Chaotic Systems Research Articles

Does a Fractional-Order Recurrent Neural Network Improve the Identification of Chaotic Dynamics?

This paper presents a quantitative study of the effects of using arbitrary-order operators in Neural Networks. It is based on a Recurrent Wavelet First-Order Neural Network (RWFONN), which can accurately identify several chaotic systems (measured by the mean square error and the coefficient of determination, also known as R-Squared, r2) under a fixed parameter scheme in the neural algorithm. Using fractional operators, we analyze whether the identification capabilities of the RWFONN are improved, and whether it can identify signals from fractional-order chaotic systems. The results presented in this paper show that using a fractional-order Neural Network does not bring significant advantages in the identification process, compared to an integer-order RWFONN. Nevertheless, the neural algorithm (modeled with an integer-order derivative) proved capable of identifying fractional-order dynamical systems, whose behavior ranges from periodic and multi-stable to chaotic oscillations. That is, the performances of the Neural Network model with an integer-order derivative and the fractional-order network are practically identical, making the use of fractional-order RWFONN-type networks meaningless. The results deepen the work previously published by the authors, and contribute to developing structures based on robust and generic neural algorithms to identify more than one chaotic oscillator without retraining the Neural Network.

Transforming chaotic and stiff systems to improve numerical accuracy

AbstractSystems of differential equations can exhibit chaotic or stiff behavior under specific conditions, posing challenges for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is shown that chaotic differential equations can indeed be transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge–Kutta solution of the Lorenz chaotic equations can be increased by several orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy. The transform method applies broadly to chaotic systems, including weather prediction and turbulence.

A continuous-discrete adaptive observer with a diagonal gain matrix and sigma-modification: applications in chaos synchronization and dynamical state estimation

In this paper, we propose an improved continuous-discrete observer with enhanced convergence properties and reduced sensitivity to persistent excitations. Our approach introduces several key innovations, including a customised diagonal gain matrix which improves the convergence speed, a sigma-modification technique that relaxes the persistent excitation conditions, and a weighting factor tailored to optimise the estimation of unknown parameters. Through comprehensive numerical experiments, we illustrate the effectiveness of our observer in two distinct scenarios. Firstly, we demonstrate its application in synchronising a 4-component chaotic dynamical system. Secondly, we showcase its utility in tracking a reduced 2-component chemical engineering process. Our simulation results reveal that the proposed observer outperforms existing methods by achieving quicker convergence and higher estimation accuracy. This study contributes to the advancement of adaptive observer design, offering valuable insights for its application across diverse fields.

FPGA Implementation for a Class of Generalized Hamiltonian Conservative Chaotic Systems Based on Integrated 4D Euler Equations with Multistability

In this paper, two newly proposed generalized Hamiltonian Conservative Chaotic Systems (CCSs) based on integrated 4D Euler equations are first discussed. Second, another generalized Hamiltonian CCS is also proposed. Furthermore, a class of generalized Hamiltonian CCSs based on integrated 4D Euler equations is given and studied. Subsequently, numerical analysis for the class of generalized Hamiltonian CCSs is further investigated to show their advantages over most of the existing CCSs, where the corresponding Lyapunov exponents’ diagrams, bifurcation diagrams and phase portraits are all given to show their multistability and complex dynamics. Finally, the class of generalized Hamiltonian CCSs is implemented by using FPGA technology, and the results observed in FPGA implementation are consistent with those observed in the numerical analysis. All these results not only show multistability from a physical point of view, but also provide new physical models for chaos applications.

Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems

Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems

Implementation of Chaotic Neural Key Generation Algorithm For IoT Devices

This paper presents a new method for generating encryption keys for Internet of Things (IoT) devices. This method combines chaos theory with neural networks to create secure and efficient keys. We use the Lorenz chaotic system to generate complex patterns and a Convolutional Neural Network (CNN) to learn and predict these patterns. This approach is designed to address the unique security challenges of IoT devices, which often have limited computing power. We tested our method with different key sizes and evaluated its performance using accuracy, loss, entropy, and correlation metrics. The results show that key sizes between 256 and 512 bits offer the best balance between model performance and security for IoT devices. We also conducted Diehard statistical tests, which our key generation method passed successfully, demonstrating its ability to produce high-quality random keys.

