September  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). Google Scholar

[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). Google Scholar

[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). Google Scholar

[4]

, C. Chicone,, Review in MathSciNet, (). Google Scholar

[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. Google Scholar

[6]

W. A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system,, Differential Integral Equations, 6 (1993), 1357. Google Scholar

[7]

F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems,", Universitext, (2006). Google Scholar

[8]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Mathematics and its Applications (Soviet Series), 18 (1988). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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., (). Google Scholar

[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). Google Scholar

[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. Google Scholar

[27]

R. Prohens and J. Torregrosa, Periodic orbits from second order perturbation via rational trigonometric integrals,, preprint, (2013). Google Scholar

[28]

F. Rothe, The periods of the Volterra-Lokta system,, J. Reine Angew. Math., 355 (1985), 129. doi: 10.1515/crll.1985.355.129. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

A. Andronov, A. Vitt and S. Khaĭkin, "Theory of Oscillations,", Pergamon Press, (1966). Google Scholar

[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). Google Scholar

[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). Google Scholar

[4]

, C. Chicone,, Review in MathSciNet, (). Google Scholar

[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. Google Scholar

[6]

W. A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system,, Differential Integral Equations, 6 (1993), 1357. Google Scholar

[7]

F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems,", Universitext, (2006). Google Scholar

[8]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Mathematics and its Applications (Soviet Series), 18 (1988). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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., (). Google Scholar

[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). Google Scholar

[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. Google Scholar

[27]

R. Prohens and J. Torregrosa, Periodic orbits from second order perturbation via rational trigonometric integrals,, preprint, (2013). Google Scholar

[28]

F. Rothe, The periods of the Volterra-Lokta system,, J. Reine Angew. Math., 355 (1985), 129. doi: 10.1515/crll.1985.355.129. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

[2]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

[3]

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

[4]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[5]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111

[6]

Rinaldo M. Colombo, Graziano Guerra. A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case. Networks & Heterogeneous Media, 2016, 11 (2) : 313-330. doi: 10.3934/nhm.2016.11.313

[7]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136

[8]

Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839

[9]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457

[10]

Josef Diblík, Radoslav Chupáč, Miroslava Růžičková. Existence of unbounded solutions of a linear homogenous system of differential equations with two delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2447-2459. doi: 10.3934/dcdsb.2014.19.2447

[11]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[12]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893

[13]

Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial & Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761

[14]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[15]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078

[16]

Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082

[17]

Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control & Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289

[18]

Yanni Xiao, Tingting Zhao, Sanyi Tang. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Mathematical Biosciences & Engineering, 2013, 10 (2) : 445-461. doi: 10.3934/mbe.2013.10.445

[19]

Nicola Gigli, Sunra Mosconi. The Abresch-Gromoll inequality in a non-smooth setting. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1481-1509. doi: 10.3934/dcds.2014.34.1481

[20]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]