The Borsuk-Ulam Theorem Elucidates Chaotic Systems

Authors: Arturo Tozzi, James F Peters

Nonlinear chaotic dynamics are widespread, both in physical and biological systems. This form of dynamics is frequently studied through logistic maps equipped with bifurcations, where intervals are dictated by the Feigenbaum constants. In such a multifaceted framework, a concept from the far-flung branch of topology, namely the Borsuk-Ulam theorem, comes into play. The theorem tells us that a continuous mapping from antipodal points with matching feature values on an n-sphere to the same real value can always be found. Here we demonstrate that embracing nonlinearity in the framework of the Borsuk-Ulam theorem means that bifurcation transformations (the antipodal points) can be described as paths or trajectories on abstract spheres equipped with a Feigenbaum dimension. Such an approach allows the evaluation of nonlinear systems through linear techniques. In conclusion, we provide a general topological mechanism which explains the elusive chaotic phenomena, cast in a physical/biological fashion which has the potential of being operationalized.

Comments: 8 Pages.

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[v1] 2016-10-19 03:34:00

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