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We uncover the basis for the validity of the Tsallis statistics at the onset of chaos in logistic maps. The dynamics within the critical attractor is found to consist of an infinite family of Mori's $q$-phase transitions of rapidly…
First, we analyze trajectories inside the Feigenbaum attractor and obtain the atypical weak sensitivity to initial conditions and loss of information associated to their dynamics. We identify the Mori singularities in its Lyapunov spectrum…
We study the dynamics of iterates at the transition to chaos in the logistic map and find that it is constituted by an infinite family of Mori's $q$-phase transitions. Starting from Feigenbaum's $\sigma $ function for the diameters ratio,…
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…
Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity $\zeta >1$ at both their pitchfork and tangent bifurcations. These…
It is pointed out that the dynamics of the order parameter at a thermal critical point obeys the precepts of the nonextensive Tsallis statistics. We arrive at this conclusion by putting together two well-defined statistical-mechanical…
This paper compares three different types of ``onset of chaos'' in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three…
We go over our finding that the dynamics at the noise-perturbed edge of chaos in logistic maps is comparable to that observed in supercooled liquids close to vitrification. That is, the three major features of glassy dynamics in structural…
We establish the existence of intermittent two-point dynamics and infinite stationary measures for a class of random circle endomorphisms with zero Lyapunov exponent, as a dynamical characterisation of the transition from synchronisation…
We analyze the properties of the self-similar network obtained from the trajectories of unimodal maps at the transition to chaos via the horizontal visibility (HV) algorithm. We first show that this network is uniquely determined by the…
The critical relations for statistical properties on saddle-node bifurcations are shown to display undulating fine structure, in addition to their known smooth dependence on the control parameter. A piecewise linear map with the type-I…
We consider regular lattices of coupled chaotic maps. Depending on lattice size, there may exist a window in parameter space where complete synchronization is eventually attained after a transient regime. Close outside this window, an…
We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity $z>1$. First we determine analytically the sensitivity to initial conditions $\xi_{t}$. Then we consider a…
Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as…
We extend existing studies of weakly sensitive points within the framework of Tsallis non-extensive thermodynamics to include weakly insensitive points at the edge of chaos. Analyzing tangent points of the logistic map we have verified that…
We consider the period-doubling and intermittency transitions in iterated nonlinear one-dimensional maps to corroborate unambiguously the validity of Tsallis' non-extensive statistics at these critical points. We study the map…
Different mechanisms for the creation of strange non-chaotic dynamics in the quasiperiodically forced logistic map are studied. These routes to strange nonchaos are characterised through the behavior of the largest nontrivial Lyapunov…
We study the probability densities of finite-time or \local Lyapunov exponents (LLEs) in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are…
The robustness of the universality class concept of the chaotic transition was investigated by analytically obtaining its critical exponent for a wide class of maps. In particular, we extended the existing one-dimensional chaotic maps,…
Critical transitions occur in a variety of dynamical systems. Here, we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for…