Rossler System Research Articles

A Lightweight Image Encryption based on Improving LED using Rossler Attractor

Lightweight image encryption refers to the use of encryption algorithms designed to secure images while considering the limitations of resources, such as processing power and memory, typically found in devices like IoT devices, mobile phones, and embedded systems. This paper proposes a lightweight image encryption based on an LED encryption algorithm and a method for a pseudo number generator called Rossler attractor. Image is split into their three-color bands (RGB) and then partitioned into a block for encryption, a three-equations model (Rossler Attractor) is used for a pseudo number generator, a specific operation applied for producing key scheduled for rounds in LED to seem as random as possible. The experiment explains the objective test to ensure the performance. The contribution of this work is to present an efficient method for lightweight encryption that is applied to color images with a pseudo number generator-based Rossler attractor dependent on a three-dimensional equation. The technique makes it sensitive to initial conditions and could seem like dynamic encryption that resisted many types of attacks. Visually test represented by the histogram of the image before and after encryption is noted that the result is uniform and differs from the input one. The correlation of the output image is a low value which means no correlation is there between pixels horizontal, vertical and diagonal. The similarity test also applied such as MSE, PSNR and SSIM which also explain no similarity between plain and encrypted images. The randomness of generated numbers was tested using the standard test NIST for evaluation of the input key

Global forecasts in reservoir computers.

A reservoir computer is a machine learning model that can be used to predict the future state(s) of time-dependent processes, e.g., dynamical systems. In practice, data in the form of an input-signal are fed into the reservoir. The trained reservoir is then used to predict the future state of this signal. We develop a new method for not only predicting the future dynamics of the input-signal but also the future dynamics starting at an arbitrary initial condition of a system. The systems we consider are the Lorenz, Rossler, and Thomas systems restricted to their attractors. This method, which creates a global forecast, still uses only a single input-signal to train the reservoir but breaks the signal into many smaller windowed signals. We examine how well this windowed method is able to forecast the dynamics of a system starting at an arbitrary point on a system's attractor and compare this to the standard method without windows. We find that the standard method has almost no ability to forecast anything but the original input-signal while the windowed method can capture the dynamics starting at most points on an attractor with significant accuracy.

A comparative study on chaos control in a fractional-order Rossler system and its time delayed feedback control system

A comparative study on chaos control in a fractional-order Rossler system and its time delayed feedback control system

Fuzzy Memory Sampled-Data Controller Design for PMSG-Based WECS With Stochastic Packet Dropouts

The fuzzy memory sampled-data control problem for permanent magnet synchronous generator (PMSG)-based wind energy conversion system (WECS) with stochastic packet dropouts is discussed in this paper. To do this, the PMSG-based WECS dynamics are firstly modelled by a set of fuzzy IF-THEN rules of the Takagi–Sugeno (T–S) fuzzy model because of high nonlinearity. And then, to minimize design conservatism, a suitable augmented looped-type Lyapunov functional along with a novel integral inequality is established by using the sampling mode information with signal transmission delay. Further, a fuzzy memory-based sampled-data controller is designed to obtain the asymptotic stability condition under the circumstances of packet dropouts, and simultaneously, disturbance reduction is confirmed. Thereafter, the associated fuzzy controller gains can be computed by solving a number of linear matrix inequalities (LMIs). Finally, the application of the suggested method to a real-world system is proved using PMSG-based WECS, and the mathematical Rossler's system demonstrates the effectiveness and superiority of the proposed technique over the existing method.

Mixformer: An improved self-attention architecture applied to multivariate chaotic time series prediction

Multivariate chaotic time series prediction has been a challenging problem, especially the coupling relationship between multiple variables need to be carefully considered. We propose a self-attention architecture with an information interaction module, called mixformer, applied to the multivariate chaotic time series prediction task. First, Based on rethinking the multivariate data structure, we compensate the deficiency of phase space reconstruction with a group of cross-convolution operator that can automatically update parameters, and propose an explicitly designed feature reconstruction module. Then, to address the problem of interactive fusion of series information and channel information, we propose an information interaction module that enables the feature communication by expanding and contracting dimensions. By breaking the communication barrier between series information and channel information, the feature representation capability is enhanced. Finally, we construct mixformer that combines locally sparse features and global context features. And notably, mixformer integrates the information interaction module and the feature reconstruction module to form a continuous solution. By comparing the results with existing models using multiple simulated data (Lorenz, Chen, and Rossler systems) and real-world application (power consumption), it is verified that our proposed model achieves surprising performance in practice.

