Abstract
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.
Highlights
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 dt
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
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 1 n 3 /8 − 1 limit cycles 10
Summary
The second part of the well-known Hilbert’s 16th problem is still unsolved since Hilbert proposed it in 1900. Based on the above studies, some of the authors of this article investigated the number of limit cycles of perturbed quintic Hamiltonian systems with different degree polynomials 14, 15. In these former articles a weakened Hilbert’s 16th problem in the following form is considered: dx dt Hy εP x, y , 1.2 dy dt −Hx εQ x, y. It has recently been discovered that processes governed by diffusion which is enhanced or hindered in some fashion are better modeled by FDEs than by integer-order differential equations These FDEs are finding numerous applications in areas ranging from financial mathematics to ocean-atmosphere dynamics to mathematical biology 16.
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