Averaging theory at any order for computing periodic orbits

作者：Gine Jaume; Grau Maite^*; Llibre Jaume

来源：Physica D: Nonlinear Phenomena , 2013, 250: 58-65.

DOI：10.1016/j.physd.2013.01.015

• First-order analytic differential equations
• Averaging theory
• Polynomial differential equations
• Limit cycles
• Periodic orbits

We provide a recurrence formula for the coefficients of the powers of a in the series expansion of the solutions around epsilon = 0 of the perturbed first-order differential equations. Using it, we give an averaging theory at any order in epsilon for the following two kinds of analytic differential equation:
dx/d theta = Sigma(k >= 1) epsilon F-k(k)(theta, x), dx/d theta = Sigma(k >= 0) epsilon F-k(k)(theta,x).
A planar polynomial differential system with a singular point at the origin can be transformed, using polar coordinates, to an equation of the previous form. Thus, we apply our results for studying the limit cycles of a planar polynomial differential systems.

• 出版日期2013-5-1