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Research Article Limit Cycles of a Class of Hilbert’s Sixteenth Problem Presented by Fractional Differential Equations G. H. Erjaee,1, 2 H. R. Z. Zangeneh,3 and N. Nyamoradi3 1

Mathematics Department, Qatar University, Doha 2713, Qatar Mathematics Department, Shiraz University, Shiraz 13797-71467, Iran 3 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran 2

Correspondence should be addressed to G. H. Erjaee, [email protected] Received 21 June 2009; Accepted 15 March 2010 Academic Editor: Mouffak Benchohra Copyright q 2010 G. H. Erjaee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The second part of Hilbert’s sixteenth problem concerned with the existence and number of the limit cycles for planer polynomial differential equations of degree n. In this article after a brief review on previous studies of a particular class of Hilbert’s sixteenth problem, we will discuss the existence and the stability of limit cycles of this class in the form of fractional differential equations.

1. Introduction The second part of the well-known Hilbert’s 16th problem is still unsolved since Hilbert proposed it in 1900. This problem is concerned with the maximum number of limit cycles and their relative distributions of the real planar polynomial systems of degree n in the form of   dx  P x, y , dt   dy  Q x, y , dt

1.1

where P x, y and Qx, y are polynomial of degree n with real coefficients. The general form of this problem, even for n  2, is yet an open problem that has attracted more researches but it is remarkably inflexible. With the development of computer’s and graphical

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Advances in Difference Equations

software, many recent new improvement results have been obtained. Some survey articles can be found in 1–5 and references therein. One of the classical methods to produce and study limit cycles in such system 1.1 is by perturbing a system which has a centre e.g., see 6, 7. In such methods the limit cycles are produced in the perturbed system from the periodic orbits of the periodic annulus of the unperturbed system. As we can see in 8 by perturbing the linear centre dx/dt  −y, dy/dt  x, using arbitrary polynomials P and Q of degree n, n − 1/2 limit cycles bifurcated with the bifurcation parameter ε of order one. Almost the same argument can be seen in 9 by perturbing the system dx/dt  −y1 x, dy/dt  x1 x with maximum n limit cycles. By perturbing the Hamiltonian centre given by H  0.5y2 xn 1 /n 1 in the polynomial differential systems of odd degree n, we can obtain n 1n 3/8 − 1 limit cycles 10. Several other similar investigations have been done using the perturbed polynomial differential systems of second, third, or even more degree. For example, see 11–13 and references therein. Based on the above studies, some of the authors of this article investigated the number