Chaotic Dynamical Systems Research Articles

State Identification Via Symbolic Time Series Analysis for Reinforcement Learning Control

Abstract This technical brief makes use of the concept of symbolic time-series analysis (STSA) for identifying discrete states from the nonlinear time response of a chaotic dynamical system for model-free reinforcement learning (RL) control. Along this line, a projection-based method is adopted to construct probabilistic finite state automata (PFSA) for identification of the current state (i.e., operational regime) of the Lorenz system; and a simple Q-map-based (and model-free) RL control strategy is formulated to reach the target state from the (identified) current state. A synergistic combination of PFSA-based state identification and RL control is demonstrated by the simulation of a numeric model of the Lorenz system, which yields very satisfactory performance to reach the target states from the current states in real-time.

Cupolets: History, Theory, and Applications

In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to form the skeleton of the associated attractors. While UPOs are insightful tools for analysis, they are naturally unstable and, as such, are difficult to find and computationally expensive to stabilize. An alternative to using UPOs is to approximate them using cupolets. Cupolets, a name derived from chaotic, unstable, periodic, orbit-lets, are a relatively new class of waveforms that represent highly accurate approximations to the UPOs of chaotic systems, but which are generated via a particular control scheme that applies tiny perturbations along Poincaré sections. Originally discovered in an application of secure chaotic communications, cupolets have since gone on to play pivotal roles in a number of theoretical and practical applications. These developments include using cupolets as wavelets for image compression, targeting in dynamical systems, a chaotic analog to quantum entanglement, an abstract reducibility classification, a basis for audio and video compression, and, most recently, their detection in a chaotic neuron model. This review will detail the historical development of cupolets, how they are generated, and their successful integration into theoretical and computational science and will also identify some unanswered questions and future directions for this work.

Robust inference of causality in high-dimensional dynamical processes from the Information Imbalance of distance ranks

We introduce an approach which allows detecting causal relationships between variables for which the time evolution is available. Causality is assessed by a variational scheme based on the Information Imbalance of distance ranks, a statistical test capable of inferring the relative information content of different distance measures. We test whether the predictability of a putative driven system Y can be improved by incorporating information from a potential driver system X, without explicitly modeling the underlying dynamics and without the need to compute probability densities of the dynamic variables. This framework makes causality detection possible even between high-dimensional systems where only few of the variables are known or measured. Benchmark tests on coupled chaotic dynamical systems demonstrate that our approach outperforms other model-free causality detection methods, successfully handling both unidirectional and bidirectional couplings. We also show that the method can be used to robustly detect causality in human electroencephalography data.

Individual chaotic behaviour of the S-stars in the Galactic centre

Located at the core of the Galactic centre, the S-star cluster serves as a remarkable illustration of chaos in dynamical systems. The long-term chaotic behaviour of this system can be studied with gravitational N-body simulations. By applying a small perturbation to the initial position of star S5, we can compare the evolution of this system to its unperturbed evolution. This results in two solutions that diverge exponentially, defined by the separation in position space δr, with an average Lyapunov timescale of ∼420 yr, corresponding to the largest positive Lyapunov exponent. Even though the general trend of the chaotic evolution is governed in part by the supermassive black hole Sagittarius A∗ (Sgr A∗), individual differences between the stars can be noted in the behaviour of their phase-space curves. We present an analysis of the individual behaviour of the stars in this Newtonian chaotic dynamical system. The individuality of their behaviour is evident from offsets in the position space separation curves of the S-stars and the black hole. We propose that the offsets originate from the initial orbital elements of the S-stars, where Sgr A∗ is considered in one of the focal points of the Keplerian orbits. Methods were considered to find a relation between these elements and the separation in position space. Symbolic regression provides the clearest diagnostics for finding an interpretable expression for the problem. Our symbolic regression model indicates that ⟨δr⟩ ∝ e2.3, implying that the time-averaged individual separation in position space increases rapidly with the initial eccentricity of the S-stars.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

Chaotic Phenomena for Generalised N-centre Problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

