Lorenz Chaotic Attractor Research Articles

Multiband absorber design and optimization with Lorenz chaotic attractor

Metamaterials are artificial materials that don’t come from nature and have unusual electromagnetic properties. It has applications in many fields such as medical, image processing, signal absorption, invisibility cloak, sensor, and antenna. In this study, signal absorption of metamaterials was discussed and a simulation study was carried out. The most important difference of the proposed structure is that the pattern obtained from the Lorenz chaotic attractor was used in the resonator’s design, unlike the geometric shaped works. The chaos pattern consists of units that have both random shapes and sizes. Chaos can be defined as the order of disorder, which is extremely sensitive to initial conditions and has a broad power spectrum like noise. In addition, this study aims to contribute to the science of chaos. The resonator structure used in the proposed structure showed similar results with the studies in the literature. Rogers RT5870 material was used as substrate in the design. The proposed design is an eight-band structure with excellent signal absorption, partially insensitive to polarization and angle of incidence. The results show that the proposed design absorbs signals in the infrared region of the spectrum at 14.24, 15.44, 19.72, 21.04, 33.92, 35.16, 36.44 and 38.79 THz. Near-perfect signal absorption at the indicated frequencies is 97.98%, 98.73%, 99.85%, 92.03%, 95.62%, 92.56%, 96.21% and 97.09%, respectively. The proposed design includes heat sensor and thermal imaging etc can be used in application areas.

Mathematical models of CBSC over wireless channels and their analysis by using the LeNN-WOA-NM algorithm

In recent studies, several chaos-based secure communication (CBSC) systems are developed and are assumed to be effective and promising tools for balancing communications in wireless applications. The reliability of a wireless communication system is dependent on the synchronization between receiving and transmitting nodes. Therefore, it is crucial to analyze the mathematical models for different enhancements through CBSC and avoid the difficulty in synchronization. We have designed a novel computational paradigm that uses weighted Legendre polynomials to construct series solutions for mathematical models of the Lorenz Chaotic Attractor (LCA) and Double Scroll Attractor (DSA) (with filter and without filter) by using Chua’s circuits. We design fitness functions for each system of differential equations. Two optimization techniques, the Whale Optimization Algorithm (WOA) and the Nelder–Mead (NM) algorithm are adopted to optimize the objective functions and calculate the best set of unknown weights. We name our soft computing paradigm, the LeNN-WOA-NM algorithm. We have compared our results with state-of-the-art, and we establish that the LeNN-WOA-NM algorithm is fast, accurate, and reliable. We present graphical and statistical analysis to further elaborate on the performance of our algorithm.

Discrete-time Kalman filter for Takagi–Sugeno fuzzy models

In this work, the Kalman Filter (KF) and Takagi–Sugeno fuzzy modeling technique are combined to extend the classical Kalman linear state estimation to the nonlinear field. The framework for such extension is given, and in this sense the discrete-time fuzzy Kalman filter (DFKF) is obtained. It will be shown that the fuzzy version gives some advantages when is compared with the Extended Kalman Filter (EKF), which is the most typical extension of the KF to the nonlinear field. The proposed approach provides a significantly smaller processing time than the processing time of the EKF while the mean square error is also reduced. Finally, some examples, such as the Lorenz chaotic attractor and under actuated mechatronic system (pendubot), are used to compare the DFKF and EKF.

Symbolic Quest into Homoclinic Chaos

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transformations are due to the emergence of Shilnikov flames originating from inclination-switch homoclinic bifurcations of codimension-two. We demonstrate how the original computational technique, based on the symbolic description and kneading invariants, can disclose the complexity and universality of parametric structures and their link with nonlocal bifurcations in this representative model.

Electromagnetic Analog of 3D Autonomous ODEs with Quadratic Nonlinearities

It is shown that the general class of three-dimensional first-order ordinary differential equations with quadratic nonlinearities can be physically interpreted as the dynamics of a charged particle in an electromagnetic field, with a constant gradient B-field. The general class of equations is derived within the Lagrangian formalism of classical mechanics. As an application of this interpretation a new way of experimentally realizing the Lorenz chaotic attractors is proposed. The actual construction of such systems could be facilitated by existing magnetic resonance imaging technology, which already makes use of constant gradient fields, and may find applications in areas such as nuclear medicine and magnetic confinement fusion devices.

