Lorenz Chaotic System Research Articles

Analysis of fractal fractional Lorenz type and financial chaotic systems with exponential decay kernels

<abstract><p>In this work, we formulate a fractal fractional chaotic system with cubic and quadratic nonlinearities. A fractal fractional chaotic Lorenz type and financial systems are studied using the Caputo Fabrizo (CF) fractal fractional derivative. This study focuses on the characterization of the chaotic nature, and the effects of the fractal fractional-order derivative in the CF sense on the evolution and behavior of each proposed systems. The stability of the equilibrium points for the both systems are investigated using the Routh-Hurwitz criterion. The numerical scheme, which includes the discretization of the CF fractal-fractional derivative, is used to depict the phase portraits of the fractal fractional chaotic Lorenz system and the fractal fractional-order financial system. The simulation results presented in both cases include the two- and three-dimensional phase portraits to evaluate the applications of the proposed operators.</p></abstract>

Investigation of numerical solutions for chaotic Lorenz system using Mathcad software

In this paper, we investigate the accuracy of numerical solution for chaotic Lorenz system. Mathcad software is used as case-study, built-in algorithms used are Runge-Kutta of fourth order (RK4), Adams backward differential formula (AdamsBDF) and developed series method. The numerical integration that are also checked involves Gauss-quadrature and Simpson’s quadrature rules. All the graphical results are showed for all the different numerical methods.

Event-triggered impulsive observers for continuous-time dynamical systems

This paper studies the issue of impulsive observers via event-triggered impulsive control (ETIC) for continuous-time dynamical systems (CDS). By using three levels of event conditions, the schemes of output-based ETIC including sampling-based ETIC are designed and full-order event-triggered impulsive observers are obtained. The impulse frequency and control strength of ETIC are also estimated. Moreover, in the case of CDS with unknown input, the notion of impulsive observer with K-asymptotic gain is proposed. And it is shown that the state from the observer can track the state of observed system within an error bound and the impulsive observer is with K-asymptotic gain. Specifically, when the unknown input trends to zero, the track can be completely achieved. Finally, based on the chaotic Lorenz system, an example is given to demonstrate the effectiveness of the theoretical results. And it is shown that for a same observer via impulsive control, the designed ETIC can achieve better performances with much lower impulse frequency and much less control strength than the classic impulsive control.

Investigation of numerical solutions for chaotic Lorenz system using Mathcad software

In this paper, we investigate the accuracy of numerical solution for chaotic Lorenz system. Mathcad software is used as case-study, built-in algorithms used are Runge-Kutta of fourth order (RK4), Adams backward differential formula (AdamsBDF) and developed series method. The numerical integration that are also checked involves Gauss-quadrature and Simpson’s quadrature rules. All the graphical results are showed for all the different numerical methods.

Enhanced synchronization due to intermittent noise

We propose a novel scheme to regulate noise infusion into the chaotic trajectories of uncoupled complex systems to achieve complete synchronization. So far the noise-induced synchronization utilize the uncontrolled noise that can be applied in the entire state space. Here, we consider the controlled (intermittent) noise which is infused in the restricted state space to realize enhanced synchronization. We find that the intermittent noise, which is applied only to a fraction of the state space, restricts the trajectories to evolve within the contraction region for a longer period of time. The basin stability of the synchronized states (SS) is found to be significantly enhanced compared to uncontrolled noise. Additionally, we uncover that the SS prevail for an extended range of noise intensity. We elucidate the results numerically in the Lorenz chaotic system, the Pikovski–Rabinovich circuit model and the Hindmarsh–Rose neuron model.

Forthcoming Issues

This paper studies synchronization of a dynamical complex network consisting of nodes being generalized Lorenz chaotic systems and connections created with transmitted synchronizing signals. The focus is on the robustness of the network synchronization with respect to its topology. The robustness is analyzed theoretically for the case of two nodes with two-sided (bidirectional) connections, and numerically for various cases with large numbers of nodes. It is shown that, unless a certain minimal coherent topology is present in the network, synchronization is always preserved. While for a minimal network where synchronization is global, the resulting synchrony reduces to semi-global if redundant connections are added.

Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing.

