| Abstract: | Engineered and natural systems often involve irregular and self-similar geometric forms,which is called fractal geometry. For instance, precision machining produces a visuallyflat surface, while which looks like a rough mountain in the nanometer scale under themicroscope. Human heart consists of a fractal network of muscle cells, Purkinje fibers,arteries and veins. Cardiac electrical activity exhibits highly nonlinear and fractalbehaviors. Although space-time dynamics occur on the fractal geometry, e.g., chemicaletching on the surface of machined parts and electrical conduction in the heart, most ofexisting works modeled space-time dynamics (e.g., reaction, diffusion and propagation) onthe Euclidean geometry (e.g., flat planes and rectangular volumes). This brings inaccurateapproximation of real-world dynamics, due to sensitive dependence of nonlinear dynamicalsystems on initial conditions. In this paper, we developed novel methods and tools for thenumerical simulation and pattern recognition of spatiotemporal dynamics on fractalsurfaces of complex systems, which include (1) characterization and modeling of fractalgeometry, (2) fractal-based simulation and modeling of spatiotemporal dynamics, (3)recognizing and quantifying spatiotemporal patterns. Experimental results show that theproposed methods outperform traditional modeling approaches based on the Euclideangeometry, and provide effective tools to model and characterize space-time dynamics onfractal surfaces of complex systems. |