Optimal Parameter Estimation Techniques for Complex Nonlinear Systems

AbstractAccurate parameter estimation and state identification within nonlinear systems are fundamental challenges addressed by optimization techniques. This paper fills a critical gap in previous research by investigating tailored optimization methods for parameter estimation in nonlinear system modeling, with a particular emphasis on chaotic dynamical systems. We introduce and compare three optimization methods: a gradient-based iterative algorithm, the Levenberg-Marquardt algorithm, and the Nelder-Mead simplex method. These methods are strategically employed to simplify complex nonlinear optimization problems, rendering them more manageable. Through a comprehensive exploration of the performance of these methods in determining parameters across diverse systems, including the van der Pol oscillator, the Rössler system, and pharmacokinetic modeling, our study revealed that the accuracy and reliability of the Nelder-Mead simplex method were consistent. The Nelder-Mead simplex algorithm emerged as a powerful tool, that consistently outperforms alternative methods in terms of root mean squared error (RMSE) and convergence reliability. Visualizations of trajectory comparisons and parameter convergence under various noise levels further emphasize the algorithm’s robustness. These studies suggest that the Nelder-Mead simplex method has potential as a valuable tool for parameter estimation in chaotic dynamical systems. Our study’s implications extend beyond theoretical considerations, offering promising insights for parameter estimation techniques in diverse scientific fields reliant on nonlinear system modeling.

Nonlinear Dynamic Behavior of Two Distinct Chaotic Systems

The nonlinear dynamics of two chaotic systems, namely the optoelectronic feedback (OEFB) and the Lu-Chen electronic chaotic system, are simulated and implemented in this work using the Berkeley Madonna software. Control parameters and initial conditions have been adjusted to demonstrate transitions from one state to another. Upon modifying certain regulating elements, both systems exhibited excessive sensitivity to initial conditions and displayed dynamic nonlinear behavior. The OEFB system reveals a homoclinic condition with a Shilnikov attractor as the feedback intensity increases. In contrast, the Lu-Chen system exhibits sensitivity to parameters a, b, and c, accompanied by multiscroll behavior, as evidenced by time series, the Fast Fourier Transform (FFT), and attractor analysis. These results offer potential applications, including data encoding, secure communications, and image processing. This research studied the properties of two different chaotic dynamical systems. These two chaotic systems are optoelectronic feedback and Chua systems. The results are analyzed, and it is found that the behavior of the Chua system changes in the time series, which in turn causes the attractor to change. The results showed a significant increase in the Chua system's bandwidth. Studying the different characteristics opens a broad scope for many applications, the most important of which is secure communications.

Learning Topological Horseshoe via Deep Neural Networks

Deep Neural Networks (DNNs) have been successfully applied to investigations of numerical dynamics of finite-dimensional nonlinear systems such as ODEs instead of finding numerical solutions to ODEs via the traditional Runge–Kutta method and its variants. To show the advantages of DNNs, in this paper, we demonstrate that the DNNs are more efficient in finding topological horseshoes in chaotic dynamical systems.

Partial Control and Beyond: Controlling Chaotic Transients with the Safety Function

Chaotic dynamical systems often exhibit transient chaos, where trajectories behave chaotically for a short amount of time before escaping to an external attractor. Sustaining transient chaotic dynamics under disturbances is challenging yet desirable for many applications. The partial control approach exploits the inherent symmetry and geometric structure of chaotic saddles, the topological object responsible of transient chaos, to enable surprising control with only small perturbations. Here, we review the latest findings in partial control techniques with the aim to sustain chaos or accelerate escapes by exploiting these intricate invariant sets. We introduce the fundamental concept of safe sets regions where orbits persist despite noise. This paper presents recent generalizations through safety functions and escape functions that automatically find the minimum control needed. Efficient numerical algorithms are presented and several examples of application are illustrated. Rather than eliminating chaos entirely, partial control techniques provide a framework to reliably control transient chaotic dynamics with minimal interventions. This approach has promising applications across diverse fields including physics, engineering, biology, and more.

A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

Chaotic dynamical system of Hopfield neural network influenced by neuron activation threshold and its image encryption

Chaotic dynamical system of Hopfield neural network influenced by neuron activation threshold and its image encryption

Nonintrusive Operator Inference for Chaotic Systems

This work explores the physics-driven machine learning technique Operator Inference (OpInf) for predicting the state of chaotic dynamical systems. OpInf provides a non-intrusive approach to infer approximations of polynomial operators in reduced space without having access to the full order operators appearing in discretized models. Datasets for the physics systems are generated using conventional numerical solvers and then projected to a low-dimensional space via Principal Component Analysis (PCA). In latent space, a least-squares problem is set to fit a quadratic polynomial operator, which is subsequently employed in a time-integration scheme in order to produce extrapolations in the same space. Once solved, the inverse PCA operation is applied to reconstruct the extrapolations in the original space. The quality of the OpInf predictions is assessed via the Normalized Root Mean Squared Error (NRMSE) metric from which the Valid Prediction Time (VPT) is computed. Numerical experiments considering the chaotic systems Lorenz 96 and the Kuramoto-Sivashinsky equation show promising forecasting capabilities of the OpInf reduced order models with VPT ranges that outperform state-of-the-art machine learning (ML) methods such as backpropagation and reservoir computing recurrent neural networks [1], as well as Markov neural operators [2].

Constructing polynomial libraries for reservoir computing in nonlinear dynamical system forecasting.

Reservoir computing is an effective model for learning and predicting nonlinear and chaotic dynamical systems; however, there remains a challenge in achieving a more dependable evolution for such systems. Based on the foundation of Koopman operator theory, considering the effectiveness of the sparse identification of nonlinear dynamics algorithm to construct candidate nonlinear libraries in the application of nonlinear data, an alternative reservoir computing method is proposed, which creates the linear Hilbert space of the nonlinear system by including nonlinear terms in the optimization process of reservoir computing, allowing for the application of linear optimization. We introduce an implementation that incorporates a polynomial transformation of arbitrary order when fitting the readout matrix. Constructing polynomial libraries with reservoir-state vectors as elements enhances the nonlinear representation of reservoir states and more easily captures the complexity of nonlinear systems. The Lorenz-63 system, the Lorenz-96 system, and the Kuramoto-Sivashinsky equation are used to validate the effectiveness of constructing polynomial libraries for reservoir states in the field of state-evolution prediction of nonlinear and chaotic dynamical systems. This study not only promotes the theoretical study of reservoir computing, but also provides a theoretical and practical method for the prediction of nonlinear and chaotic dynamical system evolution.

Learning dynamical systems from data: A simple cross-validation perspective, Part V: Sparse Kernel Flows for 132 chaotic dynamical systems

Learning equations of complex dynamical systems from data is one of the principal problems in scientific machine learning. The method of Kernel Flows (KFs) has offered an effective learning strategy that interpolates the vector-field of dynamical system with a data-adapted kernel. It is based on the premise that a kernel is good if the number of interpolation points can be halved without significant loss in accuracy. However, KFs is limited by the choice of a base kernel. In this paper, we introduce the method of Sparse Kernel Flows in order to learn the “best” kernel starting from a preset kernel library. First, we design a parameterized base kernel that is a linear combination of several existing kernel functions. Then, under the assumption of KFs that a kernel is good if the number of interpolation points can be halved without significant loss in accuracy, we design the kernel learning loss function by incorporating ℓ1 regularization. Then, Least absolute shrinkage and selection operator (LASSO) is performed to extract the fewest active terms from the base overdetermined set of candidate kernel functions, thereby estimating the weight coefficient. Furthermore, we apply our proposal to a benchmark dataset including 132 chaotic dynamical systems.

