Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.

@article{Akimoto2015GeneralizedLE,
  title={Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.},
  author={Takuma Akimoto and Masaki Nakagawa and Soya Shinkai and Yoji Aizawa},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2015},
  volume={91 1},
  pages={
          012926
        }
}
The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential… 
View on PubMed
arxiv.org
5 Citations
Background Citations
1
Methods Citations
1

Figures from this paper

5 Citations

Citation Type

Citation Type

Lyapunov Exponents and the Belousov-Zhabotinsky Oscillator: An Interactive Computational Approach
The Belousov-Zhabotinsky (BZ) chemical oscillator is the most studied oscillator. It has been modelled on the basis of single-step mechanisms which have been continuously refined since the seminal…
Exact Lyapunov exponents of the generalized Boole transformations
The generalized Boole transformations have rich behavior ranging from the \textit{mixing} phase with the Cauchy invariant measure to the \textit{dissipative} phase through the \textit{infinite…
A Simple Nonlinear Circuit Contains an Infinite Number of Functions
TLDR
This brief shows that the number of different functions that a nonlinear or chaotic circuit can implement exponentially increases as the circuit evolves in time, and this exponential increase is quantified with an exponent that is named the computing exponent.
Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise
We propose a method of estimating ergodization time of a chaotic many-particle system by monitoring equilibrium noise before and after time reversal of dynamics (Loschmidt echo). The ergodization…
Breather-like solitons, rogue waves, quasi-periodic/chaotic states for the surface elevation of water waves
In this paper, we investigate the surface elevation of water waves $$\eta $$η via an NLS equation under the coefficient constraint $$\omega ^{2}=gk$$ω2=gk, where g is the gravity acceleration, k and…

References

SHOWING 1-10 OF 90 REFERENCES
Complexity, initial condition sensitivity, dimension and weak chaos in dynamical systems
We prove quantitative relations between complexity, initial condition sensitivity and dimension which are suitable for 0-entropy dynamical systems. The orbit complexity indicator is a measure of the…
Dynamical ensembles in stationary states
We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in…
Aging generates regular motions in weakly chaotic systems
Using intermittent maps with infinite invariant measures, we investigate the universality of time-averaged observables under aging conditions. According to Aaronson's Darling-Kac theorem, in non-aged…
Observed Measures and Fluctuations in Dissipative Infinite Ergodic Systems: Randomization Theory for the Infinite-Modal Maps with Ant-Lion Property
Universal aspects of a certain class of infinite ergodic systems are studied by using the infinite-modal maps with a special interest to their observed measures. It is shown that the ant-lion (AL)…
Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
Limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied. It is known that intermittent phenomena, such as the Rayleigh-Benard convection…
Generalization of the Einstein relation for single trajectories in deterministic subdiffusion.
  • Takuma Akimoto
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
TLDR
Based on a certain transport coefficient obtained from a single trajectory, a relation is provided for the transport coefficients divided by the Lyapunov exponent and generalize the Einstein relation for single trajectories.
Subexponential instability in one-dimensional maps implies infinite invariant measure.
TLDR
It is shown that a one-dimensional, conservative, ergodic measure preserving map with subexponential instability has an infinite invariant measure, and then a generalized Lyapunov exponent is presented to characterize subexp exponential instability.
The Lempel-Ziv Complexity of Non-Stationary Chaos in Infinite Ergodic Cases
TLDR
The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/f spectral process is generated.
Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics
This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged…
Pesin-type identity for intermittent dynamics with a zero Lyaponov exponent.
TLDR
This work investigates the nonergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows deltax_{t}=deltax_{0}e;{lambda_{alpha}t;{alpha}} with 0<alpha<1.
...
1
2
3
4
5
...