Accurate parameter identification in nonlinear and chaotic dynamic systems requires optimization algorithms that can reliably balance global exploration and local refinement in complex, multimodal search landscapes. To address this challenge, a modified artificial protozoa optimizer (mAPO) is developed in this study by embedding two complementary mechanisms into the original artificial protozoa optimizer: a probabilistic random learning strategy to enhance population diversity and global search capability, and a Nelder-Mead simplex-based local refinement stage to improve exploitation and fine-scale solution adjustment. The general optimization performance and scalability of the proposed framework are first evaluated using the CEC2017 benchmark suite. Statistical analyses conducted over shifted and rotated, hybrid, and composition functions demonstrate that mAPO achieves improved mean performance and reduced variability compared with the original APO, indicating enhanced robustness in high-dimensional and complex optimization problems. The effectiveness of mAPO is then examined in nonlinear system identification applications involving chaotic dynamics. Offline and online parameter identification experiments are performed on the Rössler chaotic system and a permanent magnet synchronous motor, including scenarios with abrupt parameter variations. Comparative simulations against APO and several state-of-the-art optimizers show that mAPO consistently yields smaller objective function values, more accurate parameter estimates, and superior statistical stability. In the PMSM case, exact parameter reconstruction with zero error is achieved across all independent runs, while rapid and smooth convergence is observed under both static and time-varying conditions.

Dynamic reinforcement learning for actors.

The exploration of complex-valued chaos offers a viable pathway not only for practical applications like image encryption but also holds significant potential for simulating wave phenomena and quantum inspired process. To bridge this with nonlinear circuit elements, we introduce a novel complex-valued chaotic system by embedding a discrete memristor into a complex Gaussian map. The memristor, a component with inherent physical memory, is uniquely driven by the modulus of the complex state variable, which is a key physical quantity often associated with energy or amplitude in wave systems. This coupling induces complex nonlinear dynamics, which are physically characterized through Lyapunov exponents and bifurcation analysis, revealing an enhanced and more robust chaotic regime. The physical feasibility of this system is demonstrated by its successful hardware realization on an FPGA platform. To showcase its application potential, we leverage the system’s complex chaotic steams to engineer a dual-image encryption scheme, where the encryption process is interpreted as a physical diffusion and scrambling of information represented by a complex matrix. Our results verify that this approach not only yield a cryptosystem with high security but also provide a link between complex chaos and information security applications.

The theory underlying non-linear dynamical systems remains essential for understanding complex behaviors in science and engineering. In this study, we propose a new chaotic dynamical system that exhibits a butterfly-shaped attractor without any equilibrium point. Despite its compact structure comprising only five terms, the system demonstrates rich chaotic behavior distinct from conventional oscillator models. Detailed modeling and dynamical analyses are conducted to confirm the presence of chaos and to characterize the system’s sensitivity to initial conditions. Furthermore, synchronization of the proposed dynamical system is investigated using both identical and non-identical control algorithms. In the identical case, the activation function of the neural network is governed by the butterfly oscillator dynamics, whereas in the non-identical case, a sigmoidal activation function is employed. The proposed synchronization algorithms enable faster convergence by pinning a subset of nodes in the network. Finally, a practical implementation of the conceived dynamical system in an encryption framework is presented, with the aim to demonstrate its feasibility and potential application in secure communication systems. The results highlight the effectiveness of the proposed approach for both theoretical exploration and engineering applications involving chaotic dynamical systems.

Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is crucial for both theoretical understanding and practical implementation. The characterization of chaotic dynamical systems typically relies on conventional measures such as Lyapunov exponents and fractal dimensions. While these metrics are fundamental for describing dynamical behavior, they are often computationally expensive and may fail to capture subtle changes in the overall geometry of the attractor, limiting comparisons between systems with topologically similar structures and similar values in common chaos metrics such as the Lyapunov exponent. To address this limitation, this work proposes a geometric framework that treats chaotic attractors as spatial objects, using topological tools—specifically the α-sphere—to quantify their shape and spatial extent. The proposed method was validated using Chua’s system (including two reported variations), the Rössler system (standard and piecewise-linear), and a fractional-order multi-scroll system. A parametric characterization of the Rössler system was also performed by varying parameter b. Experimental results show that this geometric approach successfully distinguishes between attractors where classical metrics reveal no perceptible differences, in addition to being computationally simpler. Notably, we observed geometric variations of up to 80% among attractors with similar dynamics and introduced a specific index to quantify these global discrepancies. Although this geometric analysis serves as a complement rather than a substitute for chaos detection, it provides a reliable and interpretable metric for differentiating systems and selecting attractors based on their spatial properties.

This paper briefly describes the history of chaotic dynamical systems since Henri Poincaré’s original work on celestial mechanics and more particularly on the three-body problem in the 1880s–1890s. Three periods are distinguished: the first period “the dawn of chaotic dynamical systems” from 1885 to 1970 which includes the Kolmogorov–Arnold–Moser theorem on the persistence of quasi-periodic motions under small perturbations extended by Moser in 1962, with its proof achieved in 1963 by Arnold, the discovery of the Lorenz attractor in 1963, the Sharkovsky order (1964), the Shilnikov bifurcation (1965), the horseshoe and the fractals in 1967. The second period “the intermediate period 1971–1989”, begins with the definition of the strange attractors by Ruelle and Takens, and ends just before the introduction of the synchronization of chaotic systems in 1990 by Pecora and Carroll. The third period extends from 1990 to the present day, which is a flourishing time of applications in several sciences of this mathematical theory. This period corresponds to the publication of 35 volumes (415 issues) of the International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, since the 35 years of existence of this journal. This history of chaotic dynamical systems is presented in a reasoned manner, which provides some insight into the driving forces behind their evolution. The people who shaped them are also featured. An epistemological analysis structures the analysis of these three periods. It takes into account the notions of causality, determinism, predictability, unpredictability and control.

In climate systems, physiological models, optics, and many more, surrogate models are developed to reconstruct chaotic dynamical systems. We introduce four data-driven measures for low-dimensional systems using global attractor properties to evaluate the quality of the reconstruction of a given time series from a surrogate model. The measures are robust against the initial position of the chaotic system as they are based on empirical approximations of the correlation integral and the probability density function, both of which are global properties of the attractor. In contrast to previous methods, we do not need a manual fitting procedure, making the measures straightforward to evaluate. Compared with the n-dimensional Wasserstein distance, the measures are fast to evaluate, while compared with the Hausdorff distance, they perform better. Furthermore, we introduce a statistical framework inspired by hypothesis testing to systematically find and reject surrogate models whose reconstructions significantly differ from the true system. Furthermore, we show that the measures can be used as a statistical ranking metric, which, in practice, allows for hyperparameter optimization. Applying our measures to reservoir computing with a low number of nodes, we demonstrate how the measures can be used to reject poor reconstructions and use them as a tool to find an optimal spectral radius for which considerably fewer solutions are rejected and the overall quality of the solution improves.

This paper introduces a novel six-dimensional (6D) chaotic dynamic system characterized by the absence of equilibrium points and the presence of hidden attractors. The study investigates the properties of this innovative system, including the computation of Lyapunov exponents and the Lyapunov dimension. Through comprehensive computer modeling in Matlab-Simulink, phase portraits of numerous hidden attractors are obtained, providing insight into the system’s complex dynamics. To validate the theoretical findings, electronic circuits for the 6D chaotic system were designed and implemented using Multisim software. The circuit simulations exhibit behavior consistent with the Matlab-Simulink models, confirming the reliability of the proposed system’s dynamics. The paper further explores the synchronization of two identical 6D hyperchaotic systems using active control techniques. Numerical analyses compare the systems behavior before and after control implementation, demonstrating the effectiveness of the active control method in achieving synchronization. Additionally, the active control approach is applied to chaotic masking and decoding of various signals, highlighting its potential in secure communication applications. We presented a novel application of the proposed 6D system as a source of control input signals for independent navigation of multiple mobile robots, and the paths of robots become unpredictable. We investigated the influence of some external factors on the navigation of a chaotic wheeled mobile robot.

