Fluctuating dynamics at the quasiperiodic onset of chaos, Tsallis q-statistics and Moriʹs q-phase thermodynamics

We analyze the fluctuating dynamics at the golden-mean transition to chaos in the critical circle map and find that trajectories within the critical attractor consist of infinite sets of power laws mixed together. We elucidate this structure assisted by known renormalization group (RG) results. Next we proceed to weigh the new findings against Tsallis’ entropic and Moriʹs thermodynamic theoretical schemes and observe behavior to a large extent richer than previously reported. We find that the sensitivity to initial conditions ξt has the form of families of intertwined q-exponentials, of which we determine the q-indexes and the generalized Lyapunov coefficient spectra λq. Further, the dynamics within the critical attractor is found to consist of not one but a collection of Moriʹs q-phase transitions with a hierarchical structure. The value of Moriʹs ‘thermodynamic field’ variable at each transition corresponds to the same special value for the entropic index q. We discuss the relationship between the two formalisms and indicate the usefulness of the methods involved to determine the universal trajectory scaling function σ and/or the occurrence and characterization of dynamical phase transitions.