American Institute of Mathematical Sciences

November  2017, 22(9): 3259-3272. doi: 10.3934/dcdsb.2017136

Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

 1 Departamento de Matemática, ICMC-Universidade de Sãao Paulo, Avenida Trabalhador Sãao-carlense, 400, Sãao Carlos, SP, 13566-590, Brazil 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain 3 Departamento de Física, Química e Matemática, UFSCar, Sorocaba, SP, 18052-780, Brazil

Received  August 2016 Revised  March 2017 Published  April 2017

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems $\dot{x}=-y+x^2, \;\dot{y}=x+xy$, and $\dot{x}=-y+x^2y, \;\dot{y}=x+xy^2$, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively.

Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.

Citation: Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136
References:
 [1] A. Algaba and M. Reyes, Computing center conditions for vector fields with constant angular speed, J. Comput. Appl. Math., 154 (2003), 143-159.  doi: 10.1016/S0377-0427(02)00818-X. [2] A. Algaba, M. Reyes, T. Ortega and A. Bravo, Campos cuárticos con velocidad angular constante, in Actas : XVI CEDYA Congreso de Ecuaciones Diferenciales y Aplicaciones, VI CMA Congreso de Matemática Aplicada, Las Palmas de Gran Canaria, 2 (1999), 1341-1348. [3] A. Algaba, M. Reyes and A. Bravo, Geometry of the uniformly isochronous centers with polynomial commutator, Differential Equations Dynam. Systems, 10 (2002), 257-275. [4] N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation 1954 (1954), 19pp. [5] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications Appl. Math. Sci. , 163 Spring-Verlag, London, 2008. [6] A. Buicǎ and J. Llibre, Averaging methods for finding periodic orbits via Brower degree, Bull. Sci. Math., 128 (2004), 7-22. [7] J. Chavarriga, I.A. García and J. Giné, On the integrability of the differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Int. J. of Bif. and Chaos, 11 (2001), 711-722.  doi: 10.1142/S0218127401002390. [8] J. Chavarriga, J. Giné and I.A. García, Isochronous centers of cubic systems with degenerate infinity, Diff. Eq. Dyn. Sys., 7 (1999), 221-238. [9] J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70.  doi: 10.1007/BF02969404. [10] C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326.  doi: 10.1016/0022-0396(91)90142-V. [11] C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser, Boston, 2007. [12] A.G. Choudhury and P. Guha, On commuting vector fields and Darboux functions for planar differential equations, Lobachevskii J. Math., 34 (2013), 212-226.  doi: 10.1134/S1995080213030049. [13] B. Coll, A. Gasull and R. Prohens, Bifurcation of limit cycles from two families of centers, Dyn. Contin. Discrete Impuls. Syst., 12 (2005), 275-287. [14] C.B. Collins, Conditions for a center in a simple class of cubic systems, Differential and Integral Equations, 10 (1997), 333-356. [15] R. Conti, Uniformly isochronous centers of polynomial systems in $\mathbb{R}^2$, Lecture Notes in Pure and Appl. Math., 152 (1994), 21-31. [16] R. Conti, Centers of planar polynomial systems. A review, Le Matematiche, 53 (1998), 207-240. [17] J. Devlin, N.G. Lloyd and J.M. Pearson, Cubic systems and Abel equations, J. Differential Equations, 147 (1998), 435-454.  