Computation of all the coefficients for the global connections in the Z₂-symmetric Takens-Bogdanov normal forms

The goal of this paper is to obtain a description of the global connections present in the Z₂-symmetric Takens-Bogdanov normal form. The algorithm used, grounded on the nonlinear time transformation method, provides a perturbation solution up to any wanted order for the homoclinic and heteroclinic orbits, with the only restriction on the capabilities of the computer used. Some proofs are given to guarantee the existence and uniqueness of the solution found with the iterative procedure. This is possibly the first time that, for this important system, such a high-order approximation is provided for the curves of the connecting orbits in the parameter plane. Moreover, at the same time, precise approximations in the phase space for the homoclinic and heteroclinic orbits are also attained. The accuracy of our theoretical results is confirmed by numerical continuation methods.