A novel kernelized angle metric for state similarity in nonlinear dynamic systems

Modal approaches for linear and nonlinear dynamical systems with non-classical damping

Feedback-based optimization of feed-forward neural network for the modeling of complex nonlinear dynamical systems using novel APSOBP algorithm.

This article develops a model-based reinforcement learning (RL) approach to the closed-loop control of nonlinear dynamical systems with a partial nonlinear observation model. We propose an "information-state"-based approach to rigorously transform the partially observed problem into a fully observed problem where the information state consists of the past several observations and control inputs. We further show the equivalence of the transformed and the initial partially observed optimal control problems and provide the conditions to solve for the deterministic optimal solution. We develop a data-based generalization of the iterative linear quadratic regulator (ILQR) for the RL of partially observed systems using a local linear time-varying model of the information-state dynamics approximated by an autoregressive-moving-average (ARMA) model that is generated using only the input-output data. This approach allows us to design a local perturbation feedback control law that provides an optimum solution to the partially observed feedback design problem locally. The efficacy of the developed method is shown by controlling complex high-dimensional nonlinear dynamical systems in the presence of model and sensing uncertainty.

Chaotic and Quasi-Periodic Dynamics in Nonlinear Micro-Strain Wave Systems: Exact Symbolic Solutions and Symmetry-Based Analysis

Predicting nonlinear dynamic systems by causal physics-informed neural networks with ResNet blocks

The inverted pendulum on a cart is a classical benchmark used to evaluate control strategies for nonlinear and underactuated systems. Its inherent instability and strong coupling between the pendulum’s rotation and the cart’s translation make it a challenging system to stabilize. This paper proposes a fuzzy logic–based controller (FLC) designed to stabilize the pendulum in its upright position while maintaining the cart near its equilibrium point. Unlike traditional linear controllers that require precise modeling or system linearization, the FLC uses linguistic rules and triangular membership functions to manage nonlinearities and uncertainties. A nonlinear mathematical model of the pendulum–cart system is developed and implemented in MATLAB/Simulink, where the fuzzy controller computes the control force based on real-time feedback of angular and translational states. Simulation results demonstrate that the proposed FLC achieves fast stabilization with a settling time of less than 5 seconds, minimal overshoot, and smooth transient performance. The control input remains bounded and energy-efficient, and the system maintains stability under disturbances and parameter variations. Overall, the results confirm that the fuzzy logic controller provides a robust, adaptive, and interpretable solution for nonlinear dynamic systems, outperforming traditional PID and model-based controllers in terms of response speed, stability, and robustness.

This study introduces a novel approach for enhancing state estimation in non-linear dynamic systems by integrating Generative Adversarial Networks (GANs) with the Unscented Kalman Filter (UKF). While the UKF improves upon traditional Kalman Filters by using sigma points to estimate the mean and covariance in non-linear transformations, its effectiveness is limited by static parameters - specifically, the process noise covariance (Q), measurement noise covariance (R), and scaling factors (primary, secondary, and tertiary; α, κ, and β). We propose a dynamic framework in which a GAN predicts and updates these parameters in real-time, based on the UKF’s recent performance, allowing the filter to better adapt to rapidly changing system dynamics. This method is validated on real-world aircraft navigation data containing time-stamped records of position, velocity, heading, and environmental variables. Results show that the GAN-enhanced UKF significantly reduces state estimation errors compared to conventional static models. The proposed framework is generalizable and can be applied to other domains such as robotics, autonomous vehicles, and smart cities, where accurate real-time state estimation under uncertainty is critical.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-26339-9.

Reservoir computing (RC) has emerged as a promising computational paradigm for processing temporally correlated and nonlinear data with low training cost. Among various physical implementations, optoelectronic devices provide a unique opportunity to directly interface light with nonlinear dynamical systems, enriching the reservoir state space through device-intrinsic responses. Light can encode information in wavelength, intensity, and pulse duration, and stimulate multiple nodes in parallel with minimal delay or added power. Recent advances in photodiodes, optically modulated memristors, and phototransistors have revealed device-level pathways to enhance nonlinearity, temporal memory, and node diversity, moving beyond purely electrical control toward hybrid optical-electrical tuning. This review revisits these developments from a device physics perspective, highlighting mechanisms for multi-state generation, bidirectional synaptic weight modulation, and temporal response tailoring. We compare diverse excitation schemes, ranging from wavelength- and intensity-selective photocarrier modulation to con optical-assisted filament control and gate-light co-modulation. We also discuss their impact on reservoir performance in pattern recognition, time-series prediction, and dynamic signal processing. We connect material design, device architecture, and reservoir dynamics to outline emerging strategies for scaling optoelectronic RC. This review provides timely insights for researchers working at the intersection of device engineering and neuromorphic computing.

