An approximation formula and parameter-dependence of statistical quantities in low-dimensional chaotic systems

2000 ◽  
Author(s):  
Shinji Koga
Keyword(s):  
Chaotic Systems ◽  
Parameter Dependence ◽  
Approximation Formula ◽  
Low Dimensional

scholarly journals A Novel S-Box Dynamic Design Based on Nonlinear-Transform of 1D Chaotic Maps

Electronics ◽  
2021 ◽  
Vol 10 (11) ◽  
pp. 1313
Author(s):  
Wenhao Yan ◽  
Qun Ding
Keyword(s):  
Chaotic Systems ◽  
Chaotic Maps ◽  
Chaotic Map ◽  
Logistic Map ◽  
Sample Entropy ◽  
One Dimension ◽  
Linear Combinations ◽  
Dynamic Design ◽  
Construction Methods ◽  
Low Dimensional

In this paper, a method to enhance the dynamic characteristics of one-dimension (1D) chaotic maps is first presented. Linear combinations and nonlinear transform based on existing chaotic systems (LNECS) are introduced. Then, a numerical chaotic map (LCLS), based on Logistic map and Sine map, is given. Through the analysis of a bifurcation diagram, Lyapunov exponent (LE), and Sample entropy (SE), we can see that CLS has overcome the shortcomings of a low-dimensional chaotic system and can be used in the field of cryptology. In addition, the construction of eight functions is designed to obtain an S-box. Finally, five security criteria of the S-box are shown, which indicate the S-box based on the proposed in this paper has strong encryption characteristics. The research of this paper is helpful for the development of cryptography study such as dynamic construction methods based on chaotic systems.

scholarly journals Bayesian optimization for tuning chaotic systems

2014 ◽  
Vol 1 (2) ◽  
pp. 1283-1312
Author(s):  
M. Abbas ◽  
A. Ilin ◽  
A. Solonen ◽  
J. Hakkarainen ◽  
E. Oja ◽  
...  
Keyword(s):  
Chaotic Systems ◽  
Climate Models ◽  
Atmospheric Model ◽  
Bayesian Optimization ◽  
Tuning Parameters ◽  
Performance Metric ◽  
Weather And Climate ◽  
Computationally Expensive ◽  
Low Dimensional ◽  
Complex Chaotic Systems

Abstract. In this work, we consider the Bayesian optimization (BO) approach for tuning parameters of complex chaotic systems. Such problems arise, for instance, in tuning the sub-grid scale parameterizations in weather and climate models. For such problems, the tuning procedure is generally based on a performance metric which measures how well the tuned model fits the data. This tuning is often a computationally expensive task. We show that BO, as a tool for finding the extrema of computationally expensive objective functions, is suitable for such tuning tasks. In the experiments, we consider tuning parameters of two systems: a simplified atmospheric model and a low-dimensional chaotic system. We show that BO is able to tune parameters of both the systems with a low number of objective function evaluations and without the need of any gradient information.

scholarly journals DEFINING THE CONTROLLING CONDITIONS OF LOW-DIMENSIONAL CHAOTIC SYSTEMS CONTROLLED BY TIME DELAYED FEEDBACK

2001 ◽  
Vol 50 (10) ◽  
pp. 1871
Author(s):  
CHEN YAN-YAN ◽  
PENG JIAN-HUA ◽  
SHEN QI-HONG ◽  
WEI JUN-JIE
Keyword(s):  
Chaotic Systems ◽  
Delayed Feedback ◽  
Low Dimensional ◽  
Time Delayed Feedback

scholarly journals Feature-based data assimilation in geophysics

2017 ◽  
Author(s):  
Matthias Morzfeld ◽  
Jesse Adams ◽  
Spencer Lunderman ◽  
Rafael Orozco
Keyword(s):  
Computational Models ◽  
Chaotic Systems ◽  
Efficient Solutions ◽  
Effective Dimension ◽  
Dimensional Model ◽  
Computationally Efficient ◽  
Fundamental Understanding ◽  
Feature Based ◽  
Low Dimensional ◽  
Noise Models

