Uniformly valid solution of limit cycle of the Duffing-van der Pol equation

摘要

The limit cycle of the Duffing-van der Pol equation x + x + epsilon(x(2) - 1)(x) over dot + delta x(3) = 0 is studied. By considering the product of the frequency omega of the limit cycle and the coefficient epsilon as an independent parameter mu = epsilon omega, an equivalent equation is obtained and then solved by Liao's homotopy analysis method. The frequency omega is deduced as a function of mu and delta. This function provides us with an algebraic equation for omega, according to which we have an analytical approximation for the frequency. Numerical examples show that the attained approximation is very accurate. More importantly, the results are uniformly valid for all positive values of epsilon.