Normal-Size Limit Cycles of Quadratic Systems with a Structurally Unstable

Abstract

In the present paper, we consider a real quadratic system dx dt = 2∑ i+j=0 aij xiyj , dy dt = 2∑ i+j=0 bij xiyj (1) with a structurally unstable cycle. It is known that the limit cycles of system (1) can surround only one singular point, which is a focus. Since a quadratic system has at most two cycles, we have the following possible arrangements of limit cycles (if any): m and (m1, m2), where m ∈ N, m1, m2 ∈ N ∪ {0}, m1 + m2 > 0, m1 ≥ 0, m2 ≥ 0. Here m is the number of limit cycles around the focus A provided that it is unique, and m1 and m2 are the numbers of limit cycles around each of the two cycles provided that they exist. Perko [1] introduced the notion of a limit cycle of “normal size,” that is, a limit cycle that can be detected by numerical methods. Quadratic systems with limit cycles of normal size and all known so far maximum distributions 3, (3, 0), and (3, 1) were obtained in [2]. Then it is natural to consider quadratic systems with a structurally unstable focus and with the distributions 2, (2, 0), and (2, 1) of limit cycles. The aim of the present paper is to obtain a number of quadratic systems with a structurally unstable focus and with the maximum number of limit cycles of normal size for all possible configurations of singular points.