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 ² _i ,y _i+1=bx _i . Numerical experiments are carried out fora=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 one-dimensional manifold by a Cantor set.