High-dimensional Chaotic Systems Research Articles

A 5-D memristive hyperchaotic system with extreme multistability and its application in image encryption

By coupling memristors to nonlinear circuits, more complex dynamical behaviors can be induced. However, to date, there has been insufficient attention given to high-dimensional chaotic systems based on memristors. In this paper, a magnetic-controlled memristor is combined with a three-dimensional chaotic system, resulting in a five-dimensional memristive chaotic system. Through dynamic analysis and numerical simulations, the chaotic nature of the system is elucidated based on fundamental system behaviors, including Lyapunov dimension, dissipativity, stability of equilibrium points, 0–1 test, and Poincaré mapping. During the complex dynamical analysis of this system, unique dynamical behaviors are discovered, including intermittent chaos, transient chaos, extreme multistability, and offset-boosting. Moreover, the consistency between numerical calculations and the physical implementation of the actual system is verified through equivalent circuit design. Finally, this system is applied to image encryption, leading to the design of an efficient and secure hyper-chaotic image encryption algorithm, whose effectiveness is confirmed through several security tests.

Hybrid synchronisation method based on inverse generalised and inverse projected high dimensional discrete chaotic systems

High-dimensional discrete chaotic systems have a wide range of engineering applications, while the chaotic synchronization method is the key to confidential communication applications. Based on the proposed discriminant theorem for high-order polynomial chaotic mapping, in this paper, a hybrid inverse generalization and inverse projection synchronization method for high-dimensional discrete chaotic systems is constructed. The method increases the flexibility of synchronization control by designing to enable the coexistence of inverse projection synchronization and inverse generalized synchronization in chaotic systems, by specifically using an invertible and adjustable constant diagonal matrix to extend the control capability of the error system. Meanwhile, this paper proposed the validity of this hybrid synchronization scheme based on the multi-stationary chaotic system for the first time, through comparison, it is certified that the multi-stationary discrete chaotic synchronization system introduced in this paper not only has complex dynamics behavior but also has a faster synchronization speed. Meanwhile, this paper proposed the validity of this hybrid synchronization scheme based on the multi-stationary chaotic system for the first time, through comparison, it is certified that the multi-stationary discrete chaotic synchronization system introduced in this paper not only has complex dynamics behavior but also has a faster synchronization speed. Finally, in this paper, the hybrid synchronization-based encryption system is also constructed, in which the transmitter switches the chaotic system between different attractors by changing the initial conditions, and then uses different chaotic attractors to mask the plaintext information. The experimental results show that the system has higher security and larger key space.

Prediction and control of spatiotemporal chaos by learning conjugate tubular neighborhoods

I present a data-driven predictive modeling tool that is applicable to high-dimensional chaotic systems with unstable periodic orbits. The basic idea is using deep neural networks to learn coordinate transformations between the trajectories in the periodic orbits’ neighborhoods and those of low-dimensional linear systems in a latent space. I argue that the resulting models are partially interpretable since their latent-space dynamics is fully understood. To illustrate the method, I apply it to the numerical solutions of the Kuramoto–Sivashinsky partial differential equation in one dimension. Besides the forward-time predictions, I also show that these models can be leveraged for control.

Statistical physics of learning in high-dimensional chaotic systems

In many complex systems, elementary units live in a chaotic environment and need to adapt their strategies to perform a task by extracting information from the environment and controlling the feedback loop on it. One of the main examples of systems of this kind is provided by recurrent neural networks. In this case, recurrent connections between neurons drive chaotic behavior, and when learning takes place, the response of the system to a perturbation should also take into account its feedback on the dynamics of the network itself. In this work, we consider an abstract model of a high-dimensional chaotic system as a paradigmatic model and study its dynamics. We study the model under two particular settings: Hebbian driving and FORCE training. In the first case, we show that Hebbian driving can be used to tune the level of chaos in the dynamics, and this reproduces some results recently obtained in the study of more biologically realistic models of recurrent neural networks. In the latter case, we show that the dynamical system can be trained to reproduce simple periodic functions. To do this, we consider the FORCE algorithm—originally developed to train recurrent neural networks—and adapt it to our high-dimensional chaotic system. We show that this algorithm drives the dynamics close to an asymptotic attractor the larger the training time. All our results are valid in the thermodynamic limit due to an exact analysis of the dynamics through dynamical mean field theory.

