Lorenz Dynamical System Research Articles

A method to regularize optimization problems governed by chaotic dynamical systems

Long-time averages of outputs from chaotic dynamical systems exhibit high-frequency oscillations in parameter space, which makes gradient-based optimization impractical. We show that these oscillations can be eliminated by coupling the initial and final states of the trajectory to produce discontinuous periodic orbits (DPOs). This approach can be regarded as a generalization of (continuous) unstable periodic orbits, which have been shown to regularize time-averages from chaotic problems. To solve the DPO governing equations, we present a Newton–Krylov algorithm that is globalized using a line search and parameter continuation. We study the accuracy of DPO-based outputs using the Lorenz dynamical system and the Kuramoto–Sivashinsky partial differential equation, and we demonstrate the effectiveness of DPO-based optimization on two simple examples. We conclude the paper with a discussion of open questions for future research.

The Visual Boundary Recurrence Plot: A Closer Look into the Dynamics of Recurrence Plots

In this paper, we introduce a new graphical tool, the visual boundary recurrence plot, in order to provide a more detailed visualization of the well-established method of recurrence plots. In this approach, we are coloring each point of the recurrence plot depending on whether the neighboring points in the vertical and horizontal directions are recurrent or not. We use this plot to study the stability of the textures of the recurrence plots viewing them as geometrical shapes under horizontal and vertical changes. In order to compare two recurrence plots, we define the visual boundary recurrence plot rate distance between their visual boundary recurrence plots to study their closeness/similarity. Finally, we introduce a topological criterion to select an optimal threshold for constructing a recurrence plot. A threshold is optimal when the recurrence plot associated with it remains similar to the ones that are constructed for consecutive values of backwards and forwards thresholds. The results are applied to recurrence plots of time series emanating from different sources: molecular dynamics simulations, Lorenz dynamical system, random noise and daily water level measurements of a river.

Observation error covariance specification in dynamical systems for data assimilation using recurrent neural networks

Data assimilation techniques are widely used to predict complex dynamical systems with uncertainties, based on time-series observation data. Error covariance matrices modeling is an important element in data assimilation algorithms which can considerably impact the forecasting accuracy. The estimation of these covariances, which usually relies on empirical assumptions and physical constraints, is often imprecise and computationally expensive, especially for systems of large dimensions. In this work, we propose a data-driven approach based on long short term memory (LSTM) recurrent neural networks (RNN) to improve both the accuracy and the efficiency of observation covariance specification in data assimilation for dynamical systems. Learning the covariance matrix from observed/simulated time-series data, the proposed approach does not require any knowledge or assumption about prior error distribution, unlike classical posterior tuning methods. We have compared the novel approach with two state-of-the-art covariance tuning algorithms, namely DI01 and D05, first in a Lorenz dynamical system and then in a 2D shallow water twin experiments framework with different covariance parameterization using ensemble assimilation. This novel method shows significant advantages in observation covariance specification, assimilation accuracy, and computational efficiency.

An efficient image encryption scheme based on double affine substitution box and chaotic system

Privacy of information is one of the most important and unavoidable issues in our digitally advance era. A huge amount of information transmitted over different servers and networking protocols. The sphere of digitally advanced world is tied with information in different forms of facilitations, which includes online banking systems, ecommerce and many more. Providing the ease of access, anything online makes our confidential information open to different threats. Therefore, to enjoy ease of access and at the same time secure our digital information from theft, we need a robust information security system. In this article, we have designed a novel and an efficient security system, which provides secrecy to our digital information. The designed encryption scheme is fundamentally a combination of chaos and nonlinear confusion components. We have developed a new mechanism of adding confusion, namely S8 permutation of double affine transformation to construct 40320 new substitution boxes (S-boxes) having nonlinearity 112 from a single S-box. Moreover, nonlinear Lorenz dynamical system is utilized to select any three S-boxes from 40320 newly generated nonlinear components. To add diffusion in our proposed algorithm, we have utilized Chirikov discrete iterative map. The excellence of an offered digital image encryption has been examined and evaluated with standard benchmarks. The simulation results reveal that the quality of the image encryption passes all these tests and is comparable to current benchmarks.

On a topological criterion to select a recurrence threshold

In this work, a topological criterion is proposed for selecting a recurrence threshold for constructing a recurrence plot of a time series. It is based on a metric structure of the set of the recurrence plots that is defined by the recurrence plot deviation distance among recurrence plots, introduced in a previous paper by the authors. In this process for a range of threshold values, the corresponding recurrence plots are constructed. Then, a value of the threshold is considered to be optimal when the image of its recurrence plot remains close to images of the other recurrence plots constructed for close values of thresholds based on the topological criterion introduced in the present work. The results are applied both to time series emanating from the Lorenz dynamical system and to Molecular Dynamic simulations.

