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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

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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