Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term

Abstract

We study limit cycles of the following system: dx dt =y(1+x 2−ay 2)+εx(mx n+ly n−λ), dy dt =−x(1−cx 2+y 2)+εy(mx n+ly n−λ) with a> c>0, ac>1, 0<ε≪1, m,l,λ are real parameters and n is a positive integer. When n=2, J.B. Li and Z.R. Liu [Publ. Math. 35 (1991) 487] showed that the system has 11 limit cycles. When n=6, H.J. Cao, Z.R. Liu and Z.J. Jing [Chaos, Solitons & Fractals 11 (2000) 2293] showed the system has 13 limit cycles. Using the same method of detection function, we first show that the system and others four systems have the same bifurcation diagrams of limit cycle. Then we demonstrate that any one of the five systems has 14 limit cycles for n=8. The positions of the 14 limit cycles are given by numerical exploration.