An image encryption algorithm for colour images based on a cellular neural network and the Chua's chaotic system

In the internet era, image encryption technology plays a crucial role in ensuring the security of image information. Aiming at the problem that low-dimensional chaotic encryption has a relatively small key space and is vulnerable to chosen-plain attacks, this paper proposed a novel colour image block encryption algorithm based on the cellular neural network and the Chua's chaotic system. Firstly, the plain is separated into three primary colours. And Fourier transform is applied to achieve optical transformation, generating six chaotic sequences iteratively. Secondly, the chaotic sequences were combined in pairs. And then combined with the plain to form three sequences, which were used in the diffusion process. Finally, the improved Chua's chaotic system is used to scramble the image. Both diffusion and scrambling are performed synchronously during encryption. Through the above operation, we get the final colour encrypted picture. The experimental results show that the algorithm has excellent encryption effect and can resist common attacks.

Discovering dynamics and parameters of nonlinear oscillatory and chaotic systems from partial observations

Despite rapid progress in data acquisition techniques, many complex physical, chemical, and biological systems remain only partially observable, thus posing the challenge to identify valid theoretical models and estimate their parameters from an incomplete set of experimentally accessible time series. Here, we combine sensitivity methods and ranked-choice model selection to construct an automated hidden dynamics inference framework that can discover predictive nonlinear dynamical models for both observable and latent variables from noise-corrupted incomplete data in oscillatory and chaotic systems. After validating the framework for prototypical FitzHugh-Nagumo oscillations, we demonstrate its applicability to experimental data from squid neuron activity measurements and Belousov-Zhabotinsky reactions, as well as to the Lorenz system in the chaotic regime. Published by the American Physical Society 2024

Enhancing network security with hybrid feedback systems in chaotic optical communication

This paper presents a pioneering approach to bolstering network security and privacy by implementing chaotic optical communication with a hybrid optical feedback system (HOFS). The current baseline methods in network security are often less feasible for hybrid feedback systems, including limited robustness, compromised security, and synchronization challenges. Therefore, this paper proposes a hybrid approach to address these shortcomings by integrating the HOFS into chaotic optical communication systems (HOFS-COCS) to overcome the baseline challenges. This paper aims to improve network security while significantly maintaining efficient communication channels. Moreover, We designed two algorithms, one for chaotic maps generation and another for text encryption and decryption, to improve security in the hybrid feedback system. Our findings demonstrate through rigorous experimentation and analysis that the proposed (HOFS-COCS) method significantly improves network security by enabling reliable chaos generation, synchronization, and secure message transmission in chaotic optical communication systems. This research represents a significant advancement towards enhanced secrecy and synchronization in chaotic optical communication systems, promising a paradigm shift in network security protocols.

Three-Dimensional Space Chaotic Scrolls Generated by Memristive Hopfield Neural Network and Its Application in Robot Path Planning

Abstract In this paper, a novel three-dimensional space-scroll chaotic attractor is constructed by introducing piece-wise linear memristors into a three-neurons-based Hopfield neural network to simulate the electromagnetic influence on neural dynamics. The dynamical behaviors of the novel memristive Hopfield neural network are discussed by means of numerical methods including the Lyapunov exponent spectrum, bifurcation diagrams, phase plots and dynamic maps. Furthermore, we also investigated different neurons subject to the memristive electromagnetic influence, where the two-dimensional grid-scroll chaotic attractors can be generated by one neuron subjects to the electromagnetic influence and three-dimensional space-scroll chaotic attractors can be generated by all of the neurons subject to the memristive electromagnetic influence. Moreover, we present a three-dimensional space-scroll scroll attractor-based chaotic mobile robot system. Superiority of the proposed space-scroll-based chaotic mobile robot system compared to existing methods has been demonstrated through numerical simulations.

Exploring the potential of contemporary deep learning methods in purifying polluted information

Abstract Accurately predicting chaotic dynamical systems is a crucial task in various fields, and recent advancements have leveraged deep learning for this purpose. However, in the era of big data, the inevitable challenge of data contamination caused by invalid information from other interfering systems becomes increasingly prominent and complicates accurate predictions. Although contemporary deep learning methods have shown their potential, very few studies have focused on developing algorithms specifically designed to address the problem posed by such data contamination. Thus, exploring the ability of contemporary deep learning methods to purify polluted information fills an important gap in the current body of research. This study explores the performance and stability of several modern deep learning techniques for predicting chaotic systems using datasets polluted by invalid information. Our findings reveal that while most of the state-of-the-art deep learning methods exhibit reduced and unstable predictive performance owing to such contamination, the dynamical system deep learning (DSDL) method stands out, remaining unaffected by any interference. This breakthrough illustrates DSDL's unique ability to purify invalid data and uncover the inherent rules of chaotic systems. As we move forward, DSDL paves the way for a more reliable and interpretable model, ensuring that we can confidently predict chaotic systems with precision even in the most challenging environments.

