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

Previous Article
Boundary stabilization of the waves in partially rectangular domains
DCDS Home
This Issue
Next Article
Fractal bodies invisible in 2 and 3 directions

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

Abstract
Related Papers

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 - A, 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. 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. 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, (2003). Google Scholar
[4]	S. Smale, Mathematical problems for the next century,, Math. Intelligencer, 20 (1998), 7. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[17]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'', $2^{nd}$ edition, (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. 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. 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, (2003). Google Scholar
[4]	S. Smale, Mathematical problems for the next century,, Math. Intelligencer, 20 (1998), 7. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[17]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,'', $2^{nd}$ edition, (1985). Google Scholar

[1]	Amelia Álvarez, José-Luis Bravo, Manuel Fernández. The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493
[2]	José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873
[3]	Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
[4]	Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070
[5]	Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[6]	Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217
[7]	M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129
[8]	Mahmoud M. El-Borai. On some fractional differential equations in the Hilbert space. Conference Publications, 2005, 2005 (Special) : 233-240. doi: 10.3934/proc.2005.2005.233
[9]	Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[10]	Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[11]	Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[12]	Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487
[13]	Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301
[14]	Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[15]	R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435
[16]	Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019113
[17]	Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[18]	Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[19]	Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105
[20]	Feliz Minhós. Periodic solutions for some fully nonlinear fourth order differential equations. Conference Publications, 2011, 2011 (Special) : 1068-1077. doi: 10.3934/proc.2011.2011.1068

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences