Lorenz Map Research Articles

Complex dynamics in an atmospheric pressure glow discharge in helium: transition from stationary to periodic oscillatory, and fully chaotic states

Abstract This work deals with the numerical study of spontaneous temporal oscillations in an atmospheric pressure glow discharge (APGD) in helium. The transition of helium APGD from stationary to periodic oscillatory state through the Hopf bifurcation, and further from periodic to chaotic oscillations through period-doubling bifurcations is explored. The choice of the discharge and external electric circuits parameters is guided by the relevant experiments. The ballast resistance and supply voltage of the external circuit play the role of control parameters. The method is based on the stability analysis of stationary states of the discharge. 

The stability diagram predicting parameter regimes at which stable and oscillatory states of the APGD can be expected is obtained. The effects of the discharge parameters (such as the gas gap, secondary electron emission coefficients, and capacitance in the external electric circuit) on the bifurcation curves are identified. 

The Lorenz map and corresponding period-doubling bifurcation diagram characterizing transition to chaotic oscillations in helium APGD with an increase in the control parameter are derived. The value of the capacitance in the external circuit plays a critical role in the dynamical behavior of the discharge. Decreasing its value contributes to the dissipation/damping of the system, whereas increasing it enhances the irregular behavior of the system.

Chaotic Encryption Scheme for Colour Image using 3D Lorenz Chaotic Map and 3D Chen System

Chaos-based image encryption system is an encryption method that uses chaotic systems in encrypting digital images for the purpose of enhancing security. Chaos theory exhibits some distinctive characteristics, which include butterfly-like patterns, unpredictable behavior, and sensitive dependence to initial conditions. . In the past few decades, image encryption based on a single chaotic map has been a common technique in encrypting images. However, there are still unauthorized interceptors who illegally access and obtain the private information of the image. With that being the case, an encryption method that applies more than one chaotic system will contribute to creating a more complex relationship between the original and encrypted images, making it challenging for unauthorized individuals to decipher or extract the original content of the image without the appropriate decryption key. In this study, two chaotic maps, 3D Lorenz Map and 3D Chen system, are applied to generate random cryptographic keys for encrypting color images using permutation and diffusion mechanisms. The proposed algorithm that employs two chaotic maps is proven to be an effective system that achieves excellent results in security.

Analysis of dynamics of a map-based neuron model via Lorenz maps.

Modeling nerve cells can facilitate formulating hypotheses about their real behavior and improve understanding of their functioning. In this paper, we study a discrete neuron model introduced by Courbage et al. [Chaos 17, 043109 (2007)], where the originally piecewise linear function defining voltage dynamics is replaced by a cubic polynomial, with an additional parameter responsible for varying the slope. Showing that on a large subset of the multidimensional parameter space, the return map of the voltage dynamics is an expanding Lorenz map, we analyze both chaotic and periodic behavior of the system and describe the complexity of spiking patterns fired by a neuron. This is achieved by using and extending some results from the theory of Lorenz-like and expanding Lorenz mappings.

COMPARATIVE STUDY OF CHAOTIC SYSTEM FOR ENCRYPTION

Chaotic systems leverage their inherent complexity and unpredictability to generate cryptographic keys, enhancing the security of encryption algorithms. This paper presents a comparative study of 13 chaotic keymaps. Several evaluation metrics, including keyspace size, dimensions, entropy, statistical properties, sensitivity to initial conditions, security level, practical implementation, and adaptability to cloud computing, are utilized to compare the keymaps. Keymaps such as Logistic, Lorenz, and Henon demonstrate robustness and high-security levels, offering large key space sizes and resistance to attacks. Their efficient implementation in a cloud computing environment further validates their suitability for real-world encryption scenarios. The context of the study focuses on the role of the key in encryption and provides a brief specification of each map to assess the effectiveness, security, and suitability of the popular chaotic keymaps for encryption applications. The study also discusses the security assessment of resistance to the popular cryptographic attacks: brute force, known plaintext, chosen plaintext, and side channel. The findings of this comparison reveal the Lorenz Map is the best for the cloud environment based on a specific scenario.

