The number of limit cycles of a class of polynomial differential systems

Abstract

In this paper, we consider the bifurcation of limit cycles of a class of polynomial differential systems of the form x ̇ = − y ( 1 + x 4 ) + ε P n ( x , y ) , y ̇ = x ( 1 + x 4 ) + ε Q n ( x , y ) , where P n , Q n are arbitrary polynomials of degree n . We prove that there is a system of the above form having at least 3 [ n + 1 2 ] − 2 limit cycles in Hopf bifurcation. Then applying the Argument Principle, we obtain that up to first order in ε the number of limit cycles that bifurcate from period annulus surrounding the origin of this system is at most 5 [ n + 1 4 ] + [ n + 1 2 ] for n ≥ 5 .