Asynchronous robust fuzzy event-triggered control of nonlinear systems

This paper deals with the problem of event-triggered integral sliding mode control (ISMC) for the perturbed nonlinear based-Takagi-Sugeno (TS) fuzzy systems. To reduce the chattering introduced by the unmatched disturbances, a disturbance observer is designed to get the estimated unmatched disturbances. To further reduce the computational burden and communication resources, compared with previous studies, two types of sliding surfaces are established. The time-triggered-based sliding surface which is used to ensure the reachability of the system states and the event-triggered-based sliding surface that is presented to calculate the control law, directly which benefits from less updates and consequently, less chattering that enhance the system performance. Moreover, the suggested control scheme is proposed based on asynchronous trait, i.e., the premise variables of the controller are in different time scale with the fuzzy systems which also leads to communication resource reduction. As a key issue in event-triggered control, it is ensured that the proposed controlled system is Zeno-free along the implementation. By the Lyapunov theory, the appropriate conditions for the ultimate boundedness stability (UBS) of the closed-loop system with a prescribed H∞ performance are derived in new offline LMIs. In order to illustrate the effectiveness of the suggested event-triggered integral sliding mode control, the Rossler system and track-trailer system are simulated and the obtained results are evaluated with the existing methods.

A Numerical Solution and Comparative Study of the Symmetric Rossler Attractor with the Generalized Caputo Fractional Derivative via Two Different Methods

This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison between the proposed methods and Runge–Kutta Fourth Order (RK4) is made. It is discovered that the proposed methods are effective and yield solutions that are identical to the approximate solutions produced by the other methods. Therefore, we can generalize the approach to other systems and obtain more accurate results. In addition to this, it has been shown to be useful for correctly discovering examples via the demonstration of attractor chaos. In the future, the two methods can be used to find the numerical solution to a variety of models that can be used in science and engineering applications.

Explosive death transitions in complex networks of limit cycle and chaotic systems

The conditions for explosive death transitions in complex networks of oscillators having generalized network topology, coupled via mean-field diffusion, are derived analytically. The network behaviour is characterized using three order parameters that define the average amplitude of oscillations, the mean state and the fraction of dead oscillators. As the mean field coupling is changed adiabatically, the amplitude order parameter undergoes explosive death transitions and the nodes in the network collectively cease to oscillate. The transition points in the parameter space and the boundaries of amplitude death regime are derived analytically. Sub-critical Hopf bifurcations are shown to be responsible for amplitude death transition. Explosive death transitions are characterized by hysteresis on adiabatic change of the bifurcation parameter and surprisingly, the backward transition point is shown to be independent of network topology. The theoretical developments have been validated through numerical examples, involving networks of limit cycle systems — such as the Van der Pol oscillator and chaotic systems, such as the Rossler attractor for both random and small world networks.

Dynamics of coupled quasiperiodic generator and Rossler system

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic system (Rossler system) is considered. The behavior of Lyapunov exponents is studied to identify possible types of system dynamics: chaos with additional zero Lyapunov exponents, three-frequency and two-frequency quasi-periodic regimes, periodic oscillations and the mode of oscillation death. Keywords: quasi-periodic oscillations, dynamical chaos, Rossler system, Lyapunov exponents.

WACPN: A Neural Network for Pneumonia Diagnosis.

