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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.
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.
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, (2003).
|
[4] |
S. Smale, Mathematical problems for the next century,, Math. Intelligencer, 20 (1998), 7.
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.
doi: 10.1137/0521068. |
[6] |
Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions,, Nonlinearity, 13 (2000), 1337.
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.
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.
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.
doi: 10.1142/S0218127409025195. |
[10] |
M. A. M. Alwash, Polynomial differential equations with small coefficients,, Discrete Continuous Dynam. Systems - A, 25 (2009), 1129.
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.
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.
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.
|
[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.
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.
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.
|
[17] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'', $2^{nd}$ edition, (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.
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.
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, (2003).
|
[4] |
S. Smale, Mathematical problems for the next century,, Math. Intelligencer, 20 (1998), 7.
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.
doi: 10.1137/0521068. |
[6] |
Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions,, Nonlinearity, 13 (2000), 1337.
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.
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.
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.
doi: 10.1142/S0218127409025195. |
[10] |
M. A. M. Alwash, Polynomial differential equations with small coefficients,, Discrete Continuous Dynam. Systems - A, 25 (2009), 1129.
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.
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.
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.
|
[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.
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.
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.
|
[17] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'', $2^{nd}$ edition, (1985).
|
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