Retraction notice: Robust adaptive control of nonlinear dynamic systems using hybrid sliding mode regressive neural learning technique

This paper is concerned with the secure and resource-efficient cluster synchronization problem of a class of complex dynamical networks (CDNs) under random deception attacks. Each node in the CDNs is modeled by a nonlinear dynamical system with multiple time-varying delays and nonlinear couplings. The central aim is to make each cluster of nodes converge to the same reference trajectory that is distinct for each cluster regardless of the adverse effects of random deception attacks while ensuring communication efficiency for each node. Toward this aim, a distributed dynamic event-triggered mechanism is first proposed such that each node can make its own decisions to transmit or not its data of interest over the communication channel. Second, by suitably modeling the random deception attacks, secure and event-based cluster synchronization controllers are constructed, which incorporate both the effects of random deception attacks and intermittent data arrivals. Then, sufficient conditions ensuring the secure cluster synchronization of the delayed CDNs under randomly occurring deception attacks are established by constructing some appropriate Lyapunov functionals. Furthermore, tractable design criteria on the existence of desired cluster synchronization controllers are derived. Finally, an illustrative example is presented to validate the effectiveness of the main theoretical results.

ABSTRACT Purpose Importance of discovering clean, renewable energy sources, rather than being dependent on the world's finite hydrocarbon resources is growing. As a result, wind power, especially offshore wind, is an alternative gaining popularity these days. Today's offshore wind energy sector depends on robust and resilient structural design, given increased operational risks due to ambient wave loads. Floating Offshore Wind Turbines (FOWT) produce clean, renewable energy—moreover, FOWT sizes, efficiency and power output are steadily increasing. The current study has aimed to validate a novel multimodal approach for structural risk assessment, facilitating the effective extraction of pertinent statistical information from even relatively limited underlying non‐stationary datasets. Methods Excessive structural dynamics may result in either progressive or rapid structural damage, as well as accumulated fatigue damage, mostly caused by environmental in situ loads. Hydrodynamic and aerodynamic environmental covariates have been accounted for within FAST‐coupled nonlinear aero‐hydro‐servo‐elasticity software. Results The current study's methodology aimed to assist designers in assessing hazards and failure risks for complex nonlinear multimodal dynamic wind energy systems, including cases with initial manufacturing imperfections. Novelty A practical engineering design example was used to demonstrate efficiency and verify the advocated state‐of‐the‐art multimodal structural risk assessment approach. Conclusions The proposed state‐of‐the‐art multimodal structural reliability method might be beneficial for a wide range of offshore engineering applications requiring robust, durable and safe design.

Achieving precise and stable engine speed regulation in spark-ignition (SI) systems remains a challenging task because of the inherent nonlinearities, time-varying characteristics, and external disturbances of internal combustion engines (ICEs). Conventional proportional-integral-derivative (PID) controllers often fail to simultaneously ensure fast tracking and robust disturbance rejection under dynamic operating conditions. To address this limitation, a nonlinear two-degree-of-freedom (2-DOF) PID controller has been developed and optimized using the artificial lemming algorithm (ALA) which is a recent bio-inspired metaheuristic that mimics lemming population behaviors to balance exploration and exploitation adaptively through an energy-driven mechanism. The proposed controller was implemented on a detailed mathematical model of the SI engine, encompassing throttle dynamics, manifold pressure variation, combustion torque generation, and crankshaft motion. A multi-term cost function combining normalized overshoot, steady-state error, and stability coefficients was minimized to determine optimal controller gains. Extensive experiments were conducted, including statistical robustness evaluation, transient and steady-state analyses, trajectory tracking, and disturbance-rejection tests. ALA exhibited the lowest mean and standard deviation of the cost function (4.7170 and 0.1429, respectively), confirming its strong convergence stability compared to the starfish optimization algorithm, parrot optimizer, coati optimization algorithm, and dwarf mongoose optimizer. The ALA-optimized controller achieved a rise time of 0.3114s, a settling time of 2.4313s, an overshoot of only 0.0027%, and an extremely small steady-state error of 2.62 × 10⁻¹¹%. Furthermore, the controller demonstrated superior trajectory-tracking accuracy and exceptional disturbance-rejection capability, maintaining speed deviations below 0.5% under abrupt load torque perturbations. The results confirm that the ALA-based nonlinear 2-DOF PID controller provides a robust and energy-efficient solution for nonlinear engine speed regulation, outperforming recent metaheuristic-based approaches in both accuracy and reliability. Owing to its adaptive and scalable design, the proposed control framework is well-suited for integration into real-time embedded engine control units, hybrid powertrains, and other nonlinear dynamic systems requiring high-precision regulation under uncertainty.