Community-acquired pneumonia (CAP) is considered a sort of pneumonia developed outside hospitals and clinics. To diagnose community-acquired pneumonia (CAP) more efficiently, we proposed a novel neural network model. We introduce the 2-dimensional wavelet entropy (2d-WE) layer and an adaptive chaotic particle swarm optimization (ACP) algorithm to train the feed-forward neural network. The ACP uses adaptive inertia weight factor (AIWF) and Rossler attractor (RA) to improve the performance of standard particle swarm optimization. The final combined model is named WE-layer ACP-based network (WACPN), which attains a sensitivity of 91.87±1.37%, a specificity of 90.70±1.19%, a precision of 91.01±1.12%, an accuracy of 91.29±1.09%, F1 score of 91.43±1.09%, an MCC of 82.59±2.19%, and an FMI of 91.44±1.09%. The AUC of this WACPN model is 0.9577. We find that the maximum deposition level chosen as four can obtain the best result. Experiments demonstrate the effectiveness of both AIWF and RA. Finally, this proposed WACPN is efficient in diagnosing CAP and superior to six state-of-the-art models. Our model will be distributed to the cloud computing environment.

Nonintegrability of Dynamical Systems Near Degenerate Equilibria

We prove that general three- or four-dimensional systems %of differential equations are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincare-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar systems. Our result also implies that general three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations, respectively, are real-analytically nonintegrable under the weak conditions. To demonstrate these results, we give two examples for the Rossler system and coupled van der Pol oscillators.

Multi-Attn BLS: Multi-head attention mechanism with broad learning system for chaotic time series prediction

The observational 1-D signals available for realizing the highly accurate intrinsic attractor fitting of deep learning network approaches are often insufficient because of the complexity and nonlinearity of chaotic time series. Unlike deep models, a broad learning system (BLS) with the attention mechanism exhibits a unique and preeminent pattern prediction ability. Thus, this system has been applied as a practical trend in many fields. However, the application of multi-head attention fused manifold broad learning architecture to chaotic time series prediction remains inadequate. Thus, a multi-head attentional BLS (Multi-Attn BLS) for chaotic time series prediction is proposed in this study to improve the prediction accuracy of chaotic time series further. Our model develops a novel framework that combines the high computational efficiency of broad learning with the multi-head attention mechanism. First, the received data are reconstructed into fixed-size tuples. The multidimensional arrays with embedding dimensions and time delay are used as the input to a broad learning network. Subsequently, a robust BLS with a spatiotemporal multi-head attention mechanism is developed to depict the internal dynamic evolution. The Multi-Attn BLS model can capture key spatiotemporal feature information and achieve high predictive performance. It also has a good generalization ability in practical nonlinear complex systems. Comparative experiments with the traditional long short-term memory (LSTM) network and the primitive BLS show that its computing speed and generalization ability are improved. Furthermore, the network is good at capturing the spatiotemporal features of the sequence because of the multi-head attention mechanism. The experimental results show that our model outperforms BLS, ridge regression, and LSTM on the four main evaluation indicators (root mean square error, root mean square percentage error, mean absolute error, and mean absolute percentage error) in predicting classical systems (Lorenz and Rossler systems). Moreover, the model has an excellent prediction effect in the real-world chaotic system of sea clutter.

Intergroup Cascade Broad Learning System With Optimized Parameters for Chaotic Time Series Prediction

The prediction of chaotic time series can well describe the physical properties and influencing factors of a sequence of numerical data points in successive order and has played an increasingly crucial role in various scientific and engineering communities. These time series data require the prediction model to have strong dynamic feature representation ability, which often leads to excessive calculation burden. As an emerging and effective learning method, a broad learning system (BLS) can be trained quickly by incremental learning, and the system can be reconstructed without requiring a lengthy retraining process, which has been shown to facilitate superior time-saving performance. However, the historical information of time series data is often ignored, resulting in the lack of dynamic feature representation ability of the original BLS algorithm. In this article, a novel intergroup cascade BLS (ICBLS) method and its variant with optimized parameters, termed MOICBLS, are proposed, which can combine the previous knowledge with current information to determine the output results by constructing a novel intergroup cascade structure for the enhancement nodes. The new algorithms try to retain the time-saving advantage of BLS and greatly enhance the extraction ability of dynamic features of chaotic time series data. In addition, it cannot be ignored that the network structure parameters have an unquestionable impact on the performance of a BLS. In response, two conflicting goals of prediction accuracy and diversity are formulated as multiobjective optimization functions in the training phase to generate the optimized network structure parameters needed by multiple candidate models. The experimental results on two benchmark chaotic time series dataset, the Lorenz system and Rossler system, and a geomagnetic disturbance storm time chaotic time series dataset demonstrate that our proposed algorithms are effective for prediction of chaotic time series, and the prediction accuracy is better than some existing approaches.

