Related papers: Nonstandard mixing in the standard map
We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form $S_q \equiv [1-\sum_{i=1}^W p_i^q]/[q-1]$ (with $S_1=-\sum_{i=1}^Wp_i \ln p_i$) for two families of one-dimensional dissipative maps,…
We numerically investigate the sensitivity to initial conditions of asymmetric unimodal maps $x_{t+1} = 1-a|x_t|^{z_i}$ ($i=1,2$ correspond to $x_t>0$ and $x_t<0$ respectively, $z_i >1$, $0<a\leq 2$, $t=0,1,2,...$) at the edge of chaos. We…
We give a hierarchy of many-parameter families of maps of the interval [0,1] with an invariant measure and using the measure, we calculate Kolmogorov--Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps…
Phase space can be constructed for $N$ equal and distinguishable subsystems that could be (probabilistically) either {\it weakly} (or {\it "locally"}) correlated (e.g., independent, i.e., uncorrelated), or {\it strongly} (or {\it globally})…
A random chaotic interval map with noise which causes coarse-graining induces a finite-state Markov chain. For a map topologically conjugate to a piecewise-linear map with the Lebesgue measure being ergodic, we prove that the Shannon…
The Boltzmann-Gibbs celebrated entropy $S_{BG}=-k\sum_ip_i \ln p_i$ is {\it concave} (with regard to all probability distributions $\{p_i\}$) and {\it stable} (under arbitrarily small deformations of any given probability distribution). It…
Hierarchy of one-parameter families of chaotic maps with an invariant measure have been introduced, where their appropriate coupling has lead to the generation of some coupled chaotic maps with an invariant measure. It is shown that these…
Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like…
We apply renormalized entropy as a complexity measure to the logistic and sine-circle maps. In the case of logistic map, renormalized entropy decreases (increases) until the accumulation point (after the accumulation point up to the most…
Ensemble of initial conditions for nonlinear maps can be described in terms of entropy. This ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to…
A new type of deterministic chaos for a system described by iterative two-dimensional maps is reported. The series being generated by the original map has an average upward trend while the first difference, which is the series of changes…
We consider nonequilibrium probabilistic dynamics in logistic-like maps $x_{t+1}=1-a|x_t|^z$, $(z>1)$ at their chaos threshold: We first introduce many initial conditions within one among $W>>1$ intervals partitioning the phase space and…
While on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on…
We investigate the high dimensional Hamiltonian chaotic dynamics in $N$ coupled area-preserving maps. We show the existence of an enhanced trapping regime caused by trajectories performing a random walk {\em inside} the area corresponding…
The log-periodic equation for the entropy $S = - (k/a) \sum_{i=1}^{N} p_{i} \sin(a \ln p_{i})$, based on the forgotten Sharma-Taneja entropy measure, is studied for the first time with $N$ the total number of system states and $p_{i}$ the…
We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional $S_{BG}[p(x)]…
We investigate the concept of a standard map for the interaction of relativistic particles and electrostatic waves of arbitrary amplitudes, under the action of external magnetic fields. The map is adequate for physical settings where waves…
We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_\alpha$ using the full parameter range $0<…
A {\it uniformly $p$-to-one endomorphism} is a measure-preserving map with entropy log $p$ which is almost everywhere $p$-to-one and for which the conditional expectation of each preimage is precisely $1/p$. The {\it standard} example of…
Recent mathematical investigations have shown that under very general conditions exponential mixing implies the Bernoulli property. As a concrete example of a statistical mechanics which is exponentially mixing we consider a Bernoulli shift…