Heterogeneity of the attractor of the Lorenz ’96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties

It is well known that the predictability of weather and climate is strongly state-dependent. Special, easily recognisable, and extremely relevant atmospheric states like blockings are associated with anomalous instability. This reflects the general property that the attractors of chaotic systems can feature considerable heterogeneity in terms of dynamical properties, and specifically, of their instability. The attractor of a chaotic system is densely populated by unstable periodic orbits (UPOs) that can be used to approximate any forward trajectory through the so-called shadowing. Dynamical heterogeneity can lead to the presence of UPOs with different number of unstable dimensions. This phenomenon – unstable dimensions variability – is a serious breakdown of hyperbolicity and has considerable implications in terms of the structural stability of the system and of the possibility to model accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, here we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz ’96 model. By combining the Lyapunov analysis of the tangent space with the study of the shadowing of the chaotic trajectory performed by a very large set of UPOs, we show that the observed variability in the number of unstable dimensions is associated with the presence of a substantial number of finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered. The transition between regions of the attractor with different degrees of instability is associated with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing UPOs, we are able to characterise the slow fluctuations of the system between regions featuring, on the average, anomalously high and anomalously low instability. In turn, such regions are associated, respectively, with states of anomalously high and low energy, thus providing a clear link between the microscopic and thermodynamical properties of the system.

A class of 2n+1 dimensional simplest Hamiltonian conservative chaotic systems and fast image encryption schemes

Compared with dissipative systems, conservative systems do not have attractors, thus they can better cope with the security risks brought by phase space reconstruction. In this paper, a paradigm for constructing 2n+1 dimensional simplest Hamiltonian conservative chaotic systems (HCCSs) is proposed, and their Hamiltonian energy conservation, Casimir energy conservation, volume conservation, and boundedness are proved. Based on this construction paradigm, we propose a five-dimensional HCCS (FHCCS) that contains the fewest terms but has complex dynamic behaviors, including energy-controlled hyperchaotic multi-scroll flow, hyperchaos under wide parameter range, energy-induced extremely multistability, initial value enhancement of hyperchaos. To solve the problem of high-dimensional continuous chaotic systems seriously slowing down the speed of chaotic image encryption, a new fractal scrambling and batch diffusion algorithm is proposed. They only need a small number of chaotic sequences to achieve excellent encryption effects, such as strong plaintext sensitivity, strong key sensitivity, high information entropy, low correlation, and huge key space. After sufficient testing, on the Matlab platform, for high-resolution images with a resolution of 2048×2048, only 0.186886s is needed to generate all pseudorandom numbers required for encryption. In addition, they can also make full use of the SIMD technology provided by the CPU to maximize the encryption speed.

Optimizing the combination of data-driven and model-based elements in hybrid reservoir computing.

Hybrid reservoir computing combines purely data-driven machine learning predictions with a physical model to improve the forecasting of complex systems. In this study, we investigate in detail the predictive capabilities of three different architectures for hybrid reservoir computing: the input hybrid (IH), output hybrid (OH), and full hybrid (FH), which combines IH and OH. By using nine different three-dimensional chaotic model systems and the high-dimensional spatiotemporal chaotic Kuramoto-Sivashinsky system, we demonstrate that all hybrid reservoir computing approaches significantly improve the prediction results, provided that the model is sufficiently accurate. For accurate models, we find that the OH and FH results are equivalent and significantly outperform the IH results, especially for smaller reservoir sizes. For totally inaccurate models, the predictive capabilities of IH and FH may decrease drastically, while the OH architecture remains as accurate as the purely data-driven results. Furthermore, OH allows for the separation of the reservoir and the model contributions to the output predictions. This enables an interpretation of the roles played by the data-driven and model-based elements in output hybrid reservoir computing, resulting in higher explainability of the prediction results. Overall, our findings suggest that the OH approach is the most favorable architecture for hybrid reservoir computing, when taking accuracy, interpretability, robustness to model error, and simplicity into account.

