Limit cycles of some polynomial differential systems in dimension 2, 3 and 4, via averaging theory

In the qualitative study of a differential system it is important to know its limit cycles and their stability. Here through two relevant applications, we show how to study the existence of limit cycles and their stability using the averaging theory. The first application is a 4-dimensional system which is a model arising in synchronization phenomena. Under the natural assumptions of this problem, we can prove the existence of a stable limit cycle. It is known that perturbing the linear center (Formula presented.), (Formula presented.), up to first order by a family of polynomial differential systems of degree n in (Formula presented.), there are perturbed systems with (n − 1) /2 limit cycles if n is odd, and (n − 2) /2 limit cycles if n is even. The second application consists in extending this classical result to dimension 3. More precisely, perturbing the system (Formula presented.), (Formula presented.), (Formula presented.), up to first order by a family of polynomial differential systems of degree n in (Formula presented.), we can obtain at most n(n − 1) /2 limit cycles. Moreover, there are such perturbed systems having at least n(n − 1) /2 limit cycles. © 2008, Taylor & Francis Group, LLC.