Abstract. Many applications in science require that computational models and data be combined. In a Bayesian framework, this is usually done by defining likelihoods based on the mismatch of model outputs and data. However, matching model outputs and data in this way can be unnecessary or impossible. For example, using large amounts of steady state data is unnecessary because these data are redundant, it is numerically difficult to assimilate data in chaotic systems, and it is often impossible to assimilate data of a complex system into a low-dimensional model. These issues can be addressed by selecting features of the data, and defining likelihoods based on the features, rather than by the usual mismatch of model output and data. Our goal is to contribute to a fundamental understanding of such a feature-based approach that allows us to assimilate selected aspects of data into models. Specifically, we explain how the feature-based approach can be interpreted as a method for reducing an effective dimension, and derive new noise models, based on perturbed observations, that lead to computationally efficient solutions. Numerical implementations of our ideas are illustrated in four examples.

scholarly journals A new scheme of coupling and synchronizing low-dimensional dynamical systems

2021 ◽  
Vol 67 (2 Mar-Apr) ◽  
pp. 334
Author(s):  
U. Uriostegui Legorreta ◽  
E. S. Tututi Hernández ◽  
G. Arroyo-Correa
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Circuits ◽  
Bifurcation Diagrams ◽  
Partial Synchronization ◽  
Low Dimensionality ◽  
Low Dimensional ◽  
The Way

A different manner of study synchronization between chaotic systems is presented. This is done by using two different forced coupled nonlinear circuits. The way of coupling the systems under study is different from those used in the analysis of chaos in dynamical systems of low dimensionality. The study of synchronization and how to manipulate it, is carried out through the variation of the couplings by calculating the bifurcation diagrams. We observed that for rather larger values of the coupling between the circuits it is reached total synchronization, while for small values of the coupling it is obtained, in the best of the cases, partial synchronization.

Validity of Dimensional Complexity Measures of EEG Signals

1997 ◽  
Vol 07 (01) ◽  
pp. 173-186 ◽  
Author(s):  
N. Pradhan ◽  
P. K. Sadasivan
Keyword(s):  
Time Series ◽  
Gaussian Noise ◽  
Chaotic Systems ◽  
Surrogate Data ◽  
Data Sets ◽  
Eeg Signals ◽  
Complexity Measures ◽  
Dimensional Complexity ◽  
Colored Gaussian Noise ◽  
Low Dimensional

The measure of dimensional complexity has the potential for feature extraction, modeling and prediction of EEG signals. However, the nonlinear dynamics of neuronal processes is under criticism that EEG signals may have a simpler stochastic description and chaotic dynamical measures of EEG may be spurious or unnecessary. Surrogate-data testing has been propounded to detect nonlinearity and chaos in experimental time series and to differentiate it from linear stochastic processes or colored noises. The surrogate data tests of brain signals (EEG) have produced equivocal results. Therefore, we examine the surrogate testing procedure using numerical data of classical chaotic systems, mixed sine waves, white Gaussian and colored Gaussian noises and typical EEGs. White Gaussian noise and classical chaotic time series are easily discerned by the surrogate-data test. However, a colored Gaussian noise data of low correlation dimensions (D2) or mixed sine waves containing less number of sinusoids show behaviors similar to the low dimensional deterministic chaotic systems. There are significant differences in D2 values between the original and surrogate data sets. The colored Gaussian noise appears linear and stochastic only when there is an increased randomness in its pattern and the signal is high dimensional. Our results clearly indicate that the "surrogate testing" alone may not be a sufficient test for distinguishing colored noises from low dimensional chaos. The EEG time series produce finite correlation dimensions. The surrogate testing of 8 independent realizations of different forms of EEG activities produce significantly different D2 values than the original data sets. Apparently many natural phenomena follow deterministic chaos and as the dimensional complexity of the system increases (D2 > 5) it may approximate a stochastic process. Thus EEG appears unlikely to have originated from a linear system driven by white noise.

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