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April  2013, 33(4): 1603-1614. doi: 10.3934/dcds.2013.33.1603

Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations

Josep M. Olm 1, and  Xavier Ros-Oton 2,

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

Received  October 2011 Revised  May 2012 Published  October 2012

This article provides sufficient conditions for the existence of periodic solutions with nonconstant sign in a family of polynomial, non-auto-nomous, first-order differential equations that arise as a generalization of the Abel equation of the second kind.
Keywords: Hilbert's 16th problem., periodic solutions, polynomial differential equations, limit cycles, Abel differential equations.
Mathematics Subject Classification: Primary: 34C07; Secondary: 34C2.
Citation: Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603
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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