Analysis on limit cycles of [formula omitted]-equivariant polynomial vector fields with degree 3 or 4

Abstract

This paper presents a study on the limit cycles of Z q -equivariant polynomial vector fields with degree 3 or 4. Previous studies have shown that when q = 2 , cubic-order systems can have 12 small amplitude limit cycles. In this paper, particular attention is focused on the cases of q ⩾ 3 . It is shown that for cubic-order systems, when q = 3 there exist 3 small limit cycles and 1 big limit cycle; while for q = 4 , it has 4 small limit cycles and 1 big limit cycle; and when q ⩾ 5 , there is only 1 small limit cycle. For fourth-order systems, the cases for even q are the same as the cubic-order systems. When q = 5 it can have 10 small limit cycles; while for q ⩾ 7 , there exists only 1 small limit cycle. The case q = 3 is not considered in this paper. Numerical simulations are presented to illustrate the theoretical results.