Fast Color Image Encryption Algorithm Based on DNA Coding and Multi-Chaotic Systems

At present, there is a growing emphasis on safeguarding image data, yet conventional encryption methods are full of numerous limitations. In order to tackle the limitations of conventional color image encryption methodologies, such as inefficiency and insufficient security, this paper designs an expedited encryption method for color images that uses DNA coding in conjunction with multiple chaotic systems. The encryption algorithm proposed in this paper is based on three-dimensional permutation, global scrambling, one-dimensional diffusion and DNA coding. First of all, the encryption algorithm uses three-dimensional permutation algorithms to scramble the image, which disrupts the high correlation among the image pixels. Second, the RSA algorithm and the SHA-256 hashing algorithm are utilized to derive the starting value necessary for the chaotic system to produce the key. Third, the image is encrypted by using global scrambling and one-dimensional diffusion. Finally, DNA coding rules are used to perform DNA computing. The experimental results indicate that the encryption scheme exhibits a relatively weak inter-pixel correlation, uniform histogram distribution, and an information entropy value approaching eight. This shows that the proposed algorithm is able to protect the image safely and efficiently.

Design and FPGA realization of memristive multibutterfly chaotic system featuring coexisting attractors and complex multistability behavior

Design and FPGA realization of memristive multibutterfly chaotic system featuring coexisting attractors and complex multistability behavior

Block-based color image encryption algorithm by a novel memristor chaotic system and new RNA computation

The security of images is closely related to the protection of information privacy. We proposed a novel 5D memory resistive chaotic system (5D-MRCS), which exhibits good chaotic characteristics. Therefore, we employed it to design an image encryption algorithm aimed at ensuring secure image transmission. To further enhance the complexity of the algorithm and obtain more chaotic sequences, we combine the 5D-MRCS with the Hodgkin-Huxley (HH) model and use this combination in algorithm design. Initially, we combine the plain image with the hash function SHA-384 to devise and generate the secret key. Subsequently, the algorithm determines whether to pad the plain image based on different block size requirements. Then, we use multiple chaotic sequences generated by the 5D-MRCS and HH model to perform the global image permutation operation. Our designed permutation algorithm includes two parts: Block-based permutation and a new pixel-level permutation. Next, the scrambled image undergoes block-based random RNA diffusion, incorporating two newly proposed methods in the RNA operations, ultimately resulting in the ciphertext image. The algorithm’s NPCR, UACI, information entropy, and other security performance metrics are very close to the ideal values, and it possess characteristics such as resistance to differential, cutting, chosen plaintext, and noise attacks. Compared with other algorithms, it still has some advantages across multiple images and demonstrates excellent image encryption performance.

Method to discriminate between localized and chaotic quantum systems

We study whether a generic isolated quantum system initially set out of equilibrium can be considered as localized close to its initial state. Our approach considers the time evolution in the Krylov basis, which maps the system's dynamics onto that of a particle moving in a one-dimensional lattice where both the energy in the lattice sites and the tunneling from one lattice site to the next are inhomogeneous. By tying the dynamical propagation in the Krylov basis to that in the basis of microstates, we infer qualitative criteria that allow distinguishing systems that remain localized close to their initial state from systems that undergo quantum thermalization. These criteria are system-dependent and involve the expectation values and standard deviations of both the coupling strengths between Krylov states and their energy. We verify their validity by inspecting two cases: Anderson localization as a function of dimension and the out-of-equilibrium dynamics of a many-body dipolar spin system. We finally investigate the Wigner surmise and the eigenstate thermalization hypothesis that have both been proposed to characterize quantum chaotic systems. We show that when the average value of the nondiagonal terms in the Lanczos matrix is large compared to their fluctuations and to the fluctuations of the energy expectation values, which typically corresponds to delocalized quantum systems according to our criteria, there can be level repulsion of eigenenergies (also known as spectral rigidity), similar to that of the Wigner-Dyson statistics of energy levels; and we also demonstrate that in the same regime, the expectation values of typical local observables only weakly vary as a function of eigenstates, an important condition for the eigenstate thermalization hypothesis. Our demonstration assumes that, in the chaotic regime, the observable is sufficiently diagonal in the basis of Krylov states. Published by the American Physical Society 2024

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x2⟩T ⟨p2⟩T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Image encryption algorithm based on matrix projective combination-combination synchronization of an 11-dimensional time delayed hyperchaotic system