Synchronizability of Discrete Nonlinear Systems: A Master Stability Function Approach

In recent times, studies on discrete nonlinear systems received much attention among researchers because of their potential applications in real-world problems. In this study, we conducted an in-depth exploration into the stability of synchronization within discrete nonlinear systems, specifically focusing on the Hindmarsh–Rose map, the Chialvo neuron model, and the Lorenz map. Our methodology revolved around the utilization of the master stability function approach. We systematically examined all conceivable coupling configurations for each model to ascertain the stability of synchronization manifolds. The outcomes underscored that only distinct coupling schemes manifest stable synchronization manifolds, while others do not exhibit this trait. Furthermore, a comprehensive analysis of the master stability function’s behavior was performed across a diverse range of coupling strengths σ and system parameters. These findings greatly enhance our understanding of network dynamics, as discrete-time dynamical systems adeptly replicate the dynamics of continuous-time models, offering significant reductions in computational complexity.

Generalized Fibonacci shifts in the Lorenz attractor

In this article we deal with symmetric Lorenz attractors having a homoclinic loop that exhibits a well ordered orbit. We show the symmetry implies a very regular behaviour on the dynamic in the topological and metric sense. Let ([−1,1],f) be the one-dimensional reduction Lorenz map satisfying a well ordered orbit and ([−1,0],f̃) be the quotient map, given by the equivalence relation x∼−x, the dynamic of f˜ is described explicitly as a subshift of finite type which generalizes the Fibonacci shifts and this fact is used to compute topological entropy of f.Moreover we show that in general ([−1,0],f̃) is related to a factor of the k-bonacci shift. In particular we found that the 1-dimensional Lorenz map replicates an interesting duplicating behaviour of the k-bonacci shift found in Sirvent (1996, 2011).

A Fractional-Order Discrete Lorenz Map

In this paper, a discrete Lorenz map with the fractional difference is analyzed. Bifurcations of the map in commensurate-order and incommensurate-order cases are studied when an order and a parameter are varied. Hopf bifurcation and periodic-doubling cascade are found by the numerical simulations. The parameter values of Hopf bifurcation points are determined when the order is taken as a different value. It can be concluded that the parameter decreases as the order increases. Chaos control and synchronization for the fractional-order discrete Lorenz map are studied through designing the suitable controllers. The effectiveness of the controllers is illustrated by numerical simulations.

Multi Chaotic System to Generate Novel S-Box for Image Encryption

A novel method on the basis of multi chaos theory is suggested in the presented study. Also, the study used two different dimensions to generate S-Box to get a strong cipher that is difficult to break. The suggested image cryptosystem includes an identical (decryption and encryption) process, which involves a single keystream generator, shifting process (based on 3D Lorenz map) related diffusion operations, and generate S-Box (based on 2D Henon map) that related confusion operation. The comparative analysis and the simulate test show that the suggested image cryptosystem has a few properties, like high-sensitivity, fast encryption/decryption, large keyspace, excellent statistical properties related to the ciphertext, and so on. The suggested cryptosystem is considered as an alternative for practical secure communications.

Taylor-based optimized recursive extended exponential smoothed neural networks forecasting method

The development of a Time series Forecasting System is a major concern for Artificial Intelligence researchers. Commonly, existing systems only assess temporal features and analyze the behavior of the data over time, thus, resulting in uncertain forecasting accuracy. Although many forecasting systems were proposed in the literature; they have not yet answered the attending question. Hence, to overcome this problematic, we propose an innovative method called Taylor-based Optimized Recursive Extended Exponential Smoothed Neural Networks Forecasting method, abbreviated as TOREESNN. Briefly explained, the proposed technique introduces three ideas to solve this issue: First, building an innovative framework for forecasting univariate time series based on Exponential Smoothed theory. Second, designing an Elman Classifier model for uncertainty prediction in order to correct the forecasted values. And finally hybrading the two recurrent systems in one framework to obtain the final results. Experimental results demonstrate that the proposed method has a high accuracy both in training and testing data in terms of Mean Squared Error (MSE) and outperforms the state-of-the-art Recurrent Neural Networks models on Mackey-Glass, Nonlinear Auto-Regressive Moving Average time series (NARMA), Lorenz, and Henon map datasets.

F-Gintropy: An Entropic Distance Ranking Based on the Gini Index.

We consider an entropic distance analog quantity based on the density of the Gini index in the Lorenz map, i.e., gintropy. Such a quantity might be used for pairwise mapping and ranking between various countries and regions based on income and wealth inequality. Its generalization to f-gintropy, using a function of the income or wealth value, distinguishes between regional inequalities more sensitively than the original construction.

