Advanced Search
Article Contents
Article Contents

Limit cycles of non-autonomous scalar ODEs with two summands

Abstract Related Papers Cited by
  • We establish upper bounds for the number of limit cycles (isolated periodic solutions in the set of periodic solutions) of the two families of scalar ordinary differential equations $x'=(a(t) x +b(t)) f(x)$ and $x'=a(t) g(x) +b(t)f(x)$, where $f(x)$ and $g(x)$ are analytic funtions and $a(t)$, $b(t)$ are $T$--periodic continuous functions for which there exist $\alpha, \beta \in R$ such that $\alpha a(t)+\beta b(t)$ is not identically zero and does not change sign in $[0,T]$. As a consequence we obtain that generalized Abel equations $x'=a(t)x^n + b(t)x^m$, where $n> m \geq 1$ are natural numbers, have at most three limit cycles.
    Mathematics Subject Classification: Primary: 34C25; Secondary: 34C07.


    \begin{equation} \\ \end{equation}
  • [1]

    N. Alkoumi and P. J. Torres, On the number of limit cycles of a generalized Abel equation, Czech. Math. J., 61 (2011), 73-83.doi: 10.1007/s10587-011-0018-x.


    N. Alkoumi and P. J. Torres, Estimates on the number of limit cycles of a generalized Abel equation, Discrete Contin. Dyn. Syst., 31 (2011), 25-34.doi: 10.3934/dcds.2011.31.25.


    A. Álvarez, J. L. Bravo and 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.doi: 10.3934/cpaa.2009.8.1493.


    A. Álvarez, J. L. Bravo and M. Fernández, Abel-like differential equations with a unique limit cycle, Nonlinear Anal. T.M.A., 74 (2011), 3694-3702.doi: 10.1016/j.na.2011.02.049.


    A. Álvarez, J. L. Bravo and M. Fernández, Uniqueness of limit cycles for polynomial first-order differential equations, J. Math. Anal. Appl., 360 (2009), 168-189.doi: 10.1016/j.jmaa.2009.06.031.


    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.


    M. A. M. Alwash, Periodic solutions of Abel differential equations, J. Math. Anal. Appl., 329 (2007), 1161-1169.doi: 10.1016/j.jmaa.2006.07.039.


    M. A. M. Alwash and N. G. Lloyd, Nonautonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc.Edinburgh, 105A (1987), 129-152.doi: 10.1017/S0308210500021971.


    D. M. Benardete, V. W. Noonburg and B. Pollina, Qualitative tools for studying periodic solutions and bifurcations as applied to the periodically harvested logistic equation, Amer. Math. Monthly, 115 (2008) 202-219.


    J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Int. J. Bif. Chaos, 19 (2009), 3869-3876.doi: 10.1142/S0218127409025195.


    J. L. Bravo and J. Torregrosa, Abel-like equations with no periodic solutions, J. Math. Anal. Appl., 342 (2008), 931-942.doi: 10.1016/j.jmaa.2007.12.060.


    M. Chamberland and A. Gasull, Chini equations and isochronous centers in three-dimensional differential systems. Qual. Theory Dyn. Syst., 9 (2010), 29-38.doi: 10.1007/s12346-010-0019-4.


    J. Devlin, N. G. Lloyd and J. M. Pearson, Cubic systems and Abel equations, J. Differential Equations, 147 (1998), 435-454.doi: 10.1006/jdeq.1998.3420.


    A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745.doi: 10.1142/S0218127406017130.


    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.


    A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404.doi: 10.1016/j.jmaa.2004.07.030.


    A. Gasull and J. Torregrosa, Some results on rigid systems, In International Conference on Differential Equations (Equadiff-2003), World Sci. Publ., Hackensack, NJ. (2005), 340-345.


    Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc., 39 (2002), 301-354.doi: 10.1090/S0273-0979-02-00946-1.


    A. Lins Neto, On the number of solutions of the equation $\frac{d x}{d t}=\sum_{j=0} ^n a_j(t)x^j$, $0 \leq t \leq 1$, for which $x(0)=x(1)$}, Inv. Math., 59 (1980), 67-76.


    N. G. Lloyd, The number of periodic solutions of the equation $\dot z = z^N+ p_1(t) z^{N-1} +\cdots +p_N(t)$, Proc. London Math. Soc., 27 (1973), 667-700.doi: 10.1112/plms/s3-27.4.667.


    N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc., 20 (1979), 277-286.doi: 10.1112/jlms/s2-20.2.277.


    J. M. Olm, X. Ros-Oton and T. M. Seara, Periodic solutions with non-constant sign in Abel equations of the second kind, J. Math. Anal. Appl., 381 (2011), 582-589.doi: 10.1016/j.jmaa.2011.02.084.


    V. A. Pliss, "Non-Local Problems of the Theory of Oscillations," Academic Press, New York, 1966.


    Wolfram Research, Inc., "Mathematica, Version 8.0," Champaign, IL (2010).

  • 加载中

Article Metrics

HTML views() PDF downloads(130) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint