Strongly Chaotic Systems

Abstract

In this chapter we explore fast dynamo action in the limit of strong chaos. This limit was first used by Soward (1993a) who takes the pulsed Beltrami waves of Bayly & Childress (1988, 1989),

$$ u = \left\{ \begin{gathered} (0,\sin x,\cos x), (0 < t\bmod 2\tau < \tau ), \hfill \\ ( - \sin y,0,\cos y), (\tau < t\bmod 2\tau < 2\tau ), \hfill \\ \end{gathered} \right. $$

(10.0.1)

and considers the limit of very long pulses, τ ≫ 1. In the opposite limit τ → 0, the rapid pulsing of waves generates the average of the two waves, which is the Roberts’ flow of §5.1, an integrable flow. As τ is increased from zero, separatrices split, leaving thin bands of chaos in which the stretching and folding of magnetic field can be understood using the techniques of Chap. 8. As τ is increased further, the phase space becomes more chaotic, until eventually, for large τ, Poincaré sections show chaos everywhere. This is the limit of strong chaos or the anti-integrable limit, and is an important limit for proving results about chaotic systems (Aubry & Abramovici 1990). For example, using the standard map with parameter K in this limit K → ∞, Rechester & White (1980) have studied diffusion of particles, and Aubry & Abramovici (1990) have shown how periodic orbits bifurcate from a K = ∞ Bernoulli system (see §3.3.2).