An Applied Image Cryptosystem on Moore’s Automaton Operating on δ ( q k )/𝔽 2

The volume of multimedia-based image data or video frames in Web 3.0 is constantly increasing, owing to the advancement of real-time data transmission. However, security vulnerabilities frequently impair the performance of real-time applications. Many researchers have recently proposed image encryption schemes based on a high-dimensional chaotic system due to properties such as ergodicity and initial state sensitivity. Nonetheless, most schemes have suffered from excessive computational complexity, low security, and the generation of cryptographically secure random numbers. To overcome these challenges, an efficient and highly secure cryptosystem is necessary for safe multimedia transmission in Web 3.0. This article proposes a novel work on the image cryptosystem based on the Escalation function with a one-time key-oriented Moore’s Automaton over a finite field 𝔽 2 . The Escalation function is a nonlinear scrambling technique for plaintext images that goes through the confusion phase and plays an essential role in row-column permutation. To make the algorithm more secure and robust in the diffusion phase, the proposed Moore’s Automaton produced ciphertext images through a highly random key stream generated by the combination of a logistic map and cyclic group. Specifically, the proposed Moore’s Automaton operates on δ ( q k )/𝔽 2 to render random binary bits into unpredictable sequences to construct ciphertext images. Our new finding quickens the speed and provides adequate key space, and pixel distributions are more uniform, have high entropy value, and are secure against differential and statistical attacks.

Reversible Data Hiding in Encrypted Images Based on an Adaptive Recognition Strategy for Blocks

As the rapid development of third-party storage and homomorphic encryption have profoundly stimulated the desire for secure communication, reversible data hiding in encrypted images has received widespread attention, since it allows lossless data conveying and perfect image recovery. In order to obtain secure reversible data hiding with high embedding capacity, a novel block embedding method is proposed, based on an adaptive recognition strategy for combined blocks in the binary image, with which the adjacent identical blocks can be integrated into a combination to reserve more spare bits for data accommodation. Furthermore, a fully reversible data hiding method for grayscale images in the encryption domain is designed. The secret data is hidden into lower bit-planes of the image while the original bits of those embedded lower pixels are recorded into the vacated space of higher bit-planes. The original image can be reconstructed flawlessly as well as the secret data being extracted without errors. To reinforce security, the original image and the secret data are encrypted and scrambled based on sequences generated with the high-dimension chaotic system. Due to its high sensitivity of initial values, the performance such as security and robustness is guaranteed. By comparing the PSNR value of the marked decrypted image and evaluating the quality of the extracted secret image, experimental results demonstrate that the proposed method can obtain higher embedding capacity, achieving 0.2700–0.3924 bpp increment over the state-of-the-art methods, and recover the marked decrypted image with high visual symmetry/quality, and efficiently resist against potential attacks, such as the histogram analysis, differential, brute-force, JPEG attacks, and so on.

Numerical Simulation for a High-Dimensional Chaotic Lorenz System Based on Gegenbauer Wavelet Polynomials

We provide an effective simulation to investigate the solution behavior of nine-dimensional chaos for the fractional (Caputo-sense) Lorenz system using a new approximate technique of the spectral collocation method (SCM) depending on the properties of Gegenbauer wavelet polynomials (GWPs). This technique reduces the given problem to a non-linear system of algebraic equations. We satisfy the accuracy and efficiency of the proposed method by computing the residual error function. The numerical solutions obtained are compared with the results obtained by implementing the Runge–Kutta method of order four. The results show that the given procedure is an easily applied and efficient tool to simulate this model.