Abstract Chaotic dynamical systems are often characterised by a positive Lyapunov exponent, which signifies an exponential rate of separation of nearby trajectories. However, in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential in time, and the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the positive Lyapunov exponents on the system's fractal repeller can be related to the generation of metric entropy and the escape rate from the system. The escape rate, in turn, cross-links these two chaos properties to important statistical-physical quantities like the diffusion coefficient. However, no suitable generalisation of this escape rate formalism exists for weakly chaotic systems. In our paper we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, a generalisation of its Lyapunov exponent (which we call `stretching') is completely suppressed in the presence of a hole. This result is based on numerical evidence and a corresponding stochastic model. The correspondence between map and model is tested via a related partially absorbing map. We examine the structure of the map's fractal repeller, which we reconstruct via a simple algorithm. Our findings are in line with rigorous mathematical results concerning the collapse of the system's density as it evolves in time. We also examine the generation of entropy in the open map, which is shown to be consistent with the collapsed stretching. As a result, we conclude that no suitable generalisation of the escape rate formalism to weakly chaotic systems can exist.

Chaos is omnipresent in nature, and its understanding provides enormous social and economic benefits. However, the unpredictability of chaotic systems is a textbook concept due to their sensitivity to initial conditions, aperiodic behaviour, fractal dimensions, nonlinearity and strange attractors. In this work, we introduce, for the first time, chaotic learning, a novel multiscale topological paradigm that enables accurate predictions from chaotic systems. We show that seemingly random and unpredictable chaotic dynamics counterintuitively offer unprecedented quantitative predictions. Specifically, we devise multiscale topological Laplacians to embed real-world data into a family of interactive chaotic dynamical systems, modulate their dynamical behaviours and enable the accurate prediction of the input data. As a proof of concept, we consider 28 datasets from four categories of realistic problems: 10 brain waves, four benchmark protein datasets, 13 single-cell RNA sequencing datasets and an image dataset, as well as two distinct chaotic dynamical systems, namely the Lorenz and Rossler attractors. We demonstrate chaotic learning predictions of the physical properties from chaos. Our new chaotic learning paradigm profoundly changes the textbook perception of chaos and bridges topology, chaos and learning for the first time.

Abstract To further increase the intricacy of chaotic dynamical systems and investigate the regulatory mechanisms of memristors in chaotic systems, a novel five-dimensional memristive hyperchaotic system with hidden attractors is built upon a four-dimensional chaotic system. First, the investigation begins by evaluating key dynamical metrics, including Lyapunov exponents, the Kaplan-Yorke dimension, dissipation rates, fixed-point configurations, and stability behavior. Second, the system’s dynamic behaviors are investigated by using the bifurcation diagrams, spectra of Lyapunov exponents, phase trajectories, and Poincaré mappings. The results show that the developed system not only exhibits parameter-dependent periodic attractors, quasi-periodic oscillations and chaotic attractors, revealing rich dynamic behavior transitions, but also exhibits coexisting attractors sensitive to initial conditions. Furthermore, the transient chaos and offset boost control are studied, and the effective control of the chaotic attractor space dimension is realized by introducing a control variable , which enables spatial translation of the attractor while preserving the original dynamics, as evidenced by unchanged Lyapunov exponent spectra. Finally, the results of circuit simulation and hardware experiment show that the attractor generated by the system is highly consistent with the numerical simulation in topology and dynamic characteristics, which fully verifies the correctness and physical feasibility of the theoretical model.
Keywords: memristive hyperchaotic system, coexisting attractors, offset boosting control, circuit implementation.

