Attractor geometry and fractals

In the preceding chapter we discussed the dynamical side of chaos which manifests itself in the sensitive dependence of the evolution of a system on its initial conditions. This strange behaviour in time of a deterministically chaotic system has its counterpart in the geometry of the set in phase space formed by the (non-transient) trajectories of the system, the attractor.

Attractors of dissipative chaotic systems (the kind of systems we are interested in) generally have a very complicated geometry, which led people to call them strange. However, strange sets can also occur without dissipation in more general settings. As we have pointed out already in Chapter 3, a system described by autonomous differential equations (a flow) cannot be chaotic in less than three dimensions. With the same argument that trajectories are not allowed to intersect in a deterministic system we can conclude that not only the phase space but also the attractor of a chaotic flow must be more than two dimensional. However, slightly more than two dimensions is sufficient and the motion on a 2 + ∈ dimensional fractal can indeed be chaotic. As we will see, strange attractors with fractional dimensions are typical of chaotic systems. Map-like systems can of course show chaos with attractor dimensions less than two. Noninteger dimensions are assigned to geometrical objects which exhibit an unusual kind of self-similarity and which show structure on all length scales.

Example 6.1 (Self-similarity of the NMR laser attractor). Such self-similarity is demonstrated in Fig. 6.1 for an attractor reconstructed from the NMR laser time series, Appendix B.2.