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

A criterion to predict bifurcation of homoclinic orbits in strongly nonline ar 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 formal ly with the Jacobian elliptic functions to determine an approximation of th e limit cycles near homoclinicity. We then apply a criterion for predicting homoclinic orbits, based on the collision of the bifurcating limit cycle w ith the saddle equilibrium. In particular we show that this criterion leads to the same results, formally and to leading order, as the standard Melnik ov technique. Explicit applications of this criterion to quadratic or cubic nonlinearities f(x) are included.