Recurrence analysis of fractional-order Lorenz system

The fractional derivative is increasingly used in modeling of nonlinear systems. Fractional-order systems often give better fit to the experimental results, especially for the systems in which memory effects or hysteresis play a significant role. The paper presents numerical results obtained for Lorenz dynamical system, described by equations with the fractional derivative components. The impact of the fractional derivative terms on the system dynamics and stability is analyzed by using phase diagrams and recurrence plot analysis.

The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system

This paper discusses stability conditions and a chaotic behavior of the Lorenz dynamical system involving the Caputo fractional derivative of orders between 0 and 1. Contrary to some existing results on the topic, we study these problems with respect to a general (not specified) value of the Rayleigh number as a varying control parameter. Such a bifurcation analysis is known for the classical Lorenz system; we show that analysis of its fractional extension can yield different conclusions. In particular, we theoretically derive (and numerically illustrate) that nontrivial equilibria of the fractional Lorenz system become locally asymptotically stable for all values of the Rayleigh number large enough, which contradicts the behavior known from the classical case. As a main proof tool, we derive the optimal Routh–Hurwitz conditions of fractional type, i.e., necessary and sufficient conditions guaranteeing that all zeros of the corresponding characteristic polynomial are located inside the Matignon stability sector. Beside it, we perform other bifurcation investigations of the fractional Lorenz system, especially those documenting its transition from stability to chaotic behavior.

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 summaryProgram 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 TreeRunning 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.

A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter

In recent years chaotic secure communication and chaos synchronization have received ever increasing attention. In this paper, for the first time, a fractional chaotic communication method using an extended fractional Kalman filter is presented. The chaotic synchronization is implemented by the EFKF design in the presence of channel additive noise and processing noise. Encoding chaotic communication achieves a satisfactory, typical secure communication scheme. In the proposed system, security is enhanced based on spreading the signal in frequency and encrypting it in time domain. In this paper, the main advantages of using fractional order systems, increasing nonlinearity and spreading the power spectrum are highlighted. To illustrate the effectiveness of the proposed scheme, a numerical example based on the fractional Lorenz dynamical system is presented and the results are compared to the integer Lorenz system.

Statistical complexity measure of pseudorandom bit generators

Pseudorandom number generators (PRNG) are extensively used in Monte Carlo simulations, gambling machines and cryptography as substitutes of ideal random number generators (RNG). Each application imposes different statistical requirements to PRNGs. As L’Ecuyer clearly states “the main goal for Monte Carlo methods is to reproduce the statistical properties on which these methods are based whereas for gambling machines and cryptology, observing the sequence of output values for some time should provide no practical advantage for predicting the forthcoming numbers better than by just guessing at random”. In accordance with different applications several statistical test suites have been developed to analyze the sequences generated by PRNGs. In a recent paper a new statistical complexity measure [Phys. Lett. A 311 (2003) 126] has been defined. Here we propose this measure, as a randomness quantifier of a PRNGs. The test is applied to three very well known and widely tested PRNGs available in the literature. All of them are based on mathematical algorithms. Another PRNGs based on Lorenz 3D chaotic dynamical system is also analyzed. PRNGs based on chaos may be considered as a model for physical noise sources and important new results are recently reported. All the design steps of this PRNG are described, and each stage increase the PRNG randomness using different strategies. It is shown that the MPR statistical complexity measure is capable to quantify this randomness improvement. The PRNG based on the chaotic 3D Lorenz dynamical system is also evaluated using traditional digital signal processing tools for comparison.

Chaotic dynamics of the fractional Lorenz system.

In this Letter we introduce a generalization of the Lorenz dynamical system using fractional derivatives. Thus, the system can have an effective noninteger dimension Sigma defined as a sum of the orders of all involved derivatives. We found that the system with Sigma<3 can exhibit chaotic behavior. A striking finding is that there is a critical value of the effective dimension Sigma(cr), under which the system undergoes a transition from chaotic dynamics to regular one.

A METHOD FOR COMPUTING LYAPUNOV EXPONENTS SPECTRA WITHOUT REORTHOGONALIZATION

We present a method for the computation of Lyapunov exponents without reorthogon alization. In the low dimension of system(n&lt;5) ,the equations needed in present algorithm is less than those in normal methods such as QR, SVD etc. This method is applicable to both discrete systems and continuous systems, and is still valid when the Lyapunov spectra is degenerate. Numerical analysis to Lorenz dynamical system indicates that the method converges quickly and steadly for arbitrary nonzero initial state.

The proto-Lorenz system

A quotient of the Lorenz dynamical system is constructed. This “proto-Lorenz” system has the Lorenz system as a double covering, and the double covering explains the two-eared nature of the strange attractor for the Lorenz system. Arbitrary coverings of the proto-Lorenz system are possible, leading to n-eared strange attractors.

Congressional anecdotes

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.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 TreeRunning 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.

Kneading of Lorenz type for intervals and product spaces

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.