• Previous Article
    Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution
  • CPAA Home
  • This Issue
  • Next Article
    Blow-up for semilinear parabolic equations with critical Sobolev exponent
March  2013, 12(2): 1091-1102. doi: 10.3934/cpaa.2013.12.1091

Limit cycles of non-autonomous scalar ODEs with two summands

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

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.
Citation: José-Luis Bravo, Manuel Fernández. Limit cycles of non-autonomous scalar ODEs with two summands. Communications on Pure & 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. 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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), (2005), 340.

[18]

Yu. Ilyashenko, Centennial history of Hilbert's 16th problem,, Bull. Amer. Math. Soc., 39 (2002), 301. 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.

[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. 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. 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. doi: 10.1016/j.jmaa.2011.02.084.

[23]

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

[24]

Wolfram Research, Inc., "Mathematica, Version 8.0,", Champaign, (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. 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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), (2005), 340.

[18]

Yu. Ilyashenko, Centennial history of Hilbert's 16th problem,, Bull. Amer. Math. Soc., 39 (2002), 301. 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.

[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. 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. 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. doi: 10.1016/j.jmaa.2011.02.084.

[23]

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

[24]

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

[1]

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

[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 & Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493

[3]

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

[4]

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

[5]

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

[6]

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 & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[7]

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

[8]

Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014

[9]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[10]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[11]

T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277

[12]

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

[13]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091

[14]

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

[15]

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

[16]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071

[17]

Maoan Han, Yuhai Wu, Ping Bi. A new cubic system having eleven limit cycles. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 675-686. doi: 10.3934/dcds.2005.12.675

[18]

Nikolay Dimitrov. An example of rapid evolution of complex limit cycles. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 709-735. doi: 10.3934/dcds.2011.31.709

[19]

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

[20]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]