Parameter conditions for the existence of homoclinic orbits in the Lorenz equations.

Abstract

In this paper, we suggest some sufficient conditions for the existence of homoclinic orbits of the origin in the Lorenz equations. Such conditions are given by countably many bifurcation curves in the parameter space. For this purpose, we use some rigorous perturbation methods for the equivalent Lorenz system x' = k(y-x), y' = x(1-z) – εy, z' = xy – εbz; (k > 0, b > 0, 0 < ε ≪ 1) where we vary ε → 0 while k and b being fixed. For a local analysis of the unstable orbits, we study a cylindrical pendulum system and apply the "continuous dependence on initial data" theorem. For a global analysis of the solutions of Lorenz system, we use a perturbation theory for nonlinear systems of differential equations. The general procedure of perturbation theory is to insert a small parameter δ, 0 ≤ δ ≤ 1, such that when δ = 0 the problem is soluble. In this process, standard Bessel equation and Bessel functions will play some important roles for the representation of the solutions of the Lorenz system. For the asymptotic solutions of the linearized Lorenz system, WKB-theory will be useful.