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Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations
1. | Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Av. Esteve Terradas 5, 08860 Castelldefels, Spain |
2. | Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain |
References:
[1] |
P. J. Torres, Existence of closed solutions for a polynomial first order differential equation, J. Math. Anal. Applic., 328 (2007), 1108-1116.
doi: 10.1016/j.jmaa.2006.05.078. |
[2] |
M. A. M. Alwash, Periodic solutions of Abel differential equations, J. Math. Anal. Applic., 329 (2007), 1161-1169.
doi: 10.1016/j.jmaa.2006.07.039. |
[3] |
A. D. Polyanin and V. F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations,'' $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, 2003. |
[4] |
S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.
doi: 10.1007/BF03025291. |
[5] |
A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244.
doi: 10.1137/0521068. |
[6] |
Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity, 13 (2000), 1337-1342.
doi: 10.1088/0951-7715/13/4/319. |
[7] |
M. J. Álvarez, A. Gasull and H. Giacomini, A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007), 161-176.
doi: 10.1016/j.jde.2006.11.004. |
[8] |
J. L. Bravo and J. Torregrosa, Abel-like differential equations with no periodic solutions, J. Math. Anal. Applic, 342 (2008), 931-942.
doi: 10.1016/j.jmaa.2007.12.060. |
[9] |
J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3869-3876.
doi: 10.1142/S0218127409025195. |
[10] |
M. A. M. Alwash, Polynomial differential equations with small coefficients, Discrete Continuous Dynam. Systems - A, 25 (2009), 1129-1141.
doi: 10.3934/dcds.2009.25.1129. |
[11] |
N. H. M. Alkoumi and P. J. Torres, Estimates on the number of limit cycles of a generalized Abel equation, Discrete Continuous Dynam. Systems - A, 31 (2011), 25-34.
doi: 10.3934/dcds.2011.31.25. |
[12] |
J. M. Olm, X. Ros-Oton and T. M. Seara, Periodic solutions with nonconstant sign in Abel equations of the second kind, J. Math. Anal. Appl., 381 (2011), 582-589.
doi: 10.1016/j.jmaa.2011.02.084. |
[13] |
E. Fossas and J. M. Olm, Galerkin method and approximate tracking in a non-minimum phase bilinear system, Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76. |
[14] |
J. M. Olm and X. Ros-Oton, Approximate tracking of periodic references in a class of bilinear systems via stable inversion, Discrete Continuous Dynam. Systems - B, 15 (2011), 197-215.
doi: 10.3934/dcdsb.2011.15.197. |
[15] |
A. Gasull and H. Giacomini, A new criterion for controlling the number of limit cycles of some Ggeneralized Liénard equations, J. Differential Equations, 185 (2002), 54-73.
doi: 10.1006/jdeq.2002.4172. |
[16] |
A. Kelley, The stable, center-stable, center, center-unstable and unstable manifolds, J. Differential Equations, 3 (1967), 546-570. |
[17] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'' $2^{nd}$ edition, Springer-Verlag, New York, 1985. |
show all references
References:
[1] |
P. J. Torres, Existence of closed solutions for a polynomial first order differential equation, J. Math. Anal. Applic., 328 (2007), 1108-1116.
doi: 10.1016/j.jmaa.2006.05.078. |
[2] |
M. A. M. Alwash, Periodic solutions of Abel differential equations, J. Math. Anal. Applic., 329 (2007), 1161-1169.
doi: 10.1016/j.jmaa.2006.07.039. |
[3] |
A. D. Polyanin and V. F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations,'' $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, 2003. |
[4] |
S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.
doi: 10.1007/BF03025291. |
[5] |
A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244.
doi: 10.1137/0521068. |
[6] |
Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity, 13 (2000), 1337-1342.
doi: 10.1088/0951-7715/13/4/319. |
[7] |
M. J. Álvarez, A. Gasull and H. Giacomini, A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007), 161-176.
doi: 10.1016/j.jde.2006.11.004. |
[8] |
J. L. Bravo and J. Torregrosa, Abel-like differential equations with no periodic solutions, J. Math. Anal. Applic, 342 (2008), 931-942.
doi: 10.1016/j.jmaa.2007.12.060. |
[9] |
J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3869-3876.
doi: 10.1142/S0218127409025195. |
[10] |
M. A. M. Alwash, Polynomial differential equations with small coefficients, Discrete Continuous Dynam. Systems - A, 25 (2009), 1129-1141.
doi: 10.3934/dcds.2009.25.1129. |
[11] |
N. H. M. Alkoumi and P. J. Torres, Estimates on the number of limit cycles of a generalized Abel equation, Discrete Continuous Dynam. Systems - A, 31 (2011), 25-34.
doi: 10.3934/dcds.2011.31.25. |
[12] |
J. M. Olm, X. Ros-Oton and T. M. Seara, Periodic solutions with nonconstant sign in Abel equations of the second kind, J. Math. Anal. Appl., 381 (2011), 582-589.
doi: 10.1016/j.jmaa.2011.02.084. |
[13] |
E. Fossas and J. M. Olm, Galerkin method and approximate tracking in a non-minimum phase bilinear system, Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76. |
[14] |
J. M. Olm and X. Ros-Oton, Approximate tracking of periodic references in a class of bilinear systems via stable inversion, Discrete Continuous Dynam. Systems - B, 15 (2011), 197-215.
doi: 10.3934/dcdsb.2011.15.197. |
[15] |
A. Gasull and H. Giacomini, A new criterion for controlling the number of limit cycles of some Ggeneralized Liénard equations, J. Differential Equations, 185 (2002), 54-73.
doi: 10.1006/jdeq.2002.4172. |
[16] |
A. Kelley, The stable, center-stable, center, center-unstable and unstable manifolds, J. Differential Equations, 3 (1967), 546-570. |
[17] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'' $2^{nd}$ edition, Springer-Verlag, New York, 1985. |
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