ABSTRACT This paper explores the qualitative dynamics wave phenomena that arise in the ‐dimensional Chaffee–Infante equation and the Zakharov equation; both models have key physical applications in nonlinear dynamical systems. The Chaffee–Infante equation is extensively utilized to describe gas diffusion and reaction activity in diverse physical media. At the same time, the Zakharov equation governs ion‐acoustic waves in plasma fluid dynamics, with applications in plasma physics, signal processing, and electromagnetic wave theory. By utilizing the modified ‐expansion method, we attained a range of soliton solutions for both equations, including singular periodic, kink, anti‐kink, and dark solitons. The physical connection of these solutions is examined through 2D and 3D visualizations, representing the wave dynamics and the transitions between different wave structures. All symbolic computations and visualizations were carried out by utilizing Wolfram Mathematica 11. Furthermore, bifurcation and chaos phenomena are studied by changing key system parameters, revealing complex dynamical behaviors. The attained results show that the modified ‐expansion method yields superior accuracy and reliability compared to previous methods for solving nonlinear partial differential equations (NLPDEs). This study emphasizes the potential of this approach in advancing the understanding of nonlinear wave phenomena, with remarkable applications in areas such as gas diffusion modeling, plasma wave dynamics, and electromagnetic wave propagation.

As a key component of pure electric vehicles, the reducer plays a vital role in power transmission and overall drive system performance. This study investigates the nonlinear dynamic characteristics of helical gears with tooth root crack faults in high-speed reducers. A coupled bending–torsional–shaft dynamic model is developed, in which the time-varying mesh stiffness of cracked helical gears is calculated using an improved potential energy method. The system’s nonlinear dynamic responses under varying mesh error excitation, gear backlash, and damping ratio are numerically obtained via the variable-step Runge–Kutta method. The results reveal that under high input speed conditions, the motion of the faulted system evolves from single-period to quasi-periodic motion as bifurcation parameters change. In the stable state, fault characteristic signals are apparent, whereas under strong nonlinear vibrations and chaotic motion, they become difficult to distinguish in traditional time- and frequency-domain analyses. To address this limitation, the DBSCAN clustering algorithm is introduced, which applies machine learning to cluster the Poincaré cross-sections of the system under different motion states. This approach enables the effective classification and identification of crack-induced and fault-related noise, thereby improving the accuracy of fault detection in nonlinear dynamic gear systems.

Plankton dynamics lie at the core of biogeochemical cycles and ecosystem function, which makes dependable prediction essential. Neural network-based approximations show strong potential in capturing these nonlinear interactions due to their flexibility and efficiency. In this study, a dynamic nonlinear autoregressive exogenous neural network trained with the Levenberg-Marquardt algorithm (ARX-LMA) is exploited on the nonlinear carbon thermal nutrient-plankton autonomous dynamics (NCTNP-AD) system for plankton population in the marine biosphere influenced by the impact of global warming and climate change. Four nonlinear ordinary differential equations construct the asymmetric multifactor NCTNP-AD system reflected by the concentrations of carbon dioxide, temperature, nutrient, and plankton population in the marine biosphere. The Adams numerical solver is efficiently utilized to create synthetic temporals by varying rates of plankton maintaining the CO2 concentration by assimilating dissolved nutrients via membrane transporters in response to temperature, the net carbon dioxide absorption rate by the plankton population density, and the predation rate of plankton by fish within the NCTNP-AD system essentially fueling marine primary production. The neuro-computing-based ARX-LMA networks are specifically trained on these datasets to quantify, model, and anticipate the population density changes of the plankton community via a multifactor asymmetric NCTNP-AD system under global warming conditions. The novel ARX-LMA technique's efficacy is thoroughly validated across simulated reference solutions. The comparison includes error convergence graphs, training response graphs, hyperparameter state graphs, error-input correlations, error autocorrelation, regression analysis, error histograms, absolute error, and corresponding reconstruction graphs. Single- and multi-step ahead ARX-LMA predictors were expertly constructed to predict the effects of global warming on plankton population. Step-ahead and multi-step prediction errors in the range of 10-10 to 10-12 affirm the efficacy of ARX-LMA in accurately modeling and forecasting the complex NCTNP-AD system. These findings showcase that machine-learning-based surrogates can provide accurate and adaptable emulators and forecasters of coupled plankton dynamics.