Abstract Secure image transmission is critical to protect sensitive data from unauthorized access, especially in an era of increasing digital threats. Chaotic systems with their inherent complexity and unpredictability, provide a promising solution for enhancing encryption security. To contribute to this field, we investigate a new 11-dimensional hyperchaotic system by taking advantage of its complex dynamical properties to strengthen security. The high dimensional of the system intensifies chaotic behaviors such as stability, attractors and sensitive to initial conditions, making it particularly suitable for encrypted transmission.
 Time delay is an important factor to be considered affecting the control and synchronization in nonlinear system. Additionally, time delays include the effects of past states, further increasing the unpredictability of the system. To explore these dynamics, we analyze the Lyapunov exponents, stability of equilibrium points, symmetry and dissipation. A matrix projective combination-combination synchronization scheme is proposed to synchronize four identical 11-dimensional hyperchaotic systems with time delay. Nonlinear active controllers designed based on Lyapunov stability theory are used to achieve this synchronization. This work advances an important idea for encryption and decryption algorithms, which is the secure transmission of images using affine encryption. In the affine encryption algorithm, the key is based on the solution of synchronized chaotic delayed systems and the private message of the sender and receiver. This proposed encryption and decryption algorithms have been applied on plain images. Numerical simulations and security analysis including key space, histogram, information entropy and correlation analysis are conducted to validate the theoretical results and encryption algorithm. Experimental analysis and comparisons with existing literature confirm the effectiveness and security of the proposed approach for cryptographic purposes.

Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems

The spectral form factor of quantum chaotic systems has the familiar ‘ramp ++ plateau’ form. Techniques to determine its form in the semiclassical or the thermodynamic limit have been devised, in both cases based on the average over an energy range or an ensemble of systems. For a single instance, fluctuations are large, do not go away in the limit, and depend on the element of the ensemble itself, thus seeming to question the whole procedure. Considered as the modulus of a partition function in complex inverse temperature \beta_R+i\beta_IβR+iβI (\beta_I \equiv \tauβI≡τ the time), the spectral form factor has regions of Fisher zeroes, the analogue of Yang-Lee zeroes for the complex temperature plane. The large spikes in the spectral form factor are in fact a consequence of near-misses of the line parametrized by \beta_IβI to these zeroes. The largest spikes are indeed extensive and extremely sensitive to details, but we show that they are both exponentially rare and exponentially thin. Motivated by this, and inspired by the work of Derrida on the Random Energy Model, we study here a modified model of random energy levels in which we introduce level repulsion. We also check that the mechanism giving rise to spikes is the same in the SYK model.

Multi-wing chaotic system based on meminductor and its application in image encryption

Meminductor is a novel type of nonlinear device following the memristor, characterized by its memory properties. Currently, research on meminductors is still in its infancy, with their physical devices yet to be formally realized. Therefore, conducting fundamental research on their nonlinear circuit properties and applications is of great significance. In this paper, a new multi-wing chaotic system is proposed based on the mathematical model of a magnetically controlled meminductor. By varying the values of its parameters, the system can generate two-wing, three-wing, and four-wing chaotic attractors. Various analytical methods are employed to study the dynamical behaviours of the proposed chaotic system. The results demonstrate that the system is highly sensitive to its initial conditions and control parameters, which makes it suitable for image encryption. Based on the new system, we propose a new algorithm for image encryption that combines the newly established four-dimensional multi-wing chaotic system with bit plane decomposition technique, firstly, the high four-bit planes containing 94% image information are disordered by S-type permutation, then the disordered bit planes perform operation of XOR with the random matrix generated by chaotic sequences, and finally, the encrypted image is obtained by merging the bit planes.

Towards complexity in de Sitter space from the doubled-scaled Sachdev-Ye-Kitaev model

How can we define complexity in dS space from microscopic principles? Based on recent developments pointing towards a correspondence between a pair of double-scaled Sachdev-Ye-Kitaev (DSSYK) models/ 2D Liouville-de Sitter (LdS2) field theory/ 3D Schwarzschild de Sitter (SdS3) space in [1–3], we study concrete complexity proposals in the microscopic models and their dual descriptions. First, we examine the spread complexity of the maximal entropy state of the doubled DSSYK model. We show that it counts the number of entangled chord states in its doubled Hilbert space. We interpret spread complexity in terms of a time difference between antipodal observers in SdS3 space, and a boundary time difference of the dual LdS2 CFTs. This provides a new connection between entanglement and geometry in dS space. Second, Krylov complexity, which describes operator growth, is computed for physical operators on all sides of the correspondence. Their late time evolution behaves as expected for chaotic systems. Later, we define the query complexity in the LdS2 model as the number of steps in an algorithm computing n-point correlation functions of boundary operators of the corresponding antipodal points in SdS3 space. We interpret query complexity as the number of matter operator chord insertions in a cylinder amplitude in the DSSYK, and the number of junctions of Wilson lines between antipodal static patch observers in SdS3 space. Finally, we evaluate a specific proposal of Nielsen complexity for the DSSYK model and comment on its possible dual manifestations.