This paper describes a simpler version of the cryptographic system designed to secure the communications of low-power Internet of Things (IoT) devices operating in low-resource, low-computation, and low-speed environments. The system in question is based on a hybrid cryptographic model that includes a certain type of hashing with Deterministic Finite Automata (DFA) and key creation through a chaotic dynamic system, the Lorentz attractor. This model strikes a good balance of tradeoff between resource simplification and dependability. A prototype system was developed first on the Arduino Uno, then later improved on the Wokwi simulator, which enabled the simulation of the IoT devices’ actual operating environment. The automata perform a hashing which is both robust and deterministic, which helps to confirm the integrity of the data. The keys generated by the Lorentz system provide deterministic yet highly sensitive keys to perturbations ensuring total unpredictability. The encryption method used is also quite simple, and that is the XOR operation. This makes the entire system compact, fast, easily reversible, and resource efficient. This research shows how useful automatism and chaos theory are in developing cyber security measures for the Internet of Things (IoT) at the embedded system levels.

This work proposes an innovative approach using machine learning to predict extreme events in time series of chaotic dynamical systems. The research focuses on the time series of the Hénon map, a two-dimensional model known for its chaotic behavior. The method consists of identifying time windows that anticipate extreme events, using convolutional neural networks to classify the system states. By reconstructing attractors and classifying (normal and transitional) regimes, the model shows high accuracy in predicting normal regimes, although forecasting transitional regimes remains challenging, particularly for longer intervals and rarer events. The method presents a result above [Formula: see text] of success for predicting the transition regime up to three steps before the occurrence of the extreme event. Despite limitations posed by the chaotic nature of the system, the approach opens avenues for further exploration of alternative neural network architectures and broader datasets to enhance forecasting capabilities.

Machine learning methods have shown promise in learning chaotic dynamical systems, enabling model-free short-term prediction and attractor reconstruction. However, when applied to large-scale, spatiotemporally chaotic systems, purely data-driven machine learning methods often suffer from inefficiencies, as they require a large learning model size and a massive amount of training data to achieve acceptable performance. To address this challenge, we incorporate the spatial coupling structure of the target system as an inductive bias in the network design. Specifically, we introduce physics-guided clustered echo state networks, leveraging the efficiency of the echo state networks (ESNs) as a base model. Experimental results on benchmark chaotic systems demonstrate that our physics-informed method outperforms existing echo state network models in learning the target chaotic systems. Additionally, we numerically demonstrate that leveraging coupling knowledge into ESN models can enhance their robustness to variations of training and target system conditions. We further show that our proposed model remains effective even when the coupling knowledge is imperfect or extracted directly from time series data. We believe that this approach has the potential to enhance other machine learning methods.

We introduce a solvable model of a measurement-induced phase transition (MIPT) in a deterministic but chaotic dynamical system with a positive Lyapunov exponent. In this setup, an observer only has a probabilistic description of the system but mitigates chaos-induced uncertainty through repeated measurements. Using a minimal representation via a branching tree, we map this problem to the directed polymer (DP) model on the Cayley tree, although in a regime dominated by rare events. By studying the Shannon entropy of the probability distribution estimated by the observer, we demonstrate a phase transition distinguishing a chaotic phase with a reduced Lyapunov exponent from a strong-measurement phase where uncertainty remains bounded. Remarkably, the location of the MIPT transition coincides with the freezing transition of the DP, although the critical properties differ. We provide an exact, universal scaling function describing the entropy growth in the critical regime. Numerical simulations confirm our theoretical predictions, highlighting a simple yet powerful framework to explore measurement-induced transitions in classical chaotic systems.

We present a two-player game in a chaotic dynamical system where players have opposing objectives regarding the system's behavior. The game is analyzed using a methodology from the field of chaos control known as partial control. Our aim is to introduce the utility of this methodology in the scope of game theory. These algorithms enable players to devise winning strategies even when they lack complete information about their opponent's actions. To illustrate the approach, we apply it to a chaotic system-the logistic map. In this scenario, one player aims to maintain the system's trajectory within a transient chaotic region, while the opposing player seeks to expel the trajectory from this region. The methodology identifies the set of initial conditions that guarantee victory for each player, referred to as the winning sets, along with the corresponding strategies required to achieve their respective objectives.

