Homoclinic connections in strongly self-excited nonlinear oscillators: The Melnikov function and the elliptic Lindstedt-Poincare method

A criterion to predict bifurcation of homoclinic orbits in strongly nonlinear self-excited one-degree-of-freedom oscillator (x)double over dot + c(1)x+c(2) f(x) = epsilon g(mu,x,(x)over dot), is presented. The Lindstedt-Poincare perturbation method is combined formally with the Jacobian elliptic functions to determine an approximation of the limit cycles near homoclinicity. We then apply a criterion for predicting homoclinic orbits, based on the collision of the bifurcating limit cycle with the saddle equilibrium. In particular we show that this criterion leads to the same results, formally and to leading order, as the standard Melnikov technique. Explicit applications of this criterion to quadratic or cubic nonlinearities f(x) are included.