The production of Tsallis entropy in the limit of weak chaos and a new indicator of chaoticity

We study the connection between the appearance of a ‘metastable’ behaviour of weakly chaotic orbits, characterized by a constant rate of increase of the Tsallis q-entropy [C. Tsallis, Possible generalization of Boltzmann–Gibbs Statistics, Journal of Statistical Physics 52 (1988) 479–487], and the solutions of the variational equations of motion for the same orbits. We demonstrate that the variational equations yield transient solutions, lasting for long time intervals, during which the length of deviation vectors of nearby orbits grows in time almost as a power-law. The associated power exponent can be simply related to the entropic exponent for which the q-entropy exhibits a constant rate of increase. This analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits, that we call ‘Average Power Law Exponent’ (APLE). We compare the APLE with other established indicators of the literature. In particular, we give examples of application of the APLE in (a) a thin separatrix layer of the standard map, (b) the stickiness region around an island of stability in the same map, and (c) the web of resonances of a 4D symplectic map. In all these cases we identify weakly chaotic orbits exhibiting the ‘metastable’ behaviour associated with the Tsallis q-entropy.