454 research outputs found
Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos
We consider both the dynamics within and towards the supercycle attractors
along the period-doubling route to chaos to analyze the development of a
statistical-mechanical structure. In this structure the partition function
consists of the sum of the attractor position distances known as supercycle
diameters and the associated thermodynamic potential measures the rate of
approach of trajectories to the attractor. The configurational weights for
finite , and infinite , periods can be expressed
as power laws or deformed exponentials. For finite period the structure is
undeveloped in the sense that there is no true configurational degeneracy, but
in the limit this is realized together with the analog
property of a Legendre transform linking entropies of two ensembles. We also
study the partition functions for all and the action of the Central Limit
Theorem via a binomial approximation.Comment: 11 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1312.071
Sums of variables at the onset of chaos
We explain how specific dynamical properties give rise to the limit
distribution of sums of deterministic variables at the transition to chaos via
the period-doubling route. We study the sums of successive positions generated
by an ensemble of initial conditions uniformly distributed in the entire phase
space of a unimodal map as represented by the logistic map. We find that these
sums acquire their salient, multiscale, features from the repellor preimage
structure that dominates the dynamics toward the attractors along the
period-doubling cascade. And we explain how these properties transmit from the
sums to their distribution. Specifically, we show how the stationary
distribution of sums of positions at the Feigebaum point is built up from those
associated with the supercycle attractors forming a hierarchical structure with
multifractal and discrete scale invariance properties.Comment: arXiv admin note: text overlap with arXiv:1312.071
Entropies for severely contracted configuration space
We demonstrate that dual entropy expressions of the Tsallis type apply
naturally to statistical-mechanical systems that experience an exceptional
contraction of their configuration space. The entropic index
describes the contraction process, while the dual index defines the contraction dimension at which extensivity is
restored. We study this circumstance along the three routes to chaos in
low-dimensional nonlinear maps where the attractors at the transitions, between
regular and chaotic behavior, drive phase-space contraction for ensembles of
trajectories. We illustrate this circumstance for properties of systems that
find descriptions in terms of nonlinear maps. These are size-rank functions,
urbanization and similar processes, and settings where frequency locking takes
place
Critical fluctuations, intermittent dynamics and Tsallis statistics
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 developments. The first is that critical fluctuations
are correctly described by the dynamics of an intermittent nonlinear map. The
second is that intermittency in the neighborhood of a tangent bifurcation in
such map rigorously obeys nonextensive statistics. We comment on the
implications of this result. Key words: critical fluctuations, intermittency,
nonextensive statistics, anomalous stationary statesComment: Contribution to the proceedings of International Workshop on Trends
and Perspectives on Extensive and Non-Extensive Statistical Mechanics (q-60),
Angra dos Reis, Brazil, 17-21/11/2003. Submitted to Physica
Incidence of -statistics in rank distributions
We show that size-rank distributions with power-law decay (often only over a
limited extent) observed in a vast number of instances in a widespread family
of systems obey Tsallis statistics. The theoretical framework for these
distributions is analogous to that of a nonlinear iterated map near a tangent
bifurcation for which the Lyapunov exponent is negligible or vanishes. The
relevant statistical-mechanical expressions associated with these distributions
are derived from a maximum entropy principle with the use of two different
constraints, and the resulting duality of entropy indexes is seen to portray
physically relevant information. While the value of the index fixes
the distribution's power-law exponent, that for the dual index
ensures the extensivity of the deformed entropy.Comment: Santa Fe Institute working paper:
http://www.santafe.edu/media/workingpapers/14-07-024.pdf. see:
http://www.pnas.org/content/early/2014/09/03/1412093111.full.pdf+htm
Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points
The stationary distributions of sums of positions of trajectories generated
by the logistic map have been found to follow a basic renormalization group
(RG) structure: a nontrivial fixed-point multi-scale distribution at the
period-doubling onset of chaos and a Gaussian trivial fixed-point distribution
for all chaotic attractors. Here we describe in detail the crossover
distributions that can be generated at chaotic band-splitting points that
mediate between the aforementioned fixed-point distributions. Self affinity in
the chaotic region imprints scaling features to the crossover distributions
along the sequence of band splitting points. The trajectories that give rise to
these distributions are governed first by the sequential formation of
phase-space gaps when, initially uniformly-distributed, sets of trajectories
evolve towards the chaotic band attractors. Subsequently, the summation of
positions of trajectories already within the chaotic bands closes those gaps.
The possible shapes of the resultant distributions depend crucially on the
disposal of sets of early positions in the sums and the stoppage of the number
of terms retained in them
Quasiperiodic graphs at the onset of chaos
We examine the connectivity fluctuations across networks obtained when the
horizontal visibility (HV) algorithm is used on trajectories generated by
nonlinear circle maps at the quasiperiodic transition to chaos. The resultant
HV graph is highly anomalous as the degrees fluctuate at all scales with
amplitude that increases with the size of the network. We determine families of
Pesin-like identities between entropy growth rates and generalized
graph-theoretical Lyapunov exponents. An irrational winding number with pure
periodic continued fraction characterizes each family. We illustrate our
results for the so-called golden, silver and bronze numbers.Comment: arXiv admin note: text overlap with arXiv:1205.190
Manifestations of the onset of chaos in condensed matter and complex systems
We review the occurrence of the patterns of the onset of chaos in
low-dimensional nonlinear dissipative systems in leading topics of condensed
matter physics and complex systems of various disciplines. We consider the
dynamics associated with the attractors at period-doubling accumulation points
and at tangent bifurcations to describe features of glassy dynamics, critical
fluctuations and localization transitions. We recall that trajectories
pertaining to the routes to chaos form families of time series that are readily
transformed into networks via the Horizontal Visibility algorithm, and this in
turn facilitates establish connections between entropy and Renormalization
Group properties. We discretize the replicator equation of game theory to
observe the onset of chaos in familiar social dilemmas, and also to mimic the
evolution of high-dimensional ecological models. We describe an analytical
framework of nonlinear mappings that reproduce rank distributions of large
classes of data (including Zipf's law). We extend the discussion to point out a
common circumstance of drastic contraction of configuration space driven by the
attractors of these mappings. We mention the relation of generalized entropy
expressions with the dynamics along and at the period doubling, intermittency
and quasi-periodic routes to chaos. Finally, we refer to additional natural
phenomena in complex systems where these conditions may manifest.Comment: 20 pages, 7 figures. To be published in European Physical Journal
Special Topics. Special Issue: "Nonlinear Phenomena in Physics: New
Techniques and Applications
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