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March  2011, 31(1): 25-34. doi: 10.3934/dcds.2011.31.25

Estimates on the number of limit cycles of a generalized Abel equation

1. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain, Spain

Received  May 2010 Revised  March 2011 Published  June 2011

We prove new results about the number of isolated periodic solutions of a first order differential equation with a polynomial nonlinearity. Such results are applied to bound the number of limit cycles of a family of planar polynomial vector fields which generalize the so-called rigid systems.
Citation: 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
References:
[1]

A. Alvarez, J.-L. Bravo and M. Fernández, The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign,, Communications on Pure and Applied Analysis, 8 (2009), 1493. doi: 10.3934/cpaa.2009.8.1493. Google Scholar

[2]

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. Google Scholar

[3]

M. A. M. Alwash, Polynomial differential equations with small coefficients,, Discrete and Continuos Dynamical Systems, 25 (2009), 1129. doi: 10.3934/dcds.2009.25.1129. Google Scholar

[4]

M. A. M. Alwash, Periodic solutions of polynomial non-autonomous differential equations,, Electronic Journal of Differential Equations, 2005 (): 1. Google Scholar

[5]

M. Calanchi and B. Ruf, On the number of closed solutions for polynomial ODE's and a special case of Hilbert's 16th problem,, Advances in Differential Equations, 7 (2002), 197. Google Scholar

[6]

A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 3737. doi: 10.1142/S0218127406017130. Google Scholar

[7]

A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems,, Proc. Amer. Math. Soc., 133 (2005), 751. doi: 10.1090/S0002-9939-04-07542-2. Google Scholar

[8]

A. Guillamon and M. Sabatini, The number of limit cycles in planar systems and generalized Abel equations with monotonous hyperbolicity,, Nonlinear Analysis, 71 (2009), 1941. doi: 10.1016/j.na.2009.01.034. Google Scholar

[9]

Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions,, Nonlinearity, 13 (2000), 1337. doi: 10.1088/0951-7715/13/4/319. Google Scholar

[10]

P. Korman and T. Ouyang, Exact multiplicity results for two classes of periodic equations,, J. Math. Anal. Appl., 194 (1995), 763. doi: 10.1006/jmaa.1995.1328. Google Scholar

[11]

A. M. Liapunov, "Stability of Motion,", Mathematics in Science and Engineering, 30 (1966). Google Scholar

[12]

A. Lins Neto, On the number of solutions of the equation $\frac{dx}{dt}=\sum_{j=0}^na_j(t)x^j,0\leq t\leq 1,$ for which $x(0)=x(1)$,, Inv. Math., 59 (1980), 67. doi: 10.1007/BF01390315. Google Scholar

[13]

M. N. Nkashama, A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations,, J. Math. Anal. Appl., 140 (1989), 381. doi: 10.1016/0022-247X(89)90072-3. Google Scholar

[14]

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. Google Scholar

[15]

A. A. Panov, The number of periodic solutions of polynomial differential equations,, Math. Notes, 64 (1998), 622. doi: 10.1007/BF02316287. Google Scholar

[16]

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

[17]

A. Sandqvist and K. M. Andersen, On the number of closed solutions to an equation $x'=f(t,x),$ where $f_{x^n}(t,x)\geq 0$ ($n=1,2$ or $3$),, J. Math. Anal. Appl., 159 (1991), 127. doi: 10.1016/0022-247X(91)90225-O. Google Scholar

show all references

References:
[1]

A. Alvarez, J.-L. Bravo and M. Fernández, The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign,, Communications on Pure and Applied Analysis, 8 (2009), 1493. doi: 10.3934/cpaa.2009.8.1493. Google Scholar

[2]

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. Google Scholar

[3]

M. A. M. Alwash, Polynomial differential equations with small coefficients,, Discrete and Continuos Dynamical Systems, 25 (2009), 1129. doi: 10.3934/dcds.2009.25.1129. Google Scholar

[4]

M. A. M. Alwash, Periodic solutions of polynomial non-autonomous differential equations,, Electronic Journal of Differential Equations, 2005 (): 1. Google Scholar

[5]

M. Calanchi and B. Ruf, On the number of closed solutions for polynomial ODE's and a special case of Hilbert's 16th problem,, Advances in Differential Equations, 7 (2002), 197. Google Scholar

[6]

A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 3737. doi: 10.1142/S0218127406017130. Google Scholar

[7]

A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems,, Proc. Amer. Math. Soc., 133 (2005), 751. doi: 10.1090/S0002-9939-04-07542-2. Google Scholar

[8]

A. Guillamon and M. Sabatini, The number of limit cycles in planar systems and generalized Abel equations with monotonous hyperbolicity,, Nonlinear Analysis, 71 (2009), 1941. doi: 10.1016/j.na.2009.01.034. Google Scholar

[9]

Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions,, Nonlinearity, 13 (2000), 1337. doi: 10.1088/0951-7715/13/4/319. Google Scholar

[10]

P. Korman and T. Ouyang, Exact multiplicity results for two classes of periodic equations,, J. Math. Anal. Appl., 194 (1995), 763. doi: 10.1006/jmaa.1995.1328. Google Scholar

[11]

A. M. Liapunov, "Stability of Motion,", Mathematics in Science and Engineering, 30 (1966). Google Scholar

[12]

A. Lins Neto, On the number of solutions of the equation $\frac{dx}{dt}=\sum_{j=0}^na_j(t)x^j,0\leq t\leq 1,$ for which $x(0)=x(1)$,, Inv. Math., 59 (1980), 67. doi: 10.1007/BF01390315. Google Scholar

[13]

M. N. Nkashama, A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations,, J. Math. Anal. Appl., 140 (1989), 381. doi: 10.1016/0022-247X(89)90072-3. Google Scholar

[14]

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. Google Scholar

[15]

A. A. Panov, The number of periodic solutions of polynomial differential equations,, Math. Notes, 64 (1998), 622. doi: 10.1007/BF02316287. Google Scholar

[16]

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

[17]

A. Sandqvist and K. M. Andersen, On the number of closed solutions to an equation $x'=f(t,x),$ where $f_{x^n}(t,x)\geq 0$ ($n=1,2$ or $3$),, J. Math. Anal. Appl., 159 (1991), 127. doi: 10.1016/0022-247X(91)90225-O. Google Scholar

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