Interpretable predictions of chaotic dynamical systems using dynamical system deep learning

Making accurate predictions of chaotic dynamical systems is an essential but challenging task with many practical applications in various disciplines. However, the current dynamical methods can only provide short-term precise predictions, while prevailing deep learning techniques with better performances always suffer from model complexity and interpretability. Here, we propose a new dynamic-based deep learning method, namely the dynamical system deep learning (DSDL), to achieve interpretable long-term precise predictions by the combination of nonlinear dynamics theory and deep learning methods. As validated by four chaotic dynamical systems with different complexities, the DSDL framework significantly outperforms other dynamical and deep learning methods. Furthermore, the DSDL also reduces the model complexity and realizes the model transparency to make it more interpretable. We firmly believe that the DSDL framework is a promising and effective method for comprehending and predicting chaotic dynamical systems.

Estimates of constants in the limit theorems for chaotic dynamical systems

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems.

Efficient color image steganography based on new adapted chaotic dynamical system with discrete orthogonal moment transforms

In today's rapidly evolving digital landscape, safeguarding the secure transmission of sensitive information is paramount. Traditional steganography methods, while effective at data concealment, have grown vulnerable to brute force attacks in our age of powerful computing. This paper addresses the pressing need to bolster data transmission security. To address the challenges posed by limited key space and security concerns associated with 1-D chaotic maps in data transmission, we introduce a novel 1-D chaotic map, the Multi-Parameter Chebyshev Map (MCM). The MCM boasts four control parameters, each with a broad chaotic range and high chaos level, significantly enhancing security. Furthermore, we present an innovative steganography technique for color images that leverages the MCM in conjunction with discrete moment transforms. This method chaotically embeds the secret message within the moment coefficients of a cover color image using MCM and quantization rules. The expansive chaotic range and high chaos level of the MCM substantially reinforce the security of this approach. The considered moment transforms include Tchebichef, Krawtchouk, Hahn, Charlier, Meixner, Shmaliy, Racah, and Dual-Hahn. The performance of the proposed steganography method is thoroughly evaluated and compared with other state-of-the-art methods in terms of hiding capacity and image quality in both spatial and transform domains. The quantization of moment coefficients demonstrates its superior performance. This method holds crucial implications for practical applications requiring secure communication of color images, such as military and medical contexts. The combination of MCM and moment transforms in this method presents a promising avenue for future research in the field of information hiding.

Data-driven learning of chaotic dynamical systems using Discrete-Temporal Sobolev Networks

We introduce the Discrete-Temporal Sobolev Network (DTSN), a neural network loss function that assists dynamical system forecasting by minimizing variational differences between the network output and the training data via a temporal Sobolev norm. This approach is entirely data-driven, architecture agnostic, and does not require derivative information from the estimated system. The DTSN is particularly well suited to chaotic dynamical systems as it minimizes noise in the network output which is crucial for such sensitive systems. For our test cases we consider discrete approximations of the Lorenz-63 system and the Chua circuit. For the network architectures we use the Long Short-Term Memory (LSTM) and the Transformer. The performance of the DTSN is compared with the standard MSE loss for both architectures, as well as with the Physics Informed Neural Network (PINN) loss for the LSTM. The DTSN loss is shown to substantially improve accuracy for both architectures, while requiring less information than the PINN and without noticeably increasing computational time, thereby demonstrating its potential to improve neural network forecasting of dynamical systems.

Collision-free arbitrary-order chaotic path generator for differential robots

This article discusses a collision-free path generator based on arbitrary-order chaotic waveforms for differential robots. The novelty is that the robot can explore a predefined workspace with any irregular structure, avoiding obstacles and boundaries with any random shape. All positions of the differential robot are generated by the arbitrary-order chaotic dynamic system. The inverse Jacobian of the differential robot is used as control law. Therefore, the position error at each point of the chaotic path not only is iteratively reduced, but the angular velocities for each motion are also computed. Numerical simulations were performed within Matlab environment to evaluate the arbitrary-order chaotic path generator in a scenario and different starting positions. Numerical tests illustrate the usefulness of the proposed collision-free arbitrary-order chaotic path generator and it can be applied to various areas, such as household, agriculture, education, manufacturing, patrolling, medical care, military and so on.