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

The limit cycle of the Duffing–van der Pol equation x¨+x+ε(x2-1)x˙+δx3=0 is studied. By considering the product of the frequency ωω of the limit cycle and the coefficient εε as an independent parameter μ=εωμ=εω, an equivalent equation is obtained and then solved by Liao’s homotopy analysis method. The frequency ωω is deduced as a function of μμ and δδ. This function provides us with an algebraic equation for ωω, 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 εε.