A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens-Bogdanov normal form

In this paper, we present an algorithm based on the nonlinear time transformation method to approximate homoclinic orbits in planar autonomous nonlinear oscillators. With this approach, a unique perturbation solution up to any desired order can be obtained for them using trigonometric functions. To demonstrate its efficiency, the method is applied to calculate the homoclinic connection, both in the phase space and in the parameter space, of the versal unfolding of the nondegenerate Takens-Bogdanov singularity. Our approach considerably improves the results obtained so far by other methods (Melnikov, Poincare-Lindstedt, regular perturbations, multiple scales, etc.). The approximations achieved to different orders are confirmed by numerical continuation.