Abstract
We present results of numerical experimentation with a 2-D version of an equation of surface dynamics that has been derived earlier in the context of flame fronts [Frankel & Sivashinsky, 1987, 1988] and solid-liquid interfaces [Frankel, 1988]. Our observations confirm qualitative predictions of Frankel & Sivashinsky [1987, 1988]: the curves develop chaotic cellular pattern and accelerate while imbedding is sustained. However, if we allow self-intersections, in a different range of parameters the equation gives birth to remarkably complex and beautiful fractal-like structures either entirely chaotic or preserving any symmetry if inherited from the initial configuration. This accumulation of complexity is also manifested in exponential growth of the length while diameter of the set increases linearly which results in increasingly dense covering of the plane. Based on these observations we introduce concepts of self-fractalizing family and asymptotic fractal dimension, which turns out to be equal to two.
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