A Note about Certain Arbitrariness in the Solution of the Homological Equation in Deprit’s Method

Deprit’s method has been revisited in order to take advantage of certain arbitrariness arising when the inverse of the Lie operator is applied to obtain the generating function of the Lie transform. This arbitrariness is intrinsic to all perturbation techniques and can be used to demonstrate the equivalence among different perturbation methods, to remove terms from the generating function of the Lie transform, or to eliminate several angles simultaneously in the case of having a degenerate Hamiltonian.

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov–Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.

STARTING HOMOCLINIC TANGENCIES NEAR 1 : 1 RESONANCES

We construct a theory-based numerical method for starting the continuation of homoclinic tangencies near 1 : 1 resonances, for systems with arbitrary dimension ≥ 2. The core of the method is numerical center manifold reduction and flow approximation. The reduction is implemented by means of the homological equation. The starting procedure is applied in numerical examples.