Topological Characterization of Spatial-Temporal Chaos

It is well established both numerically and experimentally that nonlinear systems involving diffusion, chemotaxis, and/or convection mechanisms can generate complicated time-dependent patterns. Since this phenomenon is global in nature, obtaining a quantitative mathematical characterization that to some extent records or preserves the geometric structures of the complex patterns is difficult. Here we show that using algebraic topology, in particular homology, we can measure Lyapunov exponents that imply the existence of spatial-temporal chaos and suggest a tentative step towards the classification and/or identification of patterns within the FitzHugh-Nagumo system.