ON THE GEOMETRY OF AN ATMOSPHERIC SLOW MANIFOLD

We examine the hyperbolic structure and the invariant manifolds of a model proposed by Lorenz to introduce the concept of an atmospheric slow manifold within the framework of dynamical system theory, We address the question of the long time asymptotic behaviour of the system using the (global) geometric point of view. It is shown that the model can be reduced to the classical example of a pendulum coupled to a harmonic oscillator. The dynamical regimes of interest for the slow manifold hypothesis correspond to regions of phase space near the saddle-center fixed point of this model which were not previously explored. These phase space regions are analysed using a combination of Melnikov-type methods and ideas from singular perturbation theory. By using the reversible symmetries of the model, an extension of the Melnikov theory is derived. This extension allows us to find homoclinic orbits and determine their approximation by simply computing the zeros of a certain function, which is constructed in terms of the usual Melnikov function, Countable infinities of global homoclinic bifurcations and existence of chaotic dynamics can be shown to exist by using the new tool.