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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Kelley, The stable, center-stable, center, center-unstable and unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  Google Scholar

[17]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'' $2^{nd}$ edition, Springer-Verlag, New York, 1985.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Kelley, The stable, center-stable, center, center-unstable and unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  Google Scholar

[17]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'' $2^{nd}$ edition, Springer-Verlag, New York, 1985.  Google Scholar

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