High-dimensional Chaotic Systems Research Articles

On convergence of linear response formulae in some piecewise hyperbolic maps

Abstract When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in applications, this differentiability, which is thought to be connected to the dimensionality of the system, has remained resistant to rigorous study outside of the one-dimensional setting. To model non-uniformly hyperbolic systems in multiple dimensions, we consider a family of the mathematically tractable class of piecewise smooth hyperbolic maps, the Lozi maps. For these maps, we prove that the existence of a formal derivative of the response reduces to conditional mixing of the SRB measure on the singularity set. Heuristically, this suggests that Lozi maps, and piecewise uniformly expanding maps more generally, should have linear response.

Prediction of chaotic dynamics and extreme events: A recurrence-free quantum reservoir computing approach

In chaotic dynamical systems, extreme events manifest in time series as unpredictable large-amplitude peaks. Although deterministic, extreme events appear seemingly randomly, which makes their forecasting difficult. By learning the dynamics from observables (data), reservoir computers can time accurately predict extreme events and chaotic dynamics, but they may require many degrees of freedom (large reservoirs). In this paper, by exploiting quantum-computer ansätze and entanglement, we design reservoir computers with compact reservoirs and accurate prediction capabilities. First, we propose the recurrence-free quantum reservoir computer (RF-QRC) architecture. By developing quantum feature maps and removing recurrent connections, the RF-QRC has quantum circuits with smaller depths. This allows the RF-QRC to scale well with higher-dimensional chaotic systems, which makes it suitable for hardware implementation. Second, we forecast the temporal chaotic dynamics and their long-term statistics of low- and higher-dimensional dynamical systems. We find that RF-QRC requires smaller reservoirs than classical reservoir computers for higher-dimensional systems and the same predictability. Third, we apply the RF-QRC to the time prediction of extreme events in a model of a turbulent shear flow with turbulent bursts. We find that the RF-QRC has longer predictability than the classical reservoir computer for extreme events forecasting. The results and analyses indicate that quantum-computer ansätze offers nonlinear expressivity and computational scalability, which are useful for forecasting chaotic dynamics and extreme events. This work opens new opportunities for using quantum machine learning on near-term quantum computers. Published by the American Physical Society 2024

Double-image encryption and authentication scheme based on compressed sensing and double random phase encoding

To enhance the security of image information, a double-image encryption and authentication scheme combining compressive sensing (CS) and double random phase encoding (DRPE) is proposed. First, two plaintext images are taken as real and imaginary parts to form a complex-valued image, which is then encoded using DRPE. Next, extract the phase of the encrypted complex-valued image and encode it into authentication information. Simultaneously, the complex-valued image is sampled using the measurement matrix, which is optimized by Schmidt orthogonalization, and then quantized to form a compressed image. Subsequently, the authentication information is embedded into the compressed image, and by applying the permutation and diffusion algorithm to it, the ciphertext image is obtained. At the receiving end, the ciphertext is decrypted using inverse permutation and inverse diffusion algorithms, and the authentication information is decoded to acquire the authentication image. Finally, synchronous authentication of the two reconstructed images is achieved using a nonlinear cross-correlation method. Additionally, the keys in the proposed scheme are generated using high-dimensional chaotic systems, effectively reducing the required storage space and transmission bandwidth. Simulation results demonstrate that the proposed scheme has high image reconstruction performance at different compression ratios and possesses outstanding security and authentication capabilities.

Chaos synchronization of two coupled map lattice systems using safe reinforcement learning

When synchronizing two continuous typical chaotic systems, the state variables of the response system are usually bounded and satisfy the Lipschitz condition. This permits a universal synchronization method. In the synchronization of two high-dimensional discrete chaotic systems such as the coupled map lattice systems, the state variables of the response system often diverge, which makes it difficult to seek a universal synchronization method. To overcome this difficulty, bounded and hard constraints, which correspond to the concept of safety level III in reinforcement learning terms, on the state variables must be imposed, when the discrete response system is perturbed. We propose a universal method for synchronizing two discrete systems based on safe reinforcement learning (RL). In this method, the RL agent’s policy is used to reach the goal of synchronization and a safety layer added directly on top of the policy is used to satisfy safety level III. The safety layer consists of a one-step predictor for the perturbed response system and an action correction formulation. The one-step predictor, based on a next generation reservoir computing, is used to identify whether the next state of the perturbed system is within the chaotic domain, and if not, the action correction formula is activated to modify the corresponding perturbing force component to zero. In this way, the state of the perturbed system will remain in the bounded chaotic domain. We demonstrate that the proposed method succeeds in synchronization without divergence through a numerical example with two coupled map lattice systems, and the average synchronization time is 12.66 iteration steps over 1000 different initial conditions. The method works even when parameter mismatches and disturbances exist in the system. We compare the performance in both cases with and without the safety layer and find that successful synchronization is only possible in the former case, emphasizing the significance of the safety layer. The performance of the algorithm with optimal hyper-parameters obtained by Bayesian optimization is robust and stable.

Construction method and circuit design of a high-dimensional conservative chaotic system with binary linear combinations

Construction method and circuit design of a high-dimensional conservative chaotic system with binary linear combinations

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.

An <i>n</i>-dimensional discrete hyperchaotic system and its application in audio encryption

Discrete chaotic system, as a pseudo-random signal source, plays a very important role in realizing secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an <i>n</i>-dimensional discrete hyperchaotic system with a simple structure. Firstly, the <i>n</i>-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed based on Jacobian matrix method that the system can have the positive Lyapunov exponents. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with those of the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, and the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed. The results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. According to this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resist various common attacks. It is also validated that the proposed system can be effectively applied to the field of audio encryption.

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.

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.