Self-correction of errors in discrete dynamic systems

The ability of a discrete dynamic system for correcting functional errors is investigated. A method for enhancing the degree of self-correction is described.     

A new version of Scilab software package for the study of dynamical systems

This work presents a new version of a software package for the study of chaotic flows, maps and fractals [1]. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well-known examples are implemented, with the capability of the users inserting their own ODE or iterative equations.

New version program summary

Program title: Chaos v2.0Catalogue identifier: AEAP_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1275No. of bytes in distributed program, including test data, etc.: 7135Distribution format: tar.gzProgramming language: Scilab 5.1.1. Scilab 5.1.1 should be installed before running the program. Information about the installation can be found at http://wiki.scilab.org/howto/install/windows.Computer: PC-compatible running Scilab on MS Windows or LinuxOperating system: Windows XP, LinuxRAM: below 150 MegabytesClassification: 6.2Catalogue identifier of previous version: AEAP_v1_0Journal reference of previous version: Comput. Phys. Comm. 178 (2008) 788Does the new version supersede the previous version?: YesNature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE).Solution method:

1.

Numerical solving of ordinary differential equations for the study of chaotic flows. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincare sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropies.

2.

Numerical solving of iterative equations for the study of maps and fractals.

Reasons for new version: The program has been updated to use the new version 5.1.1 of Scilab with new graphical capabilities [2]. Moreover, new use cases have been added which make the handling of the program easier and more efficient.Summary of revisions:

1.

A new use case concerning coupled predator-prey models has been added [3].

2.

Three new use cases concerning fractals (Sierpinsky gasket, Barnsley's Fern and Tree) have been added [3].

3.

The graphical user interface (GUI) of the program has been reconstructed to include the new use cases.

4.

The program has been updated to use Scilab 5.1.1 with the new graphical capabilities.

Additional comments: The program package contains 12 subprograms.

•

interface.sce - the graphical user interface (GUI) that permits the choice of a routine as follows

•

1.sci - Lorenz dynamical system

•

2.sci - Chua dynamical system

•

3.sci - Rosler dynamical system

•

4.sci - Henon map

•

5.sci - Lyapunov exponents for Lorenz dynamical system

•

6.sci - Lyapunov exponent for the logistic map

•

7.sci - Shannon entropy for the logistic map

•

8.sci - Coupled predator-prey model

•

1f.sci - Sierpinsky gasket

•

2f.sci - Barnsley's Fern

•

3f.sci - Barnsley's Tree

Running time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE, Lyapunov exponents calculation and fractals.References:

[1]

C.C. Bordeianu, C. Besliu, Al. Jipa, D. Felea, I. V. Grossu, Comput. Phys. Comm. 178 (2008) 788.

[2]

S. Campbell, J.P. Chancelier, R. Nikoukhah, Modeling and Simulation in Scilab/Scicos, Springer, 2006.

[3]

R.H. Landau, M.J. Paez, C.C. Bordeianu, A Survey of Computational Physics, Introductory Computational Science, Princeton University Press, 2008.

     

Scilab software package for the study of dynamical systems

This work presents a new software package for the study of chaotic flows and maps. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well known examples are implemented, with the capability of the users inserting their own ODE.

Program summary

Program title: ChaosCatalogue identifier: AEAP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 885No. of bytes in distributed program, including test data, etc.: 5925Distribution format: tar.gzProgramming language: Scilab 3.1.1Computer: PC-compatible running Scilab on MS Windows or LinuxOperating system: Windows XP, LinuxRAM: below 100 MegabytesClassification: 6.2Nature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE).Solution method: Numerical solving of ordinary differential equations. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincaré sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropies.Restrictions: The package routines are normally able to handle ODE systems of high orders (up to order twelve and possibly higher), depending on the nature of the problem.Running time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE and Lyapunov exponents calculation.     