Protection-Enhanced Watermarking Scheme Combined With Non-Linear Systems

Communication of the images over the network calls for an approach to ensure security and ownership protection while exchanging images over the network. This paper proposes an improved blind watermarking scheme capable of offering image authentication, protection, and detachment avoidance between the watermark and its corresponding image. The proposed technique is composed of three modules; generator, protector, and resizer. The generator deals with embedding and extracting the watermark from the host image. Protector deals with encryption and decryption using a high dimensional chaotic system. And resizer deals with compressing and decompressing the watermarked image. The novelty related scientific literature is the encryption of Singular Value Decomposition (SVD) components, and watermark in a criss-cross manner using logistic map along with hyperchaotic system to achieve the confidentiality and avoid false positive problem (FPP). Whereas, embedding into RGB (red, green, blue components) is centered on singular value decomposition and level-2 Discrete Wavelet Transform. The joint operation is performed at sender’s side and receiver’s side, i.e., encryption, embedding, and compression is performed at sender’s side whereas decompression and decryption procedures are performed at the receiver’s side. Additionally, compressing the data reduces the higher bandwidth requirements. The performance of the proposed scheme is based on subjective and objective evaluation. Experimental results such as PSNR (peak signal-to-noise ratio), NC (normalized correlation), BER (bit error rate), CR (compression ratio), NPCR (number of pixel change rate), and SSIM (structural similarity index) indicate that the proposed technique approaches the standard main level.

Machine learning-assisted extreme events forecasting in Kerr ring resonators

Predicting complex nonlinear dynamical systems has been even more urgent because of the emergence of extreme events such as earthquakes, volcanic eruptions, extreme weather events (lightning, hurricanes/cyclones, blizzards, tornadoes), and giant oceanic rogue waves, to mention a few. The recent milestones in the machine learning framework o↵er a new prospect in this area. For a high dimensional chaotic system, increasing the system’s size causes an augmentation of the complexity and, finally, the size of the artificial neural network. Here, we propose a new supervised machine learning strategy to locally forecast bursts occurring in the turbulent regime of a fiber ring cavity.

Invariant tori in dissipative hyperchaos.

One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems, including fluid turbulence, while higher-dimensional invariant tori representing quasiperiodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; tori can be numerically identified via bifurcations of unstable periodic orbits and their parameteric continuation and characterization of stability properties are feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.

Novel One-Dimensional Chaotic System and Its Application in Image Encryption

Designing a chaotic system with a simple structure and complex dynamic behavior is one of the main tasks of chaotic cryptography. This paper designs a new 1D chaotic system called 1D two-parameters-sin-cos (1D-TPSC). Compared with high-dimensional chaotic systems, the 1D-TPSC has a simple structure and is easy to implement with software. The Lyapunov exponent analyzes the parameter space of the 1D-TPSC in a chaotic state. Furthermore, using sensitivity analysis, cobweb plot, and bifurcation diagram to verify that the sequence generated by 1D-TPSC has good performance. In addition, the 1D-TPSC has also been applied in chaotic image encryption. Arnold mapping is used to scramble the plaintext, and random XOR is used to diffuse the scrambled image. Simulation experiments show that the method can remarkably resist standard attack methods.

FPGA-Based Implementation and Synchronization Design of a New Five-Dimensional Hyperchaotic System.

Considering the security of a communication system, designing a high-dimensional complex chaotic system suitable for chaotic synchronization has become a key problem in chaotic secure communication. In this paper, a new 5-D hyperchaotic system with high order nonlinear terms was constructed and proved to be hyperchaotic by dynamical characterization characteristics, the maximum Lyapunov exponent was close to 2, and there was a better permutation entropy index, while a valid chaotic sequence could be generated in three cycles in the FPGA (Field Programmable Gate Array)-based implementation. A multivariable nonlinear feedback synchronous controller based on FPGA was proposed to design and implement synchronization of high order complex hyperchaotic systems. The results show that the error signal converged to 0 rapidly under the effect of the nonlinear feedback synchronous controller. This lays the foundation for the synchronization of high order complex chaotic systems.

