# American Institute of Mathematical Sciences

2013, 33(9): 3915-3936. doi: 10.3934/dcds.2013.33.3915

## Piecewise linear perturbations of a linear center

 1 Departamento de Matemática, Universidade Estadual Paulista, 15054-000, São José do Rio Preto, Brazil, Brazil 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received  March 2012 Revised  February 2013 Published  March 2013

This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line $\Sigma$ and the singular point of the unperturbed system is in $\Sigma$. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.
Citation: Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915
##### References:
 [1] A. Andronov, A. Vitt and S. Khaĭkin, "Theory of Oscillations,", Pergamon Press, (1966). [2] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type,, Amer. Math. Soc. Translations, 1954 (1954). [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems. Theory and Applications,", Appl. Math. Sci. Series, 163 (2008). [4] , C. Chicone,, Review in MathSciNet, (). [5] S. Coombes, Neuronal networks with gap junctions: A study of piecewise linear planar neuron models,, SIAM Applied Mathematics, 7 (2008), 1101. doi: 10.1137/070707579. [6] W. A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system,, Differential Integral Equations, 6 (1993), 1357. [7] F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems,", Universitext, (2006). [8] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Mathematics and its Applications (Soviet Series), 18 (1988). [9] J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields,, Ergod. Th. Dyn. Syst., 16 (1996), 87. doi: 10.1017/S0143385700008725. [10] J.-P. Françoise, The first derivative of the period function of a plane vector field,, Publ. Matemat., 41 (1997), 127. doi: 10.5565/PUBLMAT_41197_07. [11] J.-P. Françoise, The successive derivatives of the period function of a plane vector field,, J. Diff. Eqs., 146 (1998), 320. doi: 10.1006/jdeq.1998.3437. [12] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073. doi: 10.1142/S0218127498001728. [13] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems,, SIAM J. Applied Dynamical Systems, 11 (2012), 181. doi: 10.1137/11083928X. [14] A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles,, Rocky Mountain Journal of Mathematics, 31 (2001), 1277. doi: 10.1216/rmjm/1021249441. [15] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1755. doi: 10.1142/S0218127403007618. [16] H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles,, Nonlinearity, 9 (1996), 501. doi: 10.1088/0951-7715/9/2/013. [17] F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity,, Nonlinearity, 14 (2001), 1611. doi: 10.1088/0951-7715/14/6/311. [18] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems,, J. of Differential Equations, 248 (2010), 2399. doi: 10.1016/j.jde.2009.10.002. [19] S.-M. Huan and X.-S. Yang, The number of limit cycles in general planar piecewise linear systems,, Discrete and Continuous Dynamical Systems, 32 (2012), 2147. doi: 10.3934/dcds.2012.32.2147. [20] I. D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator,, Math. Proc. Camb. Phil. Soc., 127 (1999), 317. doi: 10.1017/S0305004199003795. [21] I. D. Iliev and L. M. Perko, Higher order bifurcations of limit cycles,, J. Differential Equations, 154 (1999), 339. doi: 10.1006/jdeq.1998.3549. [22] R. I. Leine and D. H. van Campen, Discontinuous bifurcations of periodic solutions,, Mathematical and Computing Modelling, 36 (2002), 259. doi: 10.1016/S0895-7177(02)00124-3. [23] 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 (2011), 325. [24] J. Llibre, M. A. Teixeira and J. Torregrosa, On the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation,, to appear in Internat. J. Bifur. Chaos Appl. Sci. Engrg., (). [25] R. Lum and L. O. Chua, "Global Properties of Continuous Piecewise-Linear Vector Fields. Part I. Simplest Case in $R^2$,", Memorandum UCB/ERL M90/22, (1990). [26] R. Prohens and J. Torregrosa, Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus,, Nonlinear Anal., 81 (2013), 130. doi: 10.1016/j.na.2012.10.017. [27] R. Prohens and J. Torregrosa, Periodic orbits from second order perturbation via rational trigonometric integrals,, preprint, (2013). [28] F. Rothe, The periods of the Volterra-Lokta system,, J. Reine Angew. Math., 355 (1985), 129. doi: 10.1515/crll.1985.355.129. [29] J. Villadelprat, Bifurcation of local critical periods in the generalized Loud's system,, Appl. Math. Comput., 218 (2012), 6803. doi: 10.1016/j.amc.2011.12.048. [30] Y. Zhao, The monotonicity of period function for codimension four quadratic system $Q_4$,, J. Differential Equations, 185 (2002), 370. doi: 10.1006/jdeq.2002.4175.