Reservoir computers are powerful tools for chaotic time series prediction. They can be trained to approximate phase space flows and can thus both predict future values to a high accuracy and reconstruct the general properties of a chaotic attractor without requiring a model. In this work, we show that the ability to learn the dynamics of a complex system can be extended to systems with multiple co-existing attractors, here a four-dimensional extension of the well-known Lorenz chaotic system. We demonstrate that a reservoir computer can infer entirely unexplored parts of the phase space; a properly trained reservoir computer can predict the existence of attractors that were never approached during training and, therefore, are labeled as unseen. We provide examples where attractor inference is achieved after training solely on a single noisy trajectory.

Novel hyper chaotic color image encryption based on pixel and bit level scrambling with diffusion

In this paper, a novel method for symmetric image encryption using Lorenz chaotic system is proposed. The main operations performed in the proposed system are keystream generation, permutation, and diffusion. In the permutation phase, both pixel-level and bit-level scrambling are performed to achieve the double scrambling property. The pixel-level scrambling is implemented by a novel approach using two different keystreams. Also, the bit-level scrambling is realized with a novel approach using cyclic shift operation and a keystream. Thus, the double scrambling ensures higher security in the permutation process. The images are encrypted by an enhanced diffusion process using another keystream. The permutation and diffusion processes use different keystreams generated from the Lorenz system with chaotic behavior. A novel method is adopted for the keystream generation phase. The security of the proposed cryptosystem has been assessed by various evaluation metrics like sensitivity analysis on key and cipher images, information entropy analysis, occlusion attack, and diverse statistical explorations such as correlation and histogram analyses. The experimental results show the supremacy and better performance of the proposed cryptosystem compared to several state-of-the-art techniques.

Circuit Implementation Synchronization between Two Modified Fractional-Order Lorenz Chaotic Systems via a Linear Resistor and Fractional-Order Capacitor in Parallel Coupling

In this study, a modified fractional-order Lorenz chaotic system is proposed, and the chaotic attractors are obtained. Meanwhile, we construct one electronic circuit to realize the modified fractional-order Lorenz chaotic system. Most importantly, using a linear resistor and a fractional-order capacitor in parallel coupling, we suggested one chaos synchronization scheme for this modified fractional-order Lorenz chaotic system. The electronic circuit of chaos synchronization for modified fractional-order Lorenz chaotic has been given. The simulation results verify that synchronization scheme is viable.

Research on the frequency synchronization control strategy for power system

Power system interconnection makes the modern power system become a huge and complex network, which brings great challenges to frequency stability. The pinning control strategy can reduce the control cost and has a unique advantage. It is one of the most effective method to solve the problem of complex network synchronization. Thus, based on the principle of pinning control strategy, a frequency synchronization control method is proposed to solve the problem of power grid frequency synchronization in this paper. Firstly, in the controller design phase, the pinning control model of a power system is defined based on mathematical model of hydroelectric generator. And the pinning controller is designed based on dynamic compensation LMI sliding mode control. The correctness of the controller is verified by the simulation of a Lorenz chaotic system. Secondly, to determine the controlled generator nodes, an index for evaluating the importance of the generator nodes is defined based on the node deletion method and fuzzy method. Finally, combined with the reality of power network, the external coupling matrix L is calculated based on the successive extended annular domain method and considering the ability of the nodes to receive signals. The feasibility of this method is verified by simulation of the IEEE30-bus system.

Design and FPGA implementation of TRNG based on a new multi-wing attractor in Lorenz chaotic system

Design and FPGA implementation of TRNG based on a new multi-wing attractor in Lorenz chaotic system

Stabilization and Synchronization of a Complex Hidden Attractor Chaotic System by Backstepping Technique.

In this paper, the stabilization and synchronization of a complex hidden chaotic attractor is shown. This article begins with the dynamic analysis of a complex Lorenz chaotic system considering the vector field properties of the analyzed system in the domain. Then, considering first the original domain of attraction of the complex Lorenz chaotic system in the equilibrium point, by using the required set topology of this domain of attraction, one hidden chaotic attractor is found by finding the intersection of two sets in which two of the parameters, r and b, can be varied in order to find hidden chaotic attractors. Then, a backstepping controller is derived by selecting extra state variables and establishing the required Lyapunov functionals in a recursive methodology. For the control synchronization law, a similar procedure is implemented, but this time, taking into consideration the error variable which comprise the difference of the response system and drive system, to synchronize the response system with the original drive system which is the original complex Lorenz system.

Projective synchronization of two coupled Lorenz chaotic systems in predefined time

Projective synchronization of two coupled Lorenz chaotic systems in predefined time

Cryptanalysis of hybrid secure image encryption based on Julia set fractals and three-dimensional Lorenz chaotic map

Recently a hybrid secure image encryption scheme based on Julia set and three-dimensional Lorenz chaotic system was suggested. The offered scheme was based on the multiplicative operation carried out by Julia set fractals and bitwise XOR which was implemented by the trajectories obtained from the Lorenz chaotic system. This paper presents the cryptanalysis of the scheme through a chosen-plaintext attack by taking three different chosen images. In this work, a practical chosen-plaintext attack is performed which reveals that the cryptosystem depending on only multiplication and bitwise XOR operation can be effortlessly attacked. The results of the analysis performed on the retrieved image reflect the effectiveness of the chosen-plaintext attack which fruitfully produces the original image.

Acoustic wireless communication based on parameter modulation and complex Lorenz chaotic systems with complex parameters and parametric attractors* *Project supported by the National Natural Science Foundation of China (Grant Nos. U1806202, 61773010, and 61903207), the International Collaborative Research Project of Qilu University of Technology (Grant No. QLUTGJHZ2018020), and Major Scientific and Technological Innovation Projects of Shandong Province, China (Grant Nos.

As the competition for marine resources is increasingly fierce, the security of underwater acoustic communication has attracted a great deal of attention. The information and location of the communicating platform can be leaked during the traditional underwater acoustic communication technology. According to the unique advantages of chaos communication, we put forward a novel communication scheme using complex parameter modulation and the complex Lorenz system. Firstly, we design a feedback controller and parameter update laws in a complex-variable form with rigorous mathematical proofs (while many previous references on the real-variable form were only special cases in which the imaginary part was zero), which can be realized in practical engineering; then we design a new communication scheme employing parameter modulation. The main parameter spaces of the complex Lorenz system are discussed, then they are adopted in our communication scheme. We also find that there exist parametric attractors in the complex Lorenz system. We make numerical simulations in two channels for digital signals and the simulations verify our conclusions.

Cancellable face recognition based on fractional-order Lorenz chaotic system and Haar wavelet fusion

Cancellable biometrics is the art of generating distorted or encrypted templates of original biometric templates. The evolution of cancellable biometrics is attributed to the advanced hacking technologies that can capture the original stored biometrics from databases. One of the solutions for this problem is to store cancellable biometric templates in the database rather than the original ones. This paper presents a cancellable face recognition scheme that is based on face image encryption with Fractional-Order (FO) Lorenz chaotic system. The basic idea is to generate user-specific random keys to be XORed with the red, green, and blue components of color face images. These keys are generated from the fractional-order Lorenz chaotic system. In addition, some post-processing is implemented on the encrypted color components of the face images with rotation and transposition of matrices. Finally, a wavelet fusion process is applied on these encrypted and processed face image components. The reason behind the utilization of wavelet fusion is to generate a single cancellable template for each color face image. Furthermore, wavelet fusion after post-processing of encrypted color components leads to a better degree of diffusion in the encrypted face templates. Actually, the encryption with the proposed algorithm is not full encryption, but it is appropriate for cancellable biometric applications. Moreover, the proposed scheme is secure due to the power of fractional-order Lorenz chaotic system that is very sensitive to initial conditions selected by the user. In addition, the post-processing incorporated with wavelet fusion is a non-invertible process. The validation of the proposed scheme is performed with experiments on FERET, LFW, and ORL databases. Evaluation metrics including Equal Error Rate (EER), Area under the Receiver Operating Characteristic (AROC) curve are utilized in the proposed scheme. Numerical values reveal EER levels close to zero and AROC values of 100% at low and mild noise levels.

An Encryption Architecture Suitable for on Chip Integration With Sensors

This paper demonstrates a hardware encryption algorithm that can be integrated directly with sensors on the same chip. In contrast to using power-hungry microprocessors and software implementations, the encryption algorithm was based on a Lorenz chaotic system and was implemented as time scaling chaotic shift keying (TS-CSK). The coupled differential equations of TS-CSK were implemented using low power integrators, multiplexers, switches, and passive components. Both a single-ended and differential system was developed. The single-ended based system showed no oscillation, while the differential based system demonstrated oscillation. Experimental measurements of the differential system with an integrated temperature sensor demonstrated the ability to encrypt and decrypt the transmitted signal. The differential encryption/decryption system was implemented in 180 nm technology and operated with a 1.8 V power supply and 1.5 mW of power.