Nonlinear dynamical models in psychology are widespread and testable.

Comments on the article by Fredrickson and Losada (see record 2005-11834-001). Fredrickson and Losada's claimed connection between positivity ratios in human interactions and the Lorenz chaotic attractor and the extensive critique thereof (Brown, Sokal, & Friedman, December 2013) lead me to offer several remarks about nonlinear dynamical systems (NDS) models in psychology that would-be researchers should understand if the goal is to move the science forward.

MATRIX QUANTIZATION OF TURBULENCE

Based on our recent work on Quantum Nambu Mechanics [Axenides & Floratos 2009], we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of noncommutative phase space coordinates as Hermitian N × N matrices in R3. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction, it violates the quantum commutation relations. We demonstrate that the Heisenberg–Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand, there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving nondissipative sector survive for long times.

Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations

We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.

A Watermarking Algorithm Based on Block Energy Analysis of Wavelet Transform's Coefficients and Lorenz Chaotic Attractor

Abstract Perceptual transparency and robustness are conflicting requirements. To ensure the security of the watermark,how to choose the embedding position and intensity is a difficult problem. Chaotic system has characteristics of sensitivity to initial value and unpredictability. High-dimensional chaotic systems have advantage of higher secrecy, so this paper chooses Lorenz chaotic attractor to encrypt binary watermark image. Through the statistic analysis of block energy of the host image, which is applied DWT repeatedly up to the third level, so watermark adaptively embedded into the greatest coefficients which belong to the selected blocks by using the properties of human visual system. Additionally, to realize the blind digital watermarking extraction, chaotic sequence is also applied in selected wavelet coefficients to change the original value of the inherent relationship between wavelet coefficients. Theoretical analysis and experiment results show that the proposed algorithm has high robustness and better performance of perceptual transparency.

Quasi-objective nonlinear principal component analysis

By means of mathematical analysis and numerical experimentation, this study shows that the problems of non-uniqueness of solutions and data over-fitting, that plague the multilayer feedforward neural network for NonLinear Principal Component Analysis (NLPCA), are caused by inappropriate architecture of the neural network. A simplified two-hidden-layer feedforward neural network, which has no encoding layer and no bias term in the mathematical definitions of bottleneck and output neurons, is proposed to conduct NLPCA. This new, compact NLPCA model alleviates the aforementioned problems encountered when using the more complex neural network architecture for NLPCA. The numerical experiments are based on a data set generated from a well-known nonlinear system, the Lorenz chaotic attractor. Given the same number of bottleneck neurons or reduced dimensions, the compact NLPCA model effectively characterizes and represents the Lorenz attractor with significantly fewer parameters than the relevant three-hidden-layer feedforward neural network for NLPCA.

Robust adaptive backstepping control of uncertain Lorenz system

In this paper, a novel robust adaptive control method is proposed for controlling the Lorenz chaotic attractor. A new backstepping controller for the Lorenz system based on the Lyapunov stability theorem is proposed to overcome the singularity problem that appeared in using the typical backstepping control method. By exploiting the property of the system, the resulting controller is shown to be singularity free and the closed loop system is globally stable. Due to unavailability of system states measurement in practice, the controller is selected such that only one system state is needed. To overcome the problem of parameter uncertainty, an additional term to Lyapunov function is added and an identification scheme is adopted to have a negative definite Lyapunov function derivative. The simulation results demonstrate the effectiveness of the proposed controllers and approaches.