3D Textured Model Encryption Using 2D Logistic and 3D Lorenz Chaotic Map

The widespread of recent multimedia, including various 3D model applications in different domains of areas, may lead to 3D models being stolen and attacked by hackers. Moreover, 3D models must be protected from unauthorized users and when transmitting over the internet. Nowadays the 3D model protection is a very important issue. This paper proposed a scheme that provides high protection for the textured 3D model by implementing multiple levels of security. The first level of security is achieved by encrypting the texture map based on a key generated by a 2D Logistic chaotic map. The second level of security is implemented by modifying the vertices values of the 3D mesh based on keys generated by the 3D Lorenz chaotic map. The proposed scheme was implemented on various 3D textured models varying in the number of vertices and faces. The experimental results show that the proposed scheme has a good encryption and provides high security by completely deforms the whole texture and 3D mesh of the textured 3D model into the two levels. The encryption scheme has a large key space 10135 making the scheme resists violent attacks. The Hausdorff Distance (HD) and histogram metrics are adopted to calculate the matching degree between the original and extracted model. The results show that the original and extracted model are identical through the values of HD, which are approximate to zero, and the histogram visually is similar. Index Terms— 3D textured model, encryption, 2D Logistic map, 3D Lorenz map, chaotic map

Branched manifolds for the three types of unimodal maps

Branched manifold is certainly the finest description of the structure of chaotic attractors, characterizing how the unstable periodic orbits are knotted. Many chaotic attractors produced by strongly dissipative systems were thus topologically described. In spite of this, the different possibilities for the branched manifolds which may be constructed from unimodal maps were never exhaustively listed. This is the task of the present work, starting from the folded (Logistic) map, the torn (Lorenz) map, and the less known torn away map introduced by Rössler in 1979. The case of the “reverse” horseshoe map is also discussed.

Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme

Time integration is commonly used to obtain accurate system responses, such as the limit cycle oscillations (LCOs) for an aeroelastic system with freeplay. However, the integrations that start with various initial conditions (I.C.s) are usually studied case by case, so only a few system states can possibly be focused on. This paper proposes a state space iterating (SSI) scheme to find LCO solutions using time integration by using another method. First, a large number of arbitrary I.C. cases are used for time integrations, but only a very short integration time is required for each I.C. case. Second, system behaviors are depicted visually through a method that combines a modified Poincaré map and Lorenz map, in which the LCO solutions are found as fixed points via visual inspections. To verify the SSI scheme’s ability to find LCOs, a typical plunge–pitch wing section is established numerically. Time integrations with both the classic scheme and the proposed SSI scheme are carried out. The LCO results of the SSI scheme are well-aligned with those from the classic scheme. The SSI scheme visualizes the patterns of system responses using arbitrary I.C. cases and analyzes the LCO stability, which provides more mathematical insights into an aeroelastic system with freeplay.

Chaos Based Secure Medical Image Transmission Model for IoT- Powered Healthcare Systems

Due to the extensive development of Internet of Things (IoT) in e-healthcare environment, security and integrity of the medical data especially medical images became a big issue. This paper proposes a chaotic security architecture for ensuring the security of the medical images during transmission and storage. The proposed model is built by comprising of three main stages. At first, message digest 5 algorithm is applied to the plain image for generating the seed key of Lorenz chaotic map. Subsequently Lorenz map is iterated to generate the chaotic key series utilized for further process. In second stage, dual confusion such as row-by-row and column-by-column confusion is executed on the plain image. At last, dual diffusion process is performed by applying binary reverse and compliment operation, in addition to that XOR operation is executed between diffused image and Lorenz chaotic key image. Simulation results and analysis of security level by applying different attacks indicates that the developed cryptosystem has the potential of satisfying the security requirements of IoT healthcare applications.

New Image Encryption Based on Pixel Mixing and Generating Chaos System

There was suggested new utilizing of the dispersion principle, based on the Lorenz Map, and chaos theory is known as perfect for using cryptography applications for their properties like unpredictability and sensitive initial states as one of the complex encryption approaches. to disengage the color of pixels (R, G, B). Was created this suggest with mixing all each color pixel to swap horizontally, vertically, or diagonally, finally using XOR operation. only milliseconds to encryption and decryption image, because the behavior of block cipher that leads to the period for the image to be decrypted fairly close as the images. The experiments demonstrate that the information security capabilities will be both safe more efficient, the result of our analysis of picture quality evaluation, differential attacks (NPSR), (UACI), Pixels correlation shows the quality and strength of encryption processing, Entropy show good results and closely to 8.

