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Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane
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 |
References:
[1] |
A. Andronov, A. Vitt and S. Khaĭkin, "Theory of Oscillations," Pergamon Press, Oxford, 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), 19 pp. |
[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, Springer-Verlag London, Ltd., London, 2008. |
[4] | |
[5] |
S. Coombes, Neuronal networks with gap junctions: A study of piecewise linear planar neuron models, SIAM Applied Mathematics, 7 (2008), 1101-1129.
doi: 10.1137/070707579. |
[6] |
W. A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations, 6 (1993), 1357-1365. |
[7] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems," Universitext, Springer-Verlag, Berlin, 2006. |
[8] |
A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers, Dordrecht, 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), 87-96.
doi: 10.1017/S0143385700008725. |
[10] |
J.-P. Françoise, The first derivative of the period function of a plane vector field, Publ. Matemat., 41 (1997), 127-134.
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), 320-335.
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), 2073-2097.
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), 181-211.
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), 1277-1303.
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), 1755-1765.
doi: 10.1142/S0218127403007618. |
[16] |
H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516.
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), 1611-1632.
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), 2399-2416.
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), 2147-2164.
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), 317-322.
doi: 10.1017/S0305004199003795. |
[21] |
I. D. Iliev and L. M. Perko, Higher order bifurcations of limit cycles, J. Differential Equations, 154 (1999), 339-363.
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), 259-273.
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), 325-335. |
[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, University of California at Berkeley, 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), 130-148.
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), 129-138.
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), 6803-6813.
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), 370-387.
doi: 10.1006/jdeq.2002.4175. |
show all references
References:
[1] |
A. Andronov, A. Vitt and S. Khaĭkin, "Theory of Oscillations," Pergamon Press, Oxford, 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), 19 pp. |
[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, Springer-Verlag London, Ltd., London, 2008. |
[4] | |
[5] |
S. Coombes, Neuronal networks with gap junctions: A study of piecewise linear planar neuron models, SIAM Applied Mathematics, 7 (2008), 1101-1129.
doi: 10.1137/070707579. |
[6] |
W. A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations, 6 (1993), 1357-1365. |
[7] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems," Universitext, Springer-Verlag, Berlin, 2006. |
[8] |
A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers, Dordrecht, 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), 87-96.
doi: 10.1017/S0143385700008725. |
[10] |
J.-P. Françoise, The first derivative of the period function of a plane vector field, Publ. Matemat., 41 (1997), 127-134.
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), 320-335.
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), 2073-2097.
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), 181-211.
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), 1277-1303.
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), 1755-1765.
doi: 10.1142/S0218127403007618. |
[16] |
H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516.
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), 1611-1632.
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), 2399-2416.
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), 2147-2164.
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), 317-322.
doi: 10.1017/S0305004199003795. |
[21] |
I. D. Iliev and L. M. Perko, Higher order bifurcations of limit cycles, J. Differential Equations, 154 (1999), 339-363.
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), 259-273.
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), 325-335. |
[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, University of California at Berkeley, 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), 130-148.
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), 129-138.
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), 6803-6813.
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), 370-387.
doi: 10.1006/jdeq.2002.4175. |
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