Parameter functionalization and its application to the problem of local bifurcations in dynamic systems

The method of parameter functionalization suggested by M.A. Krasnosel’skii for solving problems with parameters and continuums of fixed points is considered. A general scheme of constructing the functionals in the bifurcation problem of small solutions to operator equations is suggested. As an application, we consider problems of local bifurcations in dynamic systems that are topical for the control theory: bifurcations of double equilibrium and forced oscillations and bifurcations of cycles of discrete systems. New sufficient criteria of bifurcations are indicated, an iteration procedure for constructing the solutions and their asymptotic representations are elaborated, and new stability conditions are stated.     

Nonlinear feedback-based control of the distribution of total energy between the degrees of freedom of a mechanical system. A quantum approach

The closed mechanical system capable of spontaneous redistribution of its total energy between the degrees of freedom was shown to manifest the quantum properties.     

A lattice Boltzmann model for the Korteweg-de Vries equation with two conservation laws

A lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model to the KdV equation, our method has higher-order accuracy. Two key steps in the development of this model are the addition of a momentum conservation condition, and the construction of a correlation between the first conservation law and the second conservation law. The numerical example shows the higher-order moment method can be used to raise the truncation error of the lattice Boltzmann scheme.     

A comparison of methods for interpolating chaotic flows from discrete velocity data

In this paper we consider a variety of schemes for performing interpolation in space and time to allow particle trajectories to be integrated from a velocity field given only on a discrete collection of data points in space and time. Using a widely-studied model of chaotic advection as a test case we give a method for quantifying the quality of interpolation methods and apply this to a variety of interpolation schemes in space only and in both space and time. It is shown that the performance of a method when interpolating in space is not a reliable predictor of its performance when interpolation in time is added. It is demonstrated that a method using bicubic spatial interpolation together with third-order Lagrange polynomials in time gives excellent accuracy at very modest computational expense compared to other methods.     

Accelerating numerical solution of stochastic differential equations with CUDA

Numerical integration of stochastic differential equations is commonly used in many branches of science. In this paper we present how to accelerate this kind of numerical calculations with popular NVIDIA Graphics Processing Units using the CUDA programming environment. We address general aspects of numerical programming on stream processors and illustrate them by two examples: the noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In presented cases the measured speedup can be as high as 675× compared to a typical CPU, which corresponds to several billion integration steps per second. This means that calculations which took weeks can now be completed in less than one hour. This brings stochastic simulation to a completely new level, opening for research a whole new range of problems which can now be solved interactively.

Program summary

Program title: SDECatalogue identifier: AEFG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Gnu GPL v3No. of lines in distributed program, including test data, etc.: 978No. of bytes in distributed program, including test data, etc.: 5905Distribution format: tar.gzProgramming language: CUDA CComputer: any system with a CUDA-compatible GPUOperating system: LinuxRAM: 64 MB of GPU memoryClassification: 4.3External routines: The program requires the NVIDIA CUDA Toolkit Version 2.0 or newer and the GNU Scientific Library v1.0 or newer. Optionally gnuplot is recommended for quick visualization of the results.Nature of problem: Direct numerical integration of stochastic differential equations is a computationally intensive problem, due to the necessity of calculating multiple independent realizations of the system. We exploit the inherent parallelism of this problem and perform the calculations on GPUs using the CUDA programming environment. The GPU's ability to execute hundreds of threads simultaneously makes it possible to speed up the computation by over two orders of magnitude, compared to a typical modern CPU.Solution method: The stochastic Runge-Kutta method of the second order is applied to integrate the equation of motion. Ensemble-averaged quantities of interest are obtained through averaging over multiple independent realizations of the system.Unusual features: The numerical solution of the stochastic differential equations in question is performed on a GPU using the CUDA environment.Running time: < 1 minute     

An algorithm for determining a class of invariants in quasi-polynomial systems

In this paper an algorithm is proposed for the retrieval of a wide class of invariants in quasi-polynomial systems. The invariance properties of the algorithm under different transformations are discussed. The application of the algorithm is illustrated on physical and numerical examples. The algorithm has been implemented in the MATLAB computing environment.     

On the relation between entropy and the average complexity of trajectories in dynamical systems

If (X,T) is a measure-preserving system, a nontrivial partition of X into two sets and f a positive increasing function defined on the positive real numbers, then the limit inferior of the sequence is greater than or equal to the limit inferior of the sequence of quotients of the average complexity of trajectories of length n generated by and nf(log₂(n))/log₂(n). A similar statement also holds for the limit superior. Received: August 20, 1999.     

