Mode selection in a dendritelike nonlinear system

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Abstract
An oversimplified mathematical model that contains some of the basic features of the mode-selection problem in dendritic solidification is developed and investigated both analytically and numerically. We find that there exists a unique side-branching state of this system in which the dendrite tip is performing small oscillations about its point of marginal stability, and we argue that this state is the natural operating mode of our model. We also argue that external noise will always tend to drive the system into this state, but that purely thermal fluctuations are much too small to provide the mode-selection mechanism that is seen experimentally.