CLARIFYING CHAOS III: CHAOTIC AND STOCHASTIC PROCESSES, CHAOTIC RESONANCE, AND NUMBER THEORY

1999 ◽  
Vol 09 (05) ◽  
pp. 785-803 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Processes ◽  
Prediction Models ◽  
Prime Numbers ◽  
Chaotic Time Series ◽  
Chaotic Dynamical Systems ◽  
Estimation And Prediction ◽  
The Relationship

In this tutorial we continue our program of clarifying chaos by examining the relationship between chaotic and stochastic processes. To do this, we construct chaotic analogs of stochastic processes, stochastic differential equations, and discuss estimation and prediction models. The conclusion of this section is that from the composition of simple nonlinear periodic dynamical systems arise chaotic dynamical systems, and from the time-series of chaotic solutions of finite-difference and differential equations are formed chaotic processes, the analogs of stochastic processes. Chaotic processes are formed from chaotic dynamical systems in at least two ways. One is by the superposition of a large class of chaotic time-series. The second is through the compression of the time-scale of a chaotic time-series. As stochastic processes that arise from uniform random variables are not constructable, and chaotic processes are constructable, we conclude that chaotic processes are primary and that stochastic processes are idealizations of chaotic processes. Also, we begin to explore the relationship between the prime numbers and the possible role they may play in the formation of chaos.

scholarly journals Double image multi-encryption algorithm based on fractional chaotic time series

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Zhenghong Guo ◽  
Jie Yang ◽  
Yang Zhao
Keyword(s):  
Time Series ◽  
Chaotic Time Series ◽  
Encryption Algorithm ◽  
Brute Force ◽  
Double Image ◽  
Key Space ◽  
Plain Image ◽  
Image Pixels ◽  
Brute Force Attack ◽  
The Relationship

AbstractIn this paper, we introduce a new image encryption scheme based on fractional chaotic time series, in which shuffling the positions blocks of plain-image and changing the grey values of image pixels are combined to confuse the relationship between the plain-image and the cipher-image. Also, the experimental results demonstrate that the key space is large enough to resist the brute-force attack and the distribution of grey values of the encrypted image has a random-like behavior.

Chaotic Dynamical Systems as Automata

1987 ◽  
Vol 42 (6) ◽  
pp. 547-555 ◽  
Author(s):  
Joseph L. McCauley
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Finite Length ◽  
Discrete Maps ◽  
Chaotic Dynamical Systems ◽  
Periodically Driven ◽  
Driven System

We discuss the replacement of discrete maps by automata, algorithms for the transformation of finite length digit strings into other finite length digit strings, and then discuss what it required in order to replace chaotic phase flows that are generated by ordinary differential equations by automata without introducing unknown and uncontrollable errors. That question arises naturally in the discretization of chaotic differential equations for the purpose of computation. We discuss as examples an autonomous and a periodically driven system, and a possible connection with cellular automata is also discussed. Qualitatively, our considerations are equivalent to asking when can the solution of a chaotic set of equations be regarded as a machine, or a model of a machine.

Wavelet Reconstruction of Nonlinear Dynamics

1998 ◽  
Vol 08 (11) ◽  
pp. 2191-2201 ◽  
Author(s):  
David Allingham ◽  
Matthew West ◽  
Alistair I. Mees
Keyword(s):  
Nonlinear Dynamics ◽  
Time Series ◽  
Dynamical Systems ◽  
Wavelet Transforms ◽  
Basis Pursuit ◽  
Chaotic Dynamical Systems ◽  
Vibrating String

We investigate the reconstruction of embedded time-series from chaotic dynamical systems using wavelets. The standard wavelet transforms are not applicable because of the embedding, and we use a basis pursuit method which on its own does not perform very well. When this is combined with a continuous optimizer, however, we obtain very good models. We discuss the success of this method and apply it to some data from a vibrating string experiment.

scholarly journals Extreme values for Benedicks–Carleson quadratic maps

2008 ◽  
Vol 28 (4) ◽  
pp. 1117-1133 ◽  
Author(s):  
ANA CRISTINA MOREIRA FREITAS ◽  
JORGE MILHAZES FREITAS
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Extreme Values ◽  
Extreme Value Distribution ◽  
Random Variable ◽  
Absolutely Continuous ◽  
Quadratic Maps ◽  
Chaotic Dynamical Systems ◽  
Stationary Stochastic Processes ◽  
Absolutely Continuous Invariant

AbstractWe consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).

scholarly journals Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650014 ◽  
Author(s):  
Tiberiu Harko ◽  
Praiboon Pantaragphong ◽  
Sorin V. Sabau
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Cold Dark Matter ◽  
Evolution Equations ◽  
Dynamical Evolution ◽  
Finsler Spaces ◽  
Stability And Instability ◽  
The Relationship ◽  
Autonomous Dynamical Systems

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary [Formula: see text]-dimensional first-order differential equations. We investigate in detail the properties of the [Formula: see text]-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the [Formula: see text]-Cold Dark Matter ([Formula: see text]CDM) cosmological models, respectively.

ALGORITHM FOR EXACT LONG TIME CHAOTIC SERIES AND ITS APPLICATION TO CRYPTOSYSTEMS

2004 ◽  
Vol 14 (10) ◽  
pp. 3607-3611 ◽  
Author(s):  
SHUNJI KAWAMOTO ◽  
TAKESHI HORIUCHI
Keyword(s):  
Time Series ◽  
Logistic Map ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Chaotic Time Series ◽  
High Dependency ◽  
Key Space ◽  
Long Time ◽  
Computer Environment ◽  
Numerical Generation

It is said that the numerical generation of exact chaotic time series by iterating, for example, the logistic map, will be impossible, because chaos has a high dependency on initial values. In this letter, an algorithm to generate them without the accumulation of inevitable round-off errors caused by the iteration is proposed, where rational numbers are introduced. Also, it is shown that the period of the chaotic time series depends on the rational numbers including large prime numbers, which are fundamentally related to the Mersenne and the Fermat prime ones. Since the time series are numerically regenerated by the proposed algorithm in an usual computer environment, it could be applied to cryptosystems which do not need the synchronization, and have a large key-space by using large prime numbers.

MIXED STATE ANALYSIS OF MULTIVARIATE TIME SERIES

2001 ◽  
Vol 11 (08) ◽  
pp. 2217-2226 ◽  
Author(s):  
MARTIN WIESENFELDT ◽  
ULRICH PARLITZ ◽  
WERNER LAUTERBORN
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Mixed State ◽  
Weak Coupling ◽  
Multivariate Time Series ◽  
Generalized Synchronization ◽  
Mixed States ◽  
Chaotic Dynamical Systems ◽  
Measured Time ◽  
State Analysis

A method is presented for detecting weak coupling between (chaotic) dynamical systems below the threshold of (generalized) synchronization. This approach is based on reconstruction of mixed states consisting of delayed samples taken from simultaneously measured time series of both systems.

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