Abstract
New results are proved on the maximum number of isolated T-periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Álvarez, J. -L. Bravo, M. Fernández: The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Commun. Pure Appl. Anal. 8 (2009), 1493–1501.
M. J. Álvarez, A. Gasull, H. Giacomini: A new uniqueness criterion for the number of periodic orbits of Abel equations. J. Differ. Equations 234 (2007), 161–176.
M. J. Álvarez, A. Gasull, R. Prohens: On the number of limit cycles of some systems on the cylinder. Bull. Sci. Math. 131 (2007), 620–637.
M. A. M. Alwash: Periodic solutions of polynomial non-autonomous differential equations. Electron. J. Differ. Equ. 2005 (2005), 1–8.
M. A. M. Alwash: Periodic solutions of Abel differential equations. J. Math. Anal. Appl. 329 (2007), 1161–1169.
L A. Cherkas: Number of limit cycles of an autonomous second-order system. Differ. Equations 12 (1976), 666–668.
A. Gasull, A. Guillamon: Limit cycles for generalized Abel equations. Int. J. Bifurcation Chaos Appl. Sci. Eng. 16 (2006), 3737–3745.
A. Gasull, J. Llibre: Limit cycles for a class of Abel equations. SIAM J. Math. Anal. 21 (1990), 1235–1244.
A. Gasull, J. Torregrosa: Exact number of limit cycles for a family of rigid systems. Proc. Am. Math. Soc. 133 (2005), 751–758.
P. Korman, T. Ouyang: Exact multiplicity results for two classes of periodic equations. J. Math. Anal. Appl. 194 (1995), 763–379.
A. Lins-Neto: On the number of solutions of the equation Σ nj=0 aj(t)xj, 0 ⩽ t ⩽ 1, for which x(0) = x(1). Invent. Math. 59 (1980), 69–76.
M. N. Nkashama: A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations. J. Math. Anal. Appl. 140 (1989), 381–395.
V. A. Pliss: Nonlocal Problems of the Theory of Oscillations. Academic Press, New York, 1966.
A. Sandqvist, K. M. Andersen: On the number of closed solutions to an equation \(\dot x = f(t,x)\), where \({f_{{x^n}}}(t,x) \ge 0\) (n = 1, 2, or 3). J. Math. Anal. Appl. 159 (1991), 127–146.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by projects MTM2008-02502, Ministerio de Educación y Ciencia, Spain, and FQM2216, Junta de Andalucía.
Rights and permissions
About this article
Cite this article
Alkoumi, N., Torres, P.J. On the number of limit cycles of a generalized Abel equation. Czech Math J 61, 73–83 (2011). https://doi.org/10.1007/s10587-011-0018-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-011-0018-x