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 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.