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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

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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

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