In today's digital era, the importance of securing information has reached critical levels. Steganography is one of the methods used for this purpose by hiding sensitive data within other files. This study introduces an approach utilizing a chaotic dynamic system as a random key generator, governing both the selection of hiding locations within an image and the amount of data concealed in each location. The security of the steganography approach is considerably improved by using this random procedure. A 3D dynamic system with nine parameters influencing its behavior was carefully chosen. For each parameter, suitable interval values were determined to guarantee the system's chaotic behavior. Analysis of chaotic performance is given using the Lyapunov exponents, fractal dimension, and bifurcation diagrams. Furthermore, an algorithm is suggested to generate a random binary key, serving as the controller for the embedding process. And the randomness of the generated key was checked. Moreover, this paper introduces a technique that utilizes the generated random key to govern both the embedding process in the spatial domain and the frequency domain. The results of this study are promising and its potential applications can be extended to various fields that require discreet communication and robust data protection

Discovering governing equations that describe complex chaotic systems remains a fundamental challenge in physics and neuroscience. Here, we introduce the PEM-UDE method, which combines the prediction-error method with universal differential equations to extract interpretable mathematical expressions from chaotic dynamical systems, even with limited or noisy observations. This approach succeeds where traditional techniques fail by smoothing optimization landscapes and removing the chaotic properties during the fitting process without distorting optimal parameters. We demonstrate its efficacy by recovering hidden states in the Rossler system and reconstructing dynamics from noise-corrupted electrical circuit data, where the correct functional form of the dynamics is recovered even when one of the observed time series is corrupted by noise 5x the magnitude of the true signal. We demonstrate that this method is capable of recovering the correct dynamics, whereas direct symbolic regression methods, such as SINDy, fail to do so with the given amount of data and noise. Importantly, when applied to neural populations, our method derives novel governing equations that respect biological constraints such as network sparsity - a constraint necessary for cortical information processing yet not captured in next-generation neural mass models - while preserving microscale neuronal parameters. These equations predict an emergent relationship between connection density and both oscillation frequency and synchrony in neural circuits. We validate these predictions using three intracranial electrode recording datasets from the medial entorhinal cortex, prefrontal cortex, and orbitofrontal cortex. Our work provides a pathway to develop mechanistic, multi-scale brain models that generalize across diverse neural architectures, bridging the gap between single-neuron dynamics and macroscale brain activity.

We show how random feature maps can be used to forecast dynamical systems with excellent forecasting skill. We consider the tanh activation function and judiciously choose the internal weights in a data-driven manner such that the resulting features explore the nonlinear, non-saturated regions of the activation function. We introduce skip connections and construct a deep variant of random feature maps by combining several units. To mitigate the curse of dimensionality, we introduce localization where we learn local maps, employing conditional independence. Our modified random feature maps provide excellent forecasting skill for both single trajectory forecasts as well as long-time estimates of statistical properties, for a range of chaotic dynamical systems with dimensions up to 512. In contrast to other methods such as reservoir computers which require extensive hyperparameter tuning, we effectively need to tune only a single hyperparameter, and are able to achieve state-of-the-art forecasting skill with much smaller networks.

Using ordinal patterns for the time-series analysis of complex and chaotic dynamics in systems across a wide variety of physical, biological, and other phenomena has proven extremely powerful, with recent work exploiting metrics that also characterize symmetries and correlations underlying the complex dynamics. For that purpose we introduce here a geometrical representation for the ordinal pattern description of observed time series. We show that specific projections of this space, emphasizing time reversal symmetry, reveals information about the phase-space dynamics and identifies families of chaos and periodicity via the trajectories created as a function of parameter across this (approximate) dynamical-symmetry space. We discuss general consequences and specific applications of these observations.