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Weak chaos from Tsallis entropy
We present a geometric model-independent argument that aims to explain why the Tsallis entropy describes systems exhibiting “weak chaos” namely systems whose underlying dynamics has vanishing largest Lyapunov exponent. Our argument relies on properties of a deformation map of the reals induced by the Tsallis entropy and its conclusion agrees with all currently known results.
Vanishing largest Lyapunov exponent and Tsallis entropy
We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamic aspects which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison in metric terms by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed with respect to the corresponding Euclidean metric to zero. This conclusion is in agreement with all currently known results on systems that have a simple asymptotic behaviour and are described by the Tsallis entropy.