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

Abstract

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 , , up to first order by a family of polynomial differential systems of degree n in , 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 , , , up to first order by a family of polynomial differential systems of degree n in , 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.

Keywords:

• Limit cycle
• Synchronization
• Averaging method
• Linear center
• Polynomial differential system

1991 Mathematics Subject Classifications::

• 34C29
• 34C25
• 47H11

Acknowledgements

The first two authors are supported by the grants MCYT–Spain MTM2005-06098-C02-01 and CIRIT–Spain 2005SGR 00550. The third author is partially supported by a FAPESP–BRAZIL grant 10246–2. All authors are also supported by the joint project CAPES–MECD grant HBP2003–0017.