QUANTIFYING LOCAL INSTABILITY AND PREDICTABILITY OF CHAOTIC DYNAMICAL SYSTEMS BY MEANS OF LOCAL METRIC ENTROPY

2000 ◽  
Vol 10 (01) ◽  
pp. 135-154 ◽  
Author(s):  
MOZHENG WEI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Time Scales ◽  
Growth Rates ◽  
Metric Entropy ◽  
Random Errors ◽  
Local Instability ◽  
Instability Index ◽  
Local Growth ◽  
Chaotic Dynamical Systems

A local metric entropy (LME) is introduced and used as a measure of local instability of chaotic dynamical systems. The predictability time scale of a dynamical system during a given period of time can also be estimated with high accuracy by using the LME. It is shown that LME, at any time during the evolution of a dynamical system, can be calculated as the sum of all the positive local Lyapunov exponents (LEs). This conclusion implies that the positive local LEs represent the rates of local information changes along the directions of their respective Lyapunov vectors. LME does not depend upon the amplitudes nor the configurations of initial perturbations; it depends on the positive local LEs which are intrinsic properties of dynamical systems. In addition, the sum of all the local LEs is proven to be equal to the divergence of phase space. Thus for a general chaotic system at any time, the sum of all the local LEs is equal to the sum of all the local growth rates of either instantaneous optimal modes or normal modes. In analyzing local instability, the performance of LME is evaluated by comparing an instability index with LME, the first local LE, locally largest LE, local growth rates of the dominant instantaneous optimal mode and normal mode. When LME is used to estimate the predictability time scales of systems over specified time periods, it is found that the time scales defined by LME are generally closer to the standard predictability times than the Lyapunov times and Kolmogorov–Sinai times for most cases in the two dynamical systems we have tested. Both the instability index and standard predictability time are defined and calculated through a large number of random errors.

scholarly journals Chaotic Dynamical Systems Associated with Tilings of RN

2011 ◽  
pp. 19-41
Author(s):  
Lionel Rosier
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Homogeneous Space ◽  
Chaos Synchronization ◽  
Discrete Dynamical Systems ◽  
Dimensional Torus ◽  
Synchronization Mechanism ◽  
Chaotic Dynamical Systems ◽  
Regular Tiling

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of RN, whose most familiar example is provided by the N-dimensional torus TN. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.

scholarly journals Metric Entropy of Nonautonomous Dynamical Systems

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Christoph Kawan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Entropy ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System

AbstractWe introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (X

STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

1996 ◽  
Vol 06 (12a) ◽  
pp. 2389-2399 ◽  
Author(s):  
PHIL DIAMOND ◽  
ALEXEI POKROVSKII
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Close Agreement ◽  
Computer Experiments ◽  
Continuum State ◽  
Chaotic Dynamical Systems ◽  
The Family ◽  
Random Mappings ◽  
Statistical Laws

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings fl(x)=1−|1−2x|l,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.

Random Bit Generator Based on Non-Autonomous Chaotic Systems

2015 ◽  
pp. 203-229
Author(s):  
Christos Volos ◽  
Ioannis Kyprianidis ◽  
Ioannis Stouboulos ◽  
Sundarapandian Vaidyanathan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaos Theory ◽  
Chaotic Systems ◽  
Statistical Tests ◽  
Van Der Pol ◽  
Chaotic Dynamical Systems ◽  
Random Bit Generator ◽  
Close Relationship ◽  
Autonomous Dynamical System

In the last decade, a very interesting relationship between cryptography and chaos theory was developed. As a result of this close relationship, several chaos-based cryptosystems, especially using autonomous chaotic dynamical systems, have been put forward. However, this chapter presents a novel Chaotic Random Bit Generator (CRBG), which is based on the Poincaré map of a non-autonomous dynamical system. For this reason, the very-well known Duffing-van der Pol system has been used. The proposed CRBG also uses the X-OR function for improving the “randomness” of the produced bit streams, which are subjected to the most stringent statistical tests, the FIPS-140-2 suite tests, to detect the specific characteristics that are expected from random bit sequences.

NONLINEAR DYNAMICAL SYSTEM AND CHAOS SYNCHRONIZATION

2008 ◽  
Vol 18 (05) ◽  
pp. 1531-1537 ◽  
Author(s):  
AYUB KHAN ◽  
PREMPAL SINGH
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Numerical Techniques ◽  
Nonlinear Dynamical ◽  
Chaotic Dynamical Systems ◽  
Response System ◽  
Identical Response

Chaos synchronization of nonlinear dynamical systems has been studied through theoretical and numerical techniques. For the synchronization of two identical nonlinear chaotic dynamical systems a theorem has been constructed based on the Lyapunov function, which requires a minimal knowledge of system's structure to synchronize with an identical response system. Numerical illustrations have been provided to verify the theorem.

scholarly journals Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems

2020 ◽  
pp. 2150021
Author(s):  
Xinsheng Wang ◽  
Weisheng Wu ◽  
Yujun Zhu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Topological Entropy ◽  
Equilibrium States ◽  
Metric Entropy ◽  
Unstable Foliation ◽  
Hyperbolic Dynamical Systems ◽  
Partially Hyperbolic

Let [Formula: see text] be a [Formula: see text] random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of [Formula: see text] on the unstable foliation are introduced and investigated. A version of Shannon–McMillan–Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs [Formula: see text]-states are investigated.

scholarly journals COMPUTATIONAL MODELING OF DYNAMICAL SYSTEMS

2005 ◽  
Vol 15 (03) ◽  
pp. 471-481 ◽  
Author(s):  
JOHAN JANSSON ◽  
CLAES JOHNSON ◽  
ANDERS LOGG
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Computational Modeling ◽  
Time Scales ◽  
Time Scale ◽  
Short Note ◽  
Multiple Time Scales ◽  
Time Interval ◽  
Multiple Time ◽  
Fast Time

In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10–9 computed over the time interval [0,100], and a lattice consisting of large and small point masses.

A METHOD FOR ENCODING MESSAGES BY TIME TARGETING OF THE TRAJECTORIES OF CHAOTIC SYSTEMS

1996 ◽  
Vol 06 (11) ◽  
pp. 2119-2125 ◽  
Author(s):  
D. GLIGOROSKI ◽  
D. DIMOVSKI ◽  
L. KOCAREV ◽  
V. URUMOV ◽  
L.O. CHUA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Chaotic Systems ◽  
Main Idea ◽  
Time Constraint ◽  
Chaotic Dynamical System ◽  
Chaotic Dynamical Systems

We suggest a method for encoding messages by chaotic dynamical systems. The main idea is that by targeting the trajectories of some chaotic dynamical system with time constraint, someone can send a information to the remote recipient. The concept is based on setting receptors in the phase space of the dynamical system, and then targeting the trajectory between them. We considered the time of arriving from one receptor to another as a carrier of information obtained by searching in the table of values for arriving times.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

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