Transition from periodic to chaotic oscillations in a planar gas discharge-semiconductor system

The work studies the transition from periodic to chaotic oscillations in a dc-driven planar gas discharge with a high-ohmic electrode. The applied voltage and resistance of the semiconductor layer act as control parameters. The oscillations occur in the subnormal discharge regime. The bifurcation diagram and Lorenz map characterizing transition of this system to chaos through period-doubling bifurcations are obtained. Numerical models employed are based on the drift-diffusion theory of gas discharges. The effect of two modelling approaches, namely the ‘simple’ fluid model and the more detailed ‘extended’ fluid model, is considered. The calculations showed that the results obtained by these two approaches are different in quantitative terms, however, qualitatively similar in terms of the dynamic behavior of the system as a function of the control parameters.

Topological properties of Lorenz maps derived from unimodal maps

A symmetric Lorenz map is obtained by ‘flipping’ one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.

Speech encryption algorithm using FFT and 3D-Lorenz–logistic chaotic map

In this paper a new speech encryption method using Fast Fourier Transform (FFT) and multiple chaotic maps has been developed for secured speech communication. In order to improve the drawbacks namely residual intelligibility in encrypted signal, poor quality in decrypted signal, low key space and high computational complexity that prevail in the exist speech encryption methods, a novel speech encryption algorithm based on a three dimension (3D) Lorenz-Logistic map has been developed. A new 3D Lorenz-Logistic map has been introduced by feeding Logistic map in 3D Lorenz map to obtain three different random number sequences. Totally eight initial and controlling parameters of 3D Lorenz-Logistic map are used as key values. The behavior of chaotic map is totally changed by changing the key values which generates highly randomized number sequence. Permutation and substitution plays vital role in this method. The input speech is applied to FFT to obtain real as well as imaginary values. Two random number sequences of the 3D Lorenz-Logistic map are used to permute the real and imaginary values of input speech signal. Remaining one random number sequence is used to permute a reference speech sample which is used for substitution of permuted real values of speech signal. The reverse process of permutation and substitution is used for recovering the desired signal. The Inverse Fast Fourier Transform (IFFT) is applied to reconstruct the original signal. The performance of the proposed method is verified using histogram analysis, spectrogram analysis, correlation analysis, signal to noise ratio analysis, NSCR (Number of Samples Changing Rate) and UACI (Unified Averaged Changed Intensity) analysis, key space and key sensitivity analysis, computational complexity measure, Perceptual Evaluation of Speech Quality (PESQ) analysis and subjective evaluation of speech quality. The results evidence that the proposed speech encryption method provides better security system with robust decryption quality.

Coded Permutation Entropy: A Measure for Dynamical Changes Based on the Secondary Partitioning of Amplitude Information.

An improved permutation entropy (PE) algorithm named coded permutation entropy (CPE) is proposed in this paper to optimize the problems existing in PE based on the secondary partitioning. The principle of CPE algorithm is given, and the performance of it for dynamical change detection is analyzed using synthetic signal, logistic map and Lorenz map. The detection ability of CPE algorithm in different signal-to-noise ratios (SNR) is studied and the algorithm complexity is discussed. The results show that CPE can accurately capture minor feature information and amplify the detection results of dynamical changes compared with PE, weighted permutation entropy (WPE) and amplitude-aware permutation entropy (AAPE), but it has less robustness to noise and requires a higher computation cost than the others. Finally, we use the new algorithm to analyze the rolling bearing fault signals. The application of actual signals illustrates that CPE performs better in detecting abnormal pulse of the rolling bearing when the embedded dimension is small. From all the analyses in this paper, we find that CPE has a better performance for dynamical change detection compared with the other three algorithms when there is a larger repetition rate of permutation pattern in the position sequences.

Design of the nonlinear look-up table in the chaotic pseudorandom number generator based on augmented Lorenz map

Design of the nonlinear look-up table in the chaotic pseudorandom number generator based on augmented Lorenz map