Limit cycles of non-autonomous scalar ODEs with two summands

Previous Article
Blow-up for semilinear parabolic equations with critical Sobolev exponent
CPAA Home
This Issue
Next Article
Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution

March 2013, 12(2): 1091-1102. doi: 10.3934/cpaa.2013.12.1091

Limit cycles of non-autonomous scalar ODEs with two summands

José-Luis Bravo ^1, and Manuel Fernández ^2,

1.	Departamento de Matemáticas, Universidad de Extremadura, Badajoz, 06071
2.	Departamento de Matemáticas, Universidad de Extremadura, Facultad de Ciencias, 06071 Badajoz

Received August 2011 Revised January 2012 Published September 2012

Abstract
Related Papers

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.

Keywords: Periodic solutions, Abel equation., limit cycles.

Mathematics Subject Classification: Primary: 34C25; Secondary: 34C0.

Citation: José-Luis Bravo, Manuel Fernández. Limit cycles of non-autonomous scalar ODEs with two summands. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1091-1102. doi: 10.3934/cpaa.2013.12.1091

References:

[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.
[2]	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.
[3]	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.
[4]	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.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.
[14]	A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745. doi: 10.1142/S0218127406017130.
[15]	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.
[16]	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.
[17]	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.
[18]	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.
[19]	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.
[20]	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.
[21]	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.
[22]	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.
[23]	V. A. Pliss, "Non-Local Problems of the Theory of Oscillations," Academic Press, New York, 1966.
[24]	Wolfram Research, Inc., "Mathematica, Version 8.0," Champaign, IL (2010).

show all references

References:

[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.
[2]	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.
[3]	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.
[4]	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.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.
[14]	A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745. doi: 10.1142/S0218127406017130.
[15]	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.
[16]	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.
[17]	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.
[18]	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.
[19]	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.
[20]	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.
[21]	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.
[22]	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.
[23]	V. A. Pliss, "Non-Local Problems of the Theory of Oscillations," Academic Press, New York, 1966.
[24]	Wolfram Research, Inc., "Mathematica, Version 8.0," Champaign, IL (2010).

[1]	Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[2]	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 and Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493
[3]	Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007
[4]	José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873
[5]	Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[6]	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
[7]	Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284
[8]	Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[9]	Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
[10]	Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[11]	Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[12]	Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869
[13]	Yunming Zhou, Desheng Shang, Tonghua Zhang. Seventeen limit cycles bifurcations of a fifth system. Conference Publications, 2007, 2007 (Special) : 1070-1081. doi: 10.3934/proc.2007.2007.1070
[14]	Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091
[15]	José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020
[16]	Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070
[17]	Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67
[18]	Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071
[19]	Maoan Han. On some properties and limit cycles of Lienard systems. Conference Publications, 2001, 2001 (Special) : 426-434. doi: 10.3934/proc.2001.2001.426
[20]	Maoan Han, Yuhai Wu, Ping Bi. A new cubic system having eleven limit cycles. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 675-686. doi: 10.3934/dcds.2005.12.675

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
José-Luis Bravo Manuel Fernández
on Google Scholar
José-Luis Bravo Manuel Fernández

[Back to Top]