–252. Google Scholar

[9]

F. Dumortier, J. Llibre and J. C. Artés, Qualititive Theory of Planar Differential Systems, Springer–Verlag, Berlin, 2006.  Google Scholar

[10]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[11]

E. FreireE. Ponce and F. Torres, Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.  doi: 10.1137/11083928X.  Google Scholar

[12]

I.A. GarcíaH. GiacominiJ. Giné and J. Llibre, Analytic nilpotent centers as limits of nondegenerate centers revisited, J. Math. Anal. and Appl., 441 (2016), 893-899.  doi: 10.1016/j.jmaa.2016.04.046.  Google Scholar

[13]

B. GarcíaJ. 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.  Google Scholar

[14]

J. Giné, On the centers of planar analytic differential systems, Int. J. Bifurcation and Chaos, 17 (2007), 3061-3070.  doi: 10.1142/S0218127407018865.  Google Scholar

[15]

J. GinéM. 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.  Google Scholar

[16]

J. Giné and J. Llibre, A method for characterizing nilpotent centers, J. Math. Anal. Appl., 413 (2014), 537-545.  doi: 10.1016/j.jmaa.2013.12.013.  Google Scholar

[17]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[18]

Yu. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurcation and Chaos, 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.  Google Scholar

[19]

W. LiJ. LlibreJ. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi–homegeneous centers, J. Dyn. Diff. Equat., 21 (2009), 133-152.  doi: 10.1007/s10884-008-9126-1.  Google Scholar

[20]

H. LiangJ. 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.  Google Scholar

[21]

J. Llibre, Centers: their integrability and relations with the divergence, Appl. Math. and Nonl. Sci., 1 (2016), 79-86.  doi: 10.21042/AMNS.2016.1.00007.  Google Scholar

[22]

M. A. Lyapunov, Probléme Général de la Stabilité du Mouvement, Ann. of Math. Stud., Vol. 17, Princeton University Press, 1947.  Google Scholar

[23]

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.  Google Scholar

[24]

L. Mazzi and M. Sabatini, A characterization of centres via first integrals, J. Differential Equations, 76 (1988), 222-237.  doi: 10.1016/0022-0396(88)90072-1.  Google Scholar

[25]

R. Moussu, Symetrie et forme normale des centres et foyers degeneres, Ergodic Theory Dynam. Systems, 2 (1982), 241-251.  doi: 10.1017/s0143385700001553.  Google Scholar

[26]

R. Oliveira and Y. Zhao, Structural stability of planar quasihomogeneous vector fields, Qual. Theory Dyn. Syst., 13 (2014), 39-72.  doi: 10.1007/s12346-013-0105-5.  Google Scholar

[27]

J. M. PearsonN. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev., 38 (1996), 619-636.  doi: 10.1137/S0036144595283575.  Google Scholar

[28]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. de Mathématiques, 37 (1881), 375–422; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, 3–84. Google Scholar

[29]

H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré Ⅰ and Ⅱ, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161–191; 11 (1897), 193–239. Google Scholar

[30] G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon Press, New York, 1964.   Google Scholar
[31]

Y. TangL. Wang and X. Zhang, Center of planar quintic quasi–homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191.  doi: 10.3934/dcds.2015.35.2177.  Google Scholar

[32]

Y. Tang and X. Zhang, Global dynamics of planar quasi–homogeneous differential systems, Nonlinear Anal. Real World Appl., 49 (2019), 90-110.  doi: 10.1016/j.nonrwa.2019.02.008.  Google Scholar

[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.  doi: 10.3934/dcds.2016.36.2803.  Google Scholar

[34]

J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Bull. Sci. Math., 147 (2018), 7-25.  doi: 10.1016/j.bulsci.2018.04.001.  Google Scholar

show all references

References:
[1]

A. AlgabaE. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.  doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[2]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.  Google Scholar

[3]

W. AzizJ. 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.  Google Scholar

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008.  Google Scholar

[5]

M. di BernardoC. J. BuddA. R. ChampneysP. KowalczykA. NordmarkG. Tost and P. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.  doi: 10.1137/050625060.  Google Scholar

[6]

H. ChenS. DuanY. Tang and J. Xie, Global dynamics of a mechanical system with fry friction, J. Differential Equations, 265 (2018), 5490-5519.  doi: 10.1016/j.jde.2018.06.013.  Google Scholar

[7]

H. Chen and Y. Tang, At most two limit cycles in a piecewise linear differential system with three zones and asymmetry, Physica D, 386/387 (2019), 23-30.  doi: 10.1016/j.physd.2018.08.004.  Google Scholar

[8]

H. Dulac, Détermination et integration d'une certaine classe d'équations différentielle ayant par point singulier un centre, Bull. Sci. Math., Sér. (2), 32 (1908), 230–252. Google Scholar

[9]

F. Dumortier, J. Llibre and J. C. Artés, Qualititive Theory of Planar Differential Systems, Springer–Verlag, Berlin, 2006.  Google Scholar

[10]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[11]

E. FreireE. Ponce and F. Torres, Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.  doi: 10.1137/11083928X.  Google Scholar

[12]

I.A. GarcíaH. GiacominiJ. Giné and J. Llibre, Analytic nilpotent centers as limits of nondegenerate centers revisited, J. Math. Anal. and Appl., 441 (2016), 893-899.  doi: 10.1016/j.jmaa.2016.04.046.  Google Scholar

[13]

B. GarcíaJ. 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.  Google Scholar

[14]

J. Giné, On the centers of planar analytic differential systems, Int. J. Bifurcation and Chaos, 17 (2007), 3061-3070.  doi: 10.1142/S0218127407018865.  Google Scholar

[15]

J. GinéM. 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.  Google Scholar

[16]

J. Giné and J. Llibre, A method for characterizing nilpotent centers, J. Math. Anal. Appl., 413 (2014), 537-545.  doi: 10.1016/j.jmaa.2013.12.013.  Google Scholar

[17]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[18]

Yu. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurcation and Chaos, 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.  Google Scholar

[19]

W. LiJ. LlibreJ. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi–homegeneous centers, J. Dyn. Diff. Equat., 21 (2009), 133-152.  doi: 10.1007/s10884-008-9126-1.  Google Scholar

[20]

H. LiangJ. 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.  Google Scholar

[21]

J. Llibre, Centers: their integrability and relations with the divergence, Appl. Math. and Nonl. Sci., 1 (2016), 79-86.  doi: 10.21042/AMNS.2016.1.00007.  Google Scholar

[22]

M. A. Lyapunov, Probléme Général de la Stabilité du Mouvement, Ann. of Math. Stud., Vol. 17, Princeton University Press, 1947.  Google Scholar

[23]

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.  Google Scholar

[24]

L. Mazzi and M. Sabatini, A characterization of centres via first integrals, J. Differential Equations, 76 (1988), 222-237.  doi: 10.1016/0022-0396(88)90072-1.  Google Scholar

[25]

R. Moussu, Symetrie et forme normale des centres et foyers degeneres, Ergodic Theory Dynam. Systems, 2 (1982), 241-251.  doi: 10.1017/s0143385700001553.  Google Scholar

[26]

R. Oliveira and Y. Zhao, Structural stability of planar quasihomogeneous vector fields, Qual. Theory Dyn. Syst., 13 (2014), 39-72.  doi: 10.1007/s12346-013-0105-5.  Google Scholar

[27]

J. M. PearsonN. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev., 38 (1996), 619-636.  doi: 10.1137/S0036144595283575.  Google Scholar

[28]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. de Mathématiques, 37 (1881), 375–422; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, 3–84. Google Scholar

[29]

H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré Ⅰ and Ⅱ, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161–191; 11 (1897), 193–239. Google Scholar

[30] G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon Press, New York, 1964.   Google Scholar
[31]

Y. TangL. Wang and X. Zhang, Center of planar quintic quasi–homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191.  doi: 10.3934/dcds.2015.35.2177.  Google Scholar

[32]

Y. Tang and X. Zhang, Global dynamics of planar quasi–homogeneous differential systems, Nonlinear Anal. Real World Appl., 49 (2019), 90-110.  doi: 10.1016/j.nonrwa.2019.02.008.  Google Scholar

[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.  doi: 10.3934/dcds.2016.36.2803.  Google Scholar

[34]

J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Bull. Sci. Math., 147 (2018), 7-25.  doi: 10.1016/j.bulsci.2018.04.001.  Google Scholar

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