ANALYSIS OF A PERTURBED HAMILTONIAN SYSTEM

Abstract

In this paper we study the existence, number and distribution of limit cycles of the perturbed Hamiltonian system given by:

\[
x' = 4y\left(abx^2-by^2+1\right)
+\varepsilon x\left(
ux^n+vy^n
-b\frac{\beta+1}{\mu+1}x^\mu y^\beta
-ux^2-\lambda
\right)
\]

\[
y' = 4x\left(ax^2-aby^2-1\right)
+\varepsilon y\left(
ux^n+vy^n
+bx^\mu y^\beta
-vy^2-\lambda
\right)
\]

where \(\mu+\beta=n\), \(0<a<b<1\), \(0<\varepsilon\ll1\), \(u,v,\lambda\) are the real parameters and \(n=2k\), \(k\) integer positive.

Abelian integral method is employed, [15]. For some particular parameters we draw the distribution limit cycle diagrams.