DESIGN AND IMPLEMENTATION OF MULTI-WING BUTTERFLY CHAOTIC ATTRACTORS VIA LORENZ-TYPE SYSTEMS

Lorenz system, as the first classical chaotic system, has been intensively investigated over the last four decades. Based on the sawtooth wave function, this paper initiates a novel approach for generating multi-wing butterfly chaotic attractors from the generalized first and second kinds of Lorenz-type systems. Compared with the traditional ring-shaped multi-scroll Lorenz chaotic attractors, the proposed multi-wing butterfly chaotic attractors are much easier to be designed and implemented by analog circuits. The dynamical behaviors of these multi-wing butterfly chaotic systems are further studied. Theoretical analysis shows that every index-2 saddle-focus equilibrium corresponds to a unique wing in the butterfly attractors. Finally, a module-based unified circuit diagram is constructed for realizing various multi-wing butterfly attractors. It should be especially pointed out that this is the first time in the literature that a maximal 10-wing butterfly chaotic attractor is experimentally verified by analog circuits.

Adaptive sliding mode observer for non-linear stochastic systems with uncertainties

It is presented, in this paper, a novel adaptive sliding mode observer (ASMO) for reconstructing the states of non-linear stochastic systems with structure uncertainties, parameter perturbations and external disturbances which is presented in the Ito differential equations. The proposed ASMO uses sliding mode technique to guarantee the robustness of observation, and an adaptive law is employed to update the sliding mode gain. The estimation error of the proposed observer is theoretically proved to be mean square exponentially convergent to a limited bound. Simulation study is made on computer with MatLab for reconstructing the states of Lorenz chaotic attractor disturbed with uncertainties and polluted with noises, and the simulation results verify the effectiveness of the proposed observation strategy.

Prediction of a Lorenz chaotic attractor using two-layer perceptron neural network

This paper investigates the prediction of a Lorenz chaotic attractor having relatively high values of Lypunov's exponents. The characteristic of this time series is its rich chaotic behavior. For such dynamic reconstruction problem, regularized radial basis function (RBF) neural network (NN) models have been widely employed in the literature. However, author recommends using a two-layer multi-layer perceptron (MLP) NN-based recurrent model. When none of the available linear models have been able to learn the dynamics of this attractor, it is shown that the proposed NN-based auto regressive (AR) and auto regressive moving average (ARMA) models with regularization have not only learned the true trajectory of this attractor, but also performed much better in multi-step-ahead predictions. However, equivalent linear models seem to fail miserably in learning the dynamics of the time series, despite the low values of Akaike's final prediction error (FPE) estimate. Author proposes to employ the recurrent NN-based ARMA model with regularization which clearly outperforms all other models and thus, it is possible to obtain good results for prediction and reconstruction of the dynamics of the chaotic time series with NN-based models.

Modem‐based communications using synchronized chaos

AbstractThis paper considers the use of chaotic dynamics to enhance or add value to telecommunications systems. An application in the form of secure communications, with particular emphasis on mobile radio systems is investigated. Standard continuous time descriptions of the Lorenz and Chua chaotic attractors are transformed into the discrete time domain and used in the design of a secure chaotic modem which includes the digital modulation schemes: quadrature phase shift keying (QPSK) and π/4‐QPSK with square‐root raised cosine pulse shaping filters. The results of analytical computer simulations show that the performance of both modem systems over limited bandwidth additive white Gaussian noise (AWGN), and Rayleigh and Rician fading channels induces no degradation in the bit error probability. Finally, in order to demonstrate that the chaotic modem offers a degree of robustness to eavesdropping, a reconstruction model is derived and applied to the communication scheme. It is shown that, due to inaccurate information surrounding the state of the system, a reconstruction of the system was not possible. Copyright © 2001 John Wiley & Sons, Ltd.

A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors

We show that the fractal dimension of a chaotic attractor is bounded from above by Kaplan-Yorke-type numbers, and provide a heuristic derivation on the inequality dim frac (A) ≤ 2 - (l 1(u 0) + l 2(u 0))⧸1 3(u> 0) = 2.4013, u 0 = (0,0,0) , where A is the Lorenz chaotic attractor and l k ( u) represents local Liapunov exponents.

Construction of phenomenological models from numerical scalar time series

The problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed. Techniques are exemplified for Rössler and Lorenz chaotic attractors, with emphasis on improved algorithms with respect to those previously published. The steps which are still required in order to investigate experimental noisy data are discussed.