A Two-dimensional Mapping with a Strange Attractor

Abstract

Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a “strange attractor”. We show that the same properties can be observed in a simple mapping of the plane defined by: ${x_{i + 1}} = {y_i} + 1 - ax_i^2,{y_{i + 1}} = b{x_i}$. Numerical experiments are carried out for a =1.4, b = 0.3. Depending on the initial point (x ₀,y ₀), 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 onedimensional manifold.by a Cantor set.