Analytic Approximation of the Homoclinic Orbits of the Lorenz System at σ= 10, b = 8/3 and ρ= 13.926...

We present an iterative technique to analytically approximate the homoclinic loops of the Lorenz system for σ = 10, b = 8/3 and ρ = ρH = 13.926.... First, the local structure of the homoclinic solution for t → 0± and t → ±∞ is analyzed. Then, global approximants are used to match the local expansions. The matching procedure resembles the one used in Padé approximations. The accuracy of the approximation is improved iteratively, with each iteration providing estimates for the initial conditions of the homoclinic orbit, the value of ρH, and three undetermined constants in the local expansions. Within three iterations the error in ρH falls to the order of 0.1%. Comparisons with numerical integrations are made, and a discussion on ways to extend the technique to other types of homoclinic or heteroclinic orbits, and to improve its accuracy, is given.