PixAdapt: A novel approach to adaptive image encryption

Image encryption using genetic approach is a recent and advanced technique which has grabbed attention in recent years. Currently, most image encryption algorithms (using genetic approach) use a static set of parameters for image encryption without considering the features representative of the image. In this study, an innovative adaptive image encryption algorithm – PixAdapt is developed. The process of image encryption is being re-engineered in a way to calculate the fitness of encrypted image using UACI and adapting the respective parameters using genetic hill climb or simulated annealing. Pseudorandom numbers have been generated using the linear feedback shift register and chaos-based maps such as the Logistic map, Rossler map, Henon map and Tent map. PixAdapt algorithm also uses confusion and diffusion process to ensure that plain text image and cipher text image are completely un-related. The use of metaheuristic search techniques for optimization of image encryption parameters has been implemented for the first time. The results obtained show that the genetic hill climb algorithm encrypts the various images giving the most optimal value of UACI. The algorithm has been tested for fitness improvement, parameter evolution, statistical analysis, and quality of encryption. PixAdapt is not only unique but has proven the encryption parameter UACI to be an appropriate fitness function to encrypt an image efficiently.

Quaternion keyed Least Square Approximation for image encryption

Securing image data from prying hackers is crucial in safeguarding the secrecy of data. Over the years, this was done by encrypting the image using an algorithm and a key, where the visible image was converted into a meaningless object. It is a difficult problem to design an image encryption technique based on chaotic systems with predictable cryptographic features. In this paper, the Quaternion, along with the Rossler attractor, was used to generate the key combination. The ciphering was done using the Least Square Approximation Algorithm (LSA). The algorithm was tested on a grayscale image database. The algorithm was initially tested in software using MATLAB R2018b, and was implemented in the Cyclone II EP2C35F672C6 device FPGA. On average, for a cipher image, the Peak Signal to Noise ratio (PSNR) was 9.09303 dB and the entropy was 7.9990 bits. For the cipher image, the Number of Pixels Change Rate (NPCR) and Unified Average Change Intensity (UACI) were 99.6039 and 33.4980, respectively. This proved that the algorithm could effectively mitigate the statistical and differential attacks. The key space was 2 (M ×N ×7 ×8), which was sufficiently high and mitigated the brute force attacks. The obtained results confirm that the cipher images resulting from the proposed ciphering scheme possess good cryptographic properties in terms of entropy, PSNR, UACI, NPCR, and keyspace analysis. Furthermore, the strength of the key is evaluated by the NIST test suite.

Image Encryption using MVK Algorithm, El-Gamal and Chaotic Systems

Digital images are often transmitted in this modern era through various networks. Digital images are used in various fields like Medical, Military, etc. These images require high-security levels to prevent misusing and unauthorized access. There are several Image Encryption Techniques which are used to achieve image confidentiality; this paper presents a newer approach for Image Encryption which consists of 3 stages. MVK Algorithm is the first stage of encryption which is used to construct a permuted image and demolish the relationships among the adjacent pixels. El- Gamal Encryption Algorithm is the second stage of encryption which modifies permuted image; it is an alternative for RSA public key cipher. Third stage of encryption uses Lorenz system and Rossler system to achieve confusion and diffusion operations. To evaluate the effectiveness and security of the suggested strategy, a thorough analysis has been done. Visual and numerical results show that the proposed image cryptosystem can defend against a number of well-known attacks. Key Words: El-Gamal, Lorenz, Rossler.

Chaotic time series prediction using DTIGNet based on improved temporal-inception and GRU

To improve the prediction accuracy of chaotic time series, deep extraction of the system evolutionary patterns is a key problem in modeling. In this paper, we propose a deep learning model of automatic multi-scale feature extraction for chaotic time series prediction. A hybrid deep neural network named deep temporal-inception module and gated recurrent unit network (DTIGNet) is designed. The improved temporal-inception module stacks dilated causal convolution of different depth to increase the network adaptability to different scales and improve the network nonlinear characterization ability, and an optional 1 × 1 convolutional kernel as shared residual connection. The model is applied to the Mackey-Glass system, Rossler system, Lorenz system and sunspots time series to verify the applicability and effectiveness in chaotic time series prediction. The results show that the DTIGNet proposed has higher accuracy and better performance compared with other methods according to the 6 prediction evaluation metrics adopted.