Aiming at the dynamic characteristics and stability of smart distribution network stations under the combined effect of the uncertainty of new energy output and the control logic of power electronics, an adaptive dimensionally increasing linear dynamic modeling method based on Koopman theory is proposed. Firstly, a regional nonlinear model of an intelligent transformer integrating photovoltaic, wind power, battery, hydrogen fuel cell, and synchronous generator is constructed. The control logic of the virtual synchronous generator is then integrated to characterize the dynamic response of the power electronic interface. Secondly, by constructing a set of nonlinear observation functions, including high-order polynomials, exponents, and periodic functions, the dimensional upgrade mapping of the system state is carried out. The dynamic mode decomposition algorithm is adopted to adaptively extract the dominant dynamic modes in the dimensional upgrade space, achieving global linear approximation of complex nonlinear dynamical systems. Finally, the simulation example results show that the average RMAE error of the Koopman method proposed in this paper in voltage spatiotemporal reconstruction is 0.1419, and the maximum RMSE error is 0.1915, significantly improving the accuracy and stability of dynamic modeling.

Aiming at the problems of long computation time and complex probability density evolution rate of dynamic stochastic reliability solving methods for nonlinear dynamic systems, a finite difference method based on Karhunen-Loève (K-L) decomposition and total variation diminishing (TVD), combined with equivalent linearization, is proposed to solve the probability density evolution process, and then the entropy weight method is used to solve the dynamic stochastic reliability of nonlinear systems. The K-L decomposition method is used to determine the rate of probability density evolution of the system and reveal the statistical characteristics of random variables in the dynamic process. The probability density evolution equation of the system is solved by TVD finite difference method combined with a new initial value scheme, and the probability distribution is accurately described in the process of time evolution. The nonlinear system is linearized by the equivalent linearization method, which provides a simplified model for the analysis of complex system. In addition, the entropy weight method is used to calculate the reliability weights of each random parameter to further solve the overall reliability of the system, which provides a theoretical basis for reliability evaluation. The nonlinear gear system is taken as the research object, and the correctness of the proposed method is verified by comparing with Monte Carlo method. Furthermore, compared with the existing path integral method, the calculation time of the proposed method is reduced by more than 95%, and the calculation efficiency is significantly improved.

The control of chaotic and hyper-chaotic systems represents a crucial area of research in the field of nonlinear dynamic systems. In this study, we focus on applying chaos control techniques to a permanent magnet synchronous motor (PMSM), a system known to exhibit chaotic behavior under certain conditions. To achieve this, a sliding mode control (SMC) strategy integrated with a Lyapunov-based observer is proposed. The core concept involves designing a candidate Lyapunov function that governs the application of the control law, ensuring system stability while effectively suppressing chaotic dynamics. Through numerical simulations, the proposed sliding mode controller demonstrates its effectiveness in rapidly eliminating chaotic behavior and stabilizing the PMSM system toward a predefined reference trajectory. Notably, the system achieves error convergence within approximately 0.7 seconds under full control (four channels). When control channels are reduced to two, the system still maintains stability, showing flexibility and cost efficiency. In a further simulation, the chaotic PMSM is subjected to two unknown external disturbances, and the proposed controller continues to maintain stability with only a slight increase in convergence time. These quantitative results affirm the robustness, accuracy, and practicality of the proposed control method. This research confirms that integrating sliding mode control with a Lyapunov observer is an effective approach for chaos suppression in PMSMs, offering promising insights for the development of advanced control strategies in nonlinear electromechanical systems.

Effectively forecasting electricity generation, consumption, and pricing enhances power utilization efficiency, safeguards the stable operation of power systems, and assists power generation enterprises in formulating rational generation plans and dispatch schedules. The electricity generation, consumption, and pricing system exhibits complex chaotic dynamics. Establishing effective predictive models by leveraging the strong coupling and multi-scale uncertainty characteristics of nonlinear dynamical systems is a key challenge in grey modelling. This study leverages grey differential information to effectively transform differential equations into difference equations. Fractional-order cumulative generation operations enable more refined extraction of data characteristics. Based on the coupling and uncertainty features of electricity generation–consumption–pricing dynamics within complex power systems, three types of fractional-order multivariate grey models are established. These models both reflect the system’s dynamic relationships and expand the conceptual framework for grey prediction modelling. Simultaneously, the effectiveness of these three models is analyzed using data on generation, consumption, and prices from both new and traditional power sources within China’s electricity system. Employing identical annual data, the models are evaluated from two distinct perspectives: variations in the numbers of simulated and predicted variables. Experimental results demonstrate that all three novel models perform well. Finally, the most effective predictive application of the three models was selected to forecast electricity generation, consumption, and pricing in China. This provides a basis for China’s power system and supports national macro-level intelligent energy dispatch planning.