Analysis on limit cycles of Zq - equivariant polynomial vector fields with degree 3 or 4 ✩

This paper presents a study on the limit cycles of Zq -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 cubicorder 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. © 2005 Elsevier Inc. All rights reserved.