A image encryption algorithm based on chaotic Lorenz system and novel primitive polynomial S-boxes

We have proposed a robust, secure and efficient image encryption algorithm based on chaotic maps and algebraic structure. Nowadays, the chaotic cryptosystems gained more attention due to their efficiency, the assurance of robustness and high sensitivity corresponding to initial conditions. In literature, there are many encryption algorithms that can simply guarantees security while the schemes based on chaotic systems only promises the uncertainty, both of them can not encounter the needs of current scenario. To tackle this issue, this article proposed an image encryption algorithm based on Lorenz chaotic system and primitive irreducible polynomial substitution box. First, we have proposed 16 different S-boxes based on projective general linear group and 16 primitive irreducible polynomials of Galois field of order 256, and then utilized these S-boxes with combination of chaotic map in image encryption scheme. Three chaotic sequences can be produced by the disturbed of Lorenz chaotic system corresponding to variables x, y and z. We have constructed a new pseudo random chaotic sequence ki based on x, y and z. The plain image is encrypted by the use of chaotic sequence ki and XOR operation to get a ciphered image. To show the strength of presented image encryption, some renowned analyses are performed.

Xilinx Zynq FPGA for Hardware Implementation of a Chaos-Based Cryptosystem for Real-Time Image Protection

This paper proposes a well-optimized FPGA implementation of a chaos-based cryptosystem for real-time image encryption and decryption. A highly sensitive Pseudo-Random Number Generator (PRNG) based on the Lorenz chaotic system is designed to generate pseudo-random numbers. A high-quality encryption key is acquired by encrypting the numbers of a 128-bit counter using the PRNG. For image encryption, the image is first decomposed into 128-bit blocks. Then, the blocks are encrypted by XORing pixels with the key stream. [Formula: see text] cycles of encryption can be achieved to increase complexity. Finally, the blocks are concatenated to form an encrypted image. The algorithm is designed, implemented, and validated on the Xilinx Zynq FPGA platform using the Vivado/System Generator tool. The hardware design is well optimized for pipeline processing and low resource utilization. The experimental synthesis indicates that the provided architecture achieves high performance in terms of frequency and throughput. The encryption scheme is evaluated and analyzed by several tools and tests using different images. The experimental simulation results demonstrate that the hardware implementation is faster than a software implementation while maintaining the technique’s effectiveness. The proposed hardware implementation is extremely adopted for secret image encryption and decryption that can be used in real-time applications.

Teaching recurrent neural networks to infer global temporal structure from local examples

The ability to store and manipulate information is a hallmark of computational systems. Whereas computers are carefully engineered to represent and perform mathematical operations on structured data, neurobiological systems adapt to perform analogous functions without needing to be explicitly engineered. Recent efforts have made progress in modelling the representation and recall of information in neural systems. However, precisely how neural systems learn to modify these representations remains far from understood. Here, we demonstrate that a recurrent neural network (RNN) can learn to modify its representation of complex information using only examples, and we explain the associated learning mechanism with new theory. Specifically, we drive an RNN with examples of translated, linearly transformed or pre-bifurcated time series from a chaotic Lorenz system, alongside an additional control signal that changes value for each example. By training the network to replicate the Lorenz inputs, it learns to autonomously evolve about a Lorenz-shaped manifold. Additionally, it learns to continuously interpolate and extrapolate the translation, transformation and bifurcation of this representation far beyond the training data by changing the control signal. Furthermore, we demonstrate that RNNs can infer the bifurcation structure of normal forms and period doubling routes to chaos, and extrapolate non-dynamical, kinematic trajectories. Finally, we provide a mechanism for how these computations are learned, and replicate our main results using a Wilson–Cowan reservoir. Together, our results provide a simple but powerful mechanism by which an RNN can learn to manipulate internal representations of complex information, enabling the principled study and precise design of RNNs. Recurrent neural networks (RNNs) can learn to process temporal information, such as speech or movement. New work makes such approaches more powerful and flexible by describing theory and experiments demonstrating that RNNs can learn from a few examples to generalize and predict complex dynamics including chaotic behaviour.