Lorenz cross sections of the chaotic attractor of the double rotor

Abstract

A Lorenz cross section of an attractor with k > 0 positive Lyapunov exponents arising from a map of n variables is the transverse intersection of the attractor with an ( n − k)-dimensional plane. We describe a numerical procedure to compute Lorenz cross sections of chaotic attractors with k > 1 positive Lyapunov exponents and apply the technique to the attractor produced by the double rotor map, two of whose numerically computed Lyapunov exponents are positive and whose Lyapunov dimension is approximately 3.64. Error estimates indicate that the cross sections can be computed to high accuracy. The Lorenz cross sections suggest that the attractor for the double rotor map locally is not the cross product of two intervals and two Cantor sets. The numerically computed pointwise dimension of the Lorenz cross sections is approximately 1.64 and is independent of where the cross section plane intersects the attractor. This numerical evidence supports a conjecture that the pointwise and Lyapunov dimensions of typical attractors are equal.