Design of intelligent computing networks for nonlinear chaotic fractional Rossler system

In a recent development, the legacy of fractional-order multi-dimensional chaotic systems has attracted numerous researchers owing to their valuable applications in cryptology, control, information security, mathematics, and physical sciences. In this study, strength of artificial intelligence (AI) algorithms are explored for stochastic numerical computing for nonlinear fractional Rossler differential system by introducing state of art transformation in the radial basis neural networks (RNFN). Fractional order complex chaotic dynamical system of Rossler model represented by nonlinear coupled fractional differential equations are numerically solved with state of art fourth order fractional Runge-Kutta solver for different scenarios of control parameter and these outcomes are mensurate as reference solutions to model implementing RBFN with the different initial conditions of the chaotic dynamics. The innovative transformations are introduced to enhance the fast convergence of machine learning RBFN algorithm achieved through bimodal chaotic parameters. Performance of designed computing RBFN is authenticated efficaciously by RMSE metric for fractional order chaotic model of Rossler attractor with outcomes matching of the order of 10 to 15 decimal places in term of accuracy as compared to standard results computed from fractional Runge-Kutta solver. It is believed the proposed AI knacks based research work will open new innovative path in fractional order modelling and analysis of natural chaotic dynamic systems.

Three dimensional image encryption algorithm based on quantum random walk and multidimensional chaos

With the development of computer network technology, people’s requirements for information security is increasing day by day. However, the classical encryption technology has the defects of small key space and easy crack. The problems of image encryption technology in protecting image information security and private content need solving urgently. As a new type of quantum key generator, quantum random walk has a large key space. Compared with the classical random walk, the computing speed and security are significantly improved. This paper presents a three-dimensional image encryption algorithm that is based on quantum random walk and involves Lorenz and Rossler multidimensional chaos. Firstly, Gaussian pyramid is used to segment the image. Secondly, the Hamming distances of several sub images are calculated by using the random sequence generated by quantum random walk and the random sequence generated by Lorenz chaotic system in multi-dimensional chaos, and then synthesized, and the Euclidean distances between the three RGB channels of the image are calculated. Finally, the sequence value obtained from the remainder of Hamming distance and Euclidean distance, as an initial value is input into the Rossler system in multi-dimensional chaos to generate a random sequence which is used as the key to XOR the RGB channel of the image so as to create an encrypted image. The corresponding decryption scheme is the inverse process of the encryption process. In addition, in terms of transmission security, this paper uses a blind watermark embedding algorithm based on DCT and SVD to embed the watermark information into the encrypted image, so that the receiver can extract the watermark and judge whether the image is damaged by the attack in the transmission process according to the integrity of the watermark information. If it is not attacked maliciously, the image will be decrypted. This operation further improves the protection of image information security.The experimental results show that the peak signal-to-noise ratio of the encrypted image is stable between 7 and 9 and the encryption effect is good, the GVD score is close to 1, the correlation of the encrypted image is uniformly distributed, and the correlation coefficient is close to 0, and the key space is 2<sup>128</sup> in size and the encrypted histogram is evenly distributed, showing a high ability to resist statistical analysis attacks.

A method to detect the characteristics of intermittent generalized synchronization based on calculation of probability of the synchronous regime observation

A method to define the characteristic phases in the behavior of unidirectionally coupled systems located near the boundary of the generalized chaotic synchronization regime onset based on calculating the probability of the synchronous regime observation in an ensemble of coupled systems is proposed. Using the example of unidirectionally coupled Rossler systems in the band chaos regime, we have shown its efficiency in comparison with other known methods for detecting the characteristics of intermittent generalized synchronization. Keywords: generalized synchronization, intermittent behavior, unidirectionally coupled systems, multistability, laminar phases, probability of the laminar phase observation.