doi: 10.1006/jdeq.1998.3420. [18] F.S. Dias and L.F. Mello, The center-focus problem and small amplitude limit cycles in rigid systems, Discrete Contin. Dyn. Syst., 32 (2012), 1627-1637.  doi: 10.3934/dcds.2012.32.1627. [19] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Nauka, Moscow, 1985 (transl. Kluwer, Dordrecht, 1988). doi: 10.1007/978-94-015-7793-9. [20] G. R. Fowles and G. L. Cassidy, Analytical Mechanics, Saunders Collegs Publishing, Philadelphia, Orlando, FL, 1993. [21] A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404.  doi: 10.1016/j.jmaa.2004.07.030. [22] A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems, Proc. Amer. Math. Soc., 133 (2005), 751-758.  doi: 10.1090/S0002-9939-04-07542-2. [23] P. Guha and A. Ghose Choudhury, On planar and non-planar isochronous systems and Poisson structures, Int. J. Geom. Methods Mod. Phys., 7 (2010), 1115-1131.  doi: 10.1142/S0219887810004750. [24] J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comp. Appl. Math., 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007. [25] J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. Comp. Appl. Math., 287 (2015), 98-114.  doi: 10.1016/j.cam.2015.02.046. [26] J. Itikawa and J. Llibre, Limit cycles bifurcating from the period annulus of a uniform isochronous center in a quartic polynomial differential system, Electron. J. Differential Equations, 246 (2015), 1-11. [27] J. Itikawa, J. Llibre and D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, to appear in Revista Matemática Iberoamericana. [28] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Comput. Neurosci. , MIT Press, Cambridge, MA, 2007. [29] J. Llibre, A.C. Mereu and D.D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022. [30] J. Llibre and A.C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, J. Math. Anal. Appl., 413 (2014), 763-775.  doi: 10.1016/j.jmaa.2013.12.031. [31] N. G. Lloyd, Degree Theory, Cambridge Tracts in Math. 73 Cambridge, 1978. [32] W.S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964), 21-36. [33] 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. [34] P. Mardesic, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108.  doi: 10.1006/jdeq.1995.1122. [35] L. Peng and Z. Feng, Bifurcation of limit cycles from quartic isochronous systems, Elec. J. Differential Equations, 95 (2014), 1-14. [36] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, in World Scientific Series on Nonlinear Science A, 69, World Scientific, Singapore, 2010. [37] M. A. Teixeira, Perturbation Theory for Non-smooth Systems, in Meyers: Encyclopedia of Complexity and Systems Science, 1 (Perturbation Theory), 1325-1336, Springer, New York, 2012. [38] E.P. Volokitin and V.M. Cheresiz, Singular points and first integrals of holomorphic dynamical systems, J. Math. Sciences, 203 (2014), 605-620.  doi: 10.1007/s10958-014-2162-y.

