Isometric graphing and multidimensional scaling for reaction-diffusion modeling on regular and fractal surfaces with spatiotemporal pattern recognition

Abstract

Heterogeneous surface reactions exhibiting complex spatiotemporal dynamics and patterns can be studied as processes involving reaction-diffusion mechanisms. In many realistic situations, the surface has fractal characteristics. This situation is studied by isometric graphing and multidimensional scaling (IGMDS) of fractal surfaces for extracting geodesic distances (i.e., shortest scaled distances that obtain edges of neighboring surface nodes and their interconnections) and the results obtained used to model effects of surface diffusion with nonlinear reactions. Further analysis of evolved spatiotemporal patterns may be carried out by IGMDS because high-dimensional snapshot data can be efficiently projected to a transformed subspace with reduced dimensions. Validation of the IGMDS methodology is carried out by comparing results with reduction capabilities of conventional principal component analysis for simple situations of reaction and diffusion on surfaces. The usefulness of the IGMDS methodology is shown for analysis of complex patterns formed on both regular and fractal surfaces, and using generic nonlinear reaction-diffusion systems following FitzHugh Nagumo and cubic reaction kinetics. The studies of these systems with nonlinear kinetics and noise show that effects of surface disorder due to fractality can become very relevant. The relevance is shown by studying properties of dynamical invariants in IGMDS component space, viz., the Lyapunov exponents and the KS entropy for interesting situations of spiral formation and turbulent patterns.