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##### References:
 [1] A. Andronov, A. Vitt and S. Khaĭkin, "Theory of Oscillations,", Pergamon Press, (1966). [2] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type,, Amer. Math. Soc. Translations, 1954 (1954). [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems. Theory and Applications,", Appl. Math. Sci. Series, 163 (2008). [4] , C. Chicone,, Review in MathSciNet, (). [5] S. Coombes, Neuronal networks with gap junctions: A study of piecewise linear planar neuron models,, SIAM Applied Mathematics, 7 (2008), 1101. doi: 10.1137/070707579. [6] W. A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system,, Differential Integral Equations, 6 (1993), 1357. [7] F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems,", Universitext, (2006). [8] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Mathematics and its Applications (Soviet Series), 18 (1988). [9] J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields,, Ergod. Th. Dyn. Syst., 16 (1996), 87. doi: 10.1017/S0143385700008725. [10] J.-P. Françoise, The first derivative of the period function of a plane vector field,, Publ. Matemat., 41 (1997), 127. doi: 10.5565/PUBLMAT_41197_07. [11] J.-P. Françoise, The successive derivatives of the period function of a plane vector field,, J. Diff. Eqs., 146 (1998), 320. doi: 10.1006/jdeq.1998.3437. [12] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073. doi: 10.1142/S0218127498001728. [13] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems,, SIAM J. Applied Dynamical Systems, 11 (2012), 181. doi: 10.1137/11083928X. [14] A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles,, Rocky Mountain Journal of Mathematics, 31 (2001), 1277. doi: 10.1216/rmjm/1021249441. [15] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1755. doi: 10.1142/S0218127403007618. [16] H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles,, Nonlinearity, 9 (1996), 501. doi: 10.1088/0951-7715/9/2/013. [17] F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity,, Nonlinearity, 14 (2001), 1611. doi: 10.1088/0951-7715/14/6/311. [18] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems,, J. of Differential Equations, 248 (2010), 2399. doi: 10.1016/j.jde.2009.10.002. [19] S.-M. Huan and X.-S. Yang, The number of limit cycles in general planar piecewise linear systems,, Discrete and Continuous Dynamical Systems, 32 (2012), 2147. doi: 10.3934/dcds.2012.32.2147. [20] I. D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator,, Math. Proc. Camb. Phil. Soc., 127 (1999), 317. doi: 10.1017/S0305004199003795. [21] I. D. Iliev and L. M. Perko, Higher order bifurcations of limit cycles,, J. Differential Equations, 154 (1999), 339. doi: 10.1006/jdeq.1998.3549. [22] R. I. Leine and D. H. van Campen, Discontinuous bifurcations of periodic solutions,, Mathematical and Computing Modelling, 36 (2002), 259. doi: 10.1016/S0895-7177(02)00124-3. [23] 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 (2011), 325. [24] J. Llibre, M. A. Teixeira and J. Torregrosa, On the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation,, to appear in Internat. J. Bifur. Chaos Appl. Sci. Engrg., (). [25] R. Lum and L. O. Chua, "Global Properties of Continuous Piecewise-Linear Vector Fields. Part I. Simplest Case in $R^2$,", Memorandum UCB/ERL M90/22, (1990). [26] R. Prohens and J. Torregrosa, Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus,, Nonlinear Anal., 81 (2013), 130. doi: 10.1016/j.na.2012.10.017. [27] R. Prohens and J. Torregrosa, Periodic orbits from second order perturbation via rational trigonometric integrals,, preprint, (2013). [28] F. Rothe, The periods of the Volterra-Lokta system,, J. Reine Angew. Math., 355 (1985), 129. doi: 10.1515/crll.1985.355.129. [29] J. Villadelprat, Bifurcation of local critical periods in the generalized Loud's system,, Appl. Math. Comput., 218 (2012), 6803. doi: 10.1016/j.amc.2011.12.048. [30] Y. Zhao, The monotonicity of period function for codimension four quadratic system $Q_4$,, J. Differential Equations, 185 (2002), 370. doi: 10.1006/jdeq.2002.4175.
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