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References:
 [1] A. Algaba and M. Reyes, Computing center conditions for vector fields with constant angular speed, J. Comput. Appl. Math., 154 (2003), 143-159.  doi: 10.1016/S0377-0427(02)00818-X. [2] A. Algaba, M. Reyes, T. Ortega and A. Bravo, Campos cuárticos con velocidad angular constante, in Actas : XVI CEDYA Congreso de Ecuaciones Diferenciales y Aplicaciones, VI CMA Congreso de Matemática Aplicada, Las Palmas de Gran Canaria, 2 (1999), 1341-1348. [3] A. Algaba, M. Reyes and A. Bravo, Geometry of the uniformly isochronous centers with polynomial commutator, Differential Equations Dynam. Systems, 10 (2002), 257-275. [4] N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation 1954 (1954), 19pp. [5] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications Appl. Math. Sci. , 163 Spring-Verlag, London, 2008. [6] A. Buicǎ and J. Llibre, Averaging methods for finding periodic orbits via Brower degree, Bull. Sci. Math., 128 (2004), 7-22. [7] J. Chavarriga, I.A. García and J. Giné, On the integrability of the differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Int. J. of Bif. and Chaos, 11 (2001), 711-722.  doi: 10.1142/S0218127401002390. [8] J. Chavarriga, J. Giné and I.A. García, Isochronous centers of cubic systems with degenerate infinity, Diff. Eq. Dyn. Sys., 7 (1999), 221-238. [9] J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70.  doi: 10.1007/BF02969404. [10] C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326.  doi: 10.1016/0022-0396(91)90142-V. [11] C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser, Boston, 2007. [12] A.G. Choudhury and P. Guha, On commuting vector fields and Darboux functions for planar differential equations, Lobachevskii J. Math., 34 (2013), 212-226.  doi: 10.1134/S1995080213030049. [13] B. Coll, A. Gasull and R. Prohens, Bifurcation of limit cycles from two families of centers, Dyn. Contin. Discrete Impuls. Syst., 12 (2005), 275-287. [14] C.B. Collins, Conditions for a center in a simple class of cubic systems, Differential and Integral Equations, 10 (1997), 333-356. [15] R. Conti, Uniformly isochronous centers of polynomial systems in $\mathbb{R}^2$, Lecture Notes in Pure and Appl. Math., 152 (1994), 21-31. [16] R. Conti, Centers of planar polynomial systems. A review, Le Matematiche, 53 (1998), 207-240. [17] J. Devlin, N.G. Lloyd and J.M. Pearson, Cubic systems and Abel equations, J. Differential Equations, 147 (1998), 435-454.  doi: 10.1006/jdeq.1998.3420. [18] F.S. Dias and L.F. Mello, The center-focus problem and small amplitude limit cycles in rigid systems, Discrete Contin. Dyn. Syst., 32 (2012), 1627-1637.  doi: 10.3934/dcds.2012.32.1627. [19] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Nauka, Moscow, 1985 (transl. Kluwer, Dordrecht, 1988). doi: 10.1007/978-94-015-7793-9. [20] G. R. Fowles and G. L. Cassidy, Analytical Mechanics, Saunders Collegs Publishing, Philadelphia, Orlando, FL, 1993. [21] A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404.  doi: 10.1016/j.jmaa.2004.07.030. [22] A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems, Proc. Amer. Math. Soc., 133 (2005), 751-758.  doi: 10.1090/S0002-9939-04-07542-2. [23] P. Guha and A. Ghose Choudhury, On planar and non-planar isochronous systems and Poisson structures, Int. J. Geom. Methods Mod. Phys., 7 (2010), 1115-1131.  doi: 10.1142/S0219887810004750. [24] J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comp. Appl. Math., 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007. [25] J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. Comp. Appl. Math., 287 (2015), 98-114.  doi: 10.1016/j.cam.2015.02.046. [26] J. Itikawa and J. Llibre, Limit cycles bifurcating from the period annulus of a uniform isochronous center in a quartic polynomial differential system, Electron. J. Differential Equations, 246 (2015), 1-11. [27] J. Itikawa, J. Llibre and D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, to appear in Revista Matemática Iberoamericana. [28] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Comput. Neurosci. , MIT Press, Cambridge, MA, 2007. [29] J. Llibre, A.C. Mereu and D.D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022. [30] J. Llibre and A.C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, J. Math. Anal. Appl., 413 (2014), 763-775.  doi: 10.1016/j.jmaa.2013.12.031. [31] N. G. Lloyd, Degree Theory, Cambridge Tracts in Math. 73 Cambridge, 1978. [32] W.S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964), 21-36. [33] 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. [34] P. Mardesic, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108.  doi: 10.1006/jdeq.1995.1122. [35] L. Peng and Z. Feng, Bifurcation of limit cycles from quartic isochronous systems, Elec. J. Differential Equations, 95 (2014), 1-14. [36] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, in World Scientific Series on Nonlinear Science A, 69, World Scientific, Singapore, 2010. [37] M. A. Teixeira, Perturbation Theory for Non-smooth Systems, in Meyers: Encyclopedia of Complexity and Systems Science, 1 (Perturbation Theory), 1325-1336, Springer, New York, 2012. [38] E.P. Volokitin and V.M. Cheresiz, Singular points and first integrals of holomorphic dynamical systems, J. Math. Sciences, 203 (2014), 605-620.  doi: 10.1007/s10958-014-2162-y.
Number of limit cycles for continuous and discontinuous quadratic and cubic differential systems
 Case Number of limit cycles for system (1) system (2) Continuous 2 3 Discontinuous with 2 zones 5 7 Discontinuous with 4 zones 10 12
 Case Number of limit cycles for system (1) system (2) Continuous 2 3 Discontinuous with 2 zones 5 7 Discontinuous with 4 zones 10 12
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