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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 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).

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