Directional solidification at high speed. II. Transition to chaos

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Abstract
This paper continues our analysis of various aspects of interface dynamics in rapid solidification. The description is based on a local continuum model, relevant to both liquid crystals and conventional materials. It was derived in a preceding paper, where we dealt with primary and secondary instabilities evolving from an initially flat interface when the control parameter, a renormalized temperature gradient, is decreased. Here we focus on more complex dynamic states arising from the interaction of different oscillatory modes. We find quasiperiodic motion to occur when one of the oscillators is a (parity-breaking) drifting mode. Quasiperiodicity precedes a transition to chaos, the route to which we describe in some detail. The absence or manifestation of mode locking as well as other interesting dynamic states are discussed. A second quasiperiodic scenario, where the control parameter is the wave number of the pattern, provides evidence that the transition to chaos via intermediate quasiperiodic states is generic for systems that possess the drift instability. Both chaotic regimes are briefly characterized, and Lyapunov exponents are computed for a variety of states. We find that all chaotic states have two vanishing Lyapunov exponents, a feature that we explain as a consequence of translational invariance. An implication is that the Lyapunov dimension of chaotic attractors exceeds three. Moreover, we find attractors whose dimension is larger than four. All the considered chaotic states are purely temporal. An outlook is given on interesting and important questions related to the long-time behavior of our model on large length scales, where spatiotemporal chaos is to be expected.