Two strange attractors with a simple structure

Abstract

Numerical computations have shown that, for a range of values of the parameters, the Lorenz system of three non linear ordinary differential equations of first order has a strange attractor whose structure may be understood quite easily.

We show that the same properties can be observed in a simple mapping of the plane defined by: x_i+1 = y_i + 1 - a x ²_i , y_i+1 = b x_i. Numerical experiments are carried out for a = 1.4, b = 0.3. Depending on the initial point (x_o, y_o), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a one-dimensional manifold by a Cantor set. This strange attractor has basically the same structure than a plane section of the attractor found for the Lorenz system.