Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

Hala Abd-Elmageed,Sundarapandian Vaidyanathan,Mufda Alrawashdeh,Sahar Ahmed Idris,Amor Menaceur

doi:10.1155/2021/8178729

Abstract

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

Highlights

Introduction and Statement of the MainResultHilbert in 1900 was interested in the maximum number of the limit cycles that a polynomial differential system of a given degree can have. is problem is the well-known 16th Hilbert problem, which together with the Riemann conjecture are the two problems of the famous list of 23 problems of Hilbert which remain open
In [8], the authors improved the result of the maximum number of limit cycles for a class of polynomial differential systems which bifurcate from the period annulus surrounding the origin of the system: u_ v − v(u − y + a)(u + v + a)
In [9], the authors improved the result of the maximum number of limit cycles of sixth polynomial differential systems which bifurcate from the period annulus surrounding the origin of the system:

Summary

Introduction and Statement of the Main Result

Hilbert in 1900 was interested in the maximum number of the limit cycles that a polynomial differential system of a given degree can have. is problem is the well-known 16th Hilbert problem, which together with the Riemann conjecture are the two problems of the famous list of 23 problems of Hilbert which remain open. In [8], the authors improved the result of the maximum number of limit cycles for a class of polynomial differential systems which bifurcate from the period annulus surrounding the origin of the system: u_ v − v(u − y + a)(u + v + a),. In [9], the authors improved the result of the maximum number of limit cycles of sixth polynomial differential systems which bifurcate from the period annulus surrounding the origin of the system:. For the sufficiently small |ε| and the polynom√ia ls P(u, v) and Q(u, v) having degree 6, suppose that |a| > 2, system equation (3) has at most 15 limit cycles bifurcating from the period annulus surrounding the origin of cubic polynomial differential system equation (1) using averaging theory of first order (Figures 1 and 2)

The Averaging Theory of First Order

Proof of Theorem 1

Findings

Conclusion