A parallel image encryption algorithm using intra bitplane scrambling

More and more privacy-sensitive images have been produced daily. Chaotic encryption guarantees that only the targeted receiver can obtain the raw image. Many works perform encryption on bit-level for better permutation but incur an unbearable time consumption caused by generating additional 3D coordinates. High-dimensional chaotic system, suitable for encrypting images, needs to iterate repeatedly for several rounds, resulting in low encrypting efficiency. This paper proposes a parallel image encryption algorithm using intra bitplane scrambling. Four threads are utilized to encrypt the images in bit-level during the permutation, where each thread scrambles two bitplanes. In diffusion, we build multiple threads to generate keystream segments to reduce encryption time. We make cryptographic analyses and compare the encryption speed with the advanced works, demonstrating that our scheme is secure and efficient.

Anti-control of Hopf bifurcation for high-dimensional chaotic system with coexisting attractors

Anti-control of Hopf bifurcation for high-dimensional chaotic system with coexisting attractors

Counteracting the dynamic degradation of high-dimensional digital chaotic systems via a stochastic jump mechanism

To analyze the periodic distribution and state-mapping structures of high-dimensional digital chaotic systems in fixed-point arithmetic domain. A chaotic cycle-finding algorithm (CCFA) can be designed to precisely calculate the period length of the orbit of digital chaos under different finite calculation accuracy. Furthermore, to test the versatility and effectiveness of our proposed algorithm, the periodic distribution and security of the digital Sprott D system are analyzed in detail. Numerical simulation results show that the chaotic system have potential security risks, such as short period and low sequence complexity. In addition, we put forward a new resistance method based on automatic cycle detection and random jump of chaotic trajectories for the objective of enhancing the dynamical properties of high-dimensional digital chaotic systems. This scheme can construct a new chaotic model with many desirable properties, such as long period, high state-space utilization, low autocorrelation, and good ergodicity.

Precursors-driven machine learning prediction of chaotic extreme pulses in Kerr resonators

Machine learning algorithms have opened a breach in the prediction's fortress of high-dimensional chaotic systems. Their ability to find hidden correlations in data can be exploited to perform model-free forecasting of spatiotemporal chaos and extreme events. However, the extensive feature of these evolutions makes up a critical limitation for full-size forecasting processes. Hence, the main challenge for forecasting relevant events is to establish the set of pertinent information. Here, we identify precursors from the transfer entropy of the system and a deep Long Short-Term Memory network to forecast the complex dynamics of a system evolving in a high-dimensional spatiotemporal chaotic regime. Performances of this triggerable model-free prediction protocol based on the information flowing map are tested from experimental data originating from a passive resonator operating in such a complex nonlinear regime. We have been able to predict the occurrence of extreme events up to 9 round trips after the detection of precursor, i.e., 3 times the horizon provided by Lyapunov exponents, with 92% of true positive predictions leading to 60% of accuracy.We have implemented a process to forecast extreme events in a fully developed turbulent flow. The novelty of our strategy lies in the information's use theory method to detect precursors and use them as the input of a neural network to infer the incoming extreme events. Our process is suitable for all the extended dissipative systems that can only be partially observed or real-world data.

Non-hyperbolicity at large scales of a high-dimensional chaotic system

The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively uniformly hyperbolic, which implies many felicitous statistical properties. We obtain direct and reliable numerical evidence, contrary to the chaotic hypothesis, of the existence of non-hyperbolic large-scale dynamical structures in a mean-field coupled system. To do this, we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev basis transfer operator discretization. This enables us to obtain a high-precision estimate of a homoclinic tangency, implying a failure of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be a priori assumed to hold in all systems, and a better understanding of the domain of its validity is required.