Evidence of the correlation between positive Lyapunov exponents and good chaotic random number sequences

Using a new method to extract the data from various one-dimensional chaotic maps, we show that there is a nice correlation between the sign of the Lyapunov exponent of the maps and whether the extracted data form a good set of pseudo-random numbers using various well-known criteria.     

离散混沌系统的同步与参数识别

研究了驱动系统为参数未知的离散混沌系统时的同步问题,通过设计响应系统中的参数与控制器,实现驱动-响应系统的同步,识别出驱动系统的未知参数。基于离散系统的Lyapunov稳定性定理,给出了理论推导。以离散Hénon混沌系统为例进行了数值仿真,验证了该方法的有效性。     

随机混沌动力系统组及序列加密算法

简要分析了已有混沌加密算法的特点。为提供更有效的加密方法,提出了随机混沌动力系统组的概念。该系统组在一定条件下能构造出动力行为复杂的混沌子系统序列。基于此设计的序列加密算法,其加密过程受密钥、明文、系统组随机特征等多重因素影响,具有较高安全性。实验表明,该算法加密效果较好、密钥空间较大且易于实现。     

A model for roundoff and collapse in computation of chaotic dynamical systems

Computer simulations of dynamical systems contain discretizations, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the main features of this discretization are stochastically related to the parameters both, of the underlying continuous system and of the computer arithmetic. A model of this process is required to describe and analyze its statistical properties and this is carried out for the family of mappings f_l(x) = 1 − |1 − 2x|^l, x ∈ [0, 1], l > 2. Computer modeling results are presented.     

A classification of slow convergence near parametric periodic points of discrete dynamical systems

We study the phenomenon of slow convergence in families of discrete dynamical systems where the iteration function has a Puiseux series representation. Such occurrence consists in the slow convergence of orbits near non-hyperbolic parametric periodic points. We provide a precise new definition of the slowness of convergence which is based on literature results for the critical exponents associated with parametric periodic points. Such exponents establish a general classification for slow systems and provide a measure of rates of convergence. For dynamical systems whose iteration functions have Taylor series expansions, the new definition is natural with wider applicability. However, it can be also used for iteration functions where a more sophisticated approach, such as a Lagrange expansion, is needed. In addition, we show that even for such iteration functions, the critical exponent can be easily computed. The presented theoretical results are illustrated by numerical examples having different rates of convergence.     

Nonlinear parity relations: A logic-dynamic approach

Consideration was given to construction of the parity relations for systems described by the nonlinear dynamic models. To solve this problem, a logic-dynamic approach was proposed, and the realizability conditions providing insensitivity to the perturbing actions were given for it.     

Modeling and analyzing social network dynamics using stochastic discrete graphical dynamical systems

Motivated by applications such as the spread of epidemics and the propagation of influence in social networks, we propose a formal model for analyzing the dynamics of such networks. Our model is a stochastic version of discrete graphical dynamical systems. Using this model, we formulate and study the computational complexity of two fundamental problems (called reachability and predecessor existence problems) which arise in the context of social networks. We also address other problems that deal with the time evolution of such stochastic dynamical systems. Further, we point out the implications of our results to problems for other computational models such as Hopfield networks, communicating finite state machines and systolic arrays. In particular, our polynomial time algorithms for the predecessor existence problem for stochastic dynamical systems imply similar results for one-dimensional finite cellular automata.     

Improved piece-wise linear and nonlinear synchronization of a class of discrete chaotic systems

This paper deals with the chaotic synchronization of a discrete chaotic system. Based on a stable analysis, improved piece-wise linear feedback and nonlinear feedback control methods were given. Both the methods can achieve chaotic synchronization efficiently with one control input. The range of coupling coefficient is calculated in both the methods. Compared with nonlinear controls, the improved piece-wise linear controls have simpler structures and can be obtained more easily. Finally, the numerical simulation results verified the effectiveness of both the methods.