Nonlinear Dynamical Systems Research Articles

A novel method for response probability density of nonlinear stochastic dynamic systems

A novel method for response probability density of nonlinear stochastic dynamic systems

Identification of nonlinear dynamical system based on adaptive radial basis function neural networks

Identification of nonlinear dynamical system based on adaptive radial basis function neural networks

Effect of Interstage Torsional Damping on Nonlinear Dynamic Characteristics of Two-Stage Planetary Gear System

Abstract This paper explores the influence of interstage torsional damping on the nonlinear dynamics of a two-stage planetary gear transmission system. A comprehensive nonlinear dynamic model is developed, incorporating interstage torsional damping, stiffness, time-varying meshing parameters, damping, and tooth side clearance. The nonlinear equations are derived and solved using the fourth-order Runge–Kutta method. The investigation reveals that the changes in torsional damping between stages significantly affect the nonlinear dynamic characteristics of the system, especially under load fluctuations. Bifurcation characteristics are analyzed using phase diagrams, Poincaré diagrams, time history diagrams, and frequency–domain spectrograms. The simulation results demonstrate that heightened torsional damping between stages distinctly affects the motion states across excitation frequencies. Notably, it mitigates the unstable motion state caused by high-frequency excitation when tooth side clearance is minimal.

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Analyzing physical activity impact based on ubiquitous nonlinear dynamics and electroencephalography data.

Understanding complex systems is made easier with the tools provided by the theory of nonlinear dynamic systems. It provides novel ideas, algorithms, and techniques for signal processing, analysis, and classification. Presently, these ideas are being applied to the investigation of how physiological signals evolve. The study applies nonlinear dynamics theory to electroencephalogram (EEG) signals to better comprehend the range of alcoholic mental states. One of the main contributions of this paper is an algorithm for automatically distinguishing between sober and drunken EEG signals based on their salient features. The study utilized various entropy-based features, including ApEn, SampEn, Shannon and Renyi entropies, PE, TS, FE, WE, and KSE, to extract information from EEG signals. To identify the most relevant features, the study employed ranking methods like T-test, Wilcoxon, and Bhattacharyya, and trained SVM classifiers with the selected features. The Bhattacharyya ranking method was found to be the most effective in achieving high classification accuracy, sensitivity, and specificity. Classification accuracy of 95.89%, the sensitivity of 94.43%, and specificity of 96.67% are achieved by the SVM classifier with radial basis function (RBF) for polynomial Kernel using the Bhattacharyya ranking method. From the result, it is clear that the model serves as a cost-effective and accurate decision-support tool for doctors in diagnosing alcoholism and for rehabilitation centres to monitor the effectiveness of interventions aimed at mitigating or reversing brain damage caused by alcoholism.

The uniqueness and existence of solutions in a new complex function space for Kannan nonlinear dynamical systems

The uniqueness and existence of solutions in a new complex function space for Kannan nonlinear dynamical systems

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

On the investigation of fractional coupled nonlinear integrable dynamical system: Dynamics of soliton solutions

The primary focus of this paper is the investigation of the truncated M fractional Kuralay equation, which finds applicability in various domains such as engineering, nonlinear optics, ferromagnetic materials, signal processing, and optical fibers. As a result of its capacity to elucidate a vast array of complex physical phenomena and unveil more dynamic structures of localized wave solutions, the Kuralay equation has received considerable interest in the scientific community. To extract the solutions, the recently developed integration method, referred to as the modified generalized Riccati equation mapping (mGREM) approach, is utilized as the solving tool. Multiple types of optical solitons, including mixed, dark, singular, bright-dark, bright, complex, and combined solitons, are extracted. Furthermore, solutions that are periodic, hyperbolic, and exponential are produced. To acquire a valuable understanding of the solution dynamics, the research employs numerical simulations to examine and investigate the exact soliton solutions. Graphs in both two and three dimensions are presented. The graphical representations offer significant insights into the patterns of voltage propagation within the system. The aforementioned results make a valuable addition to the current body of knowledge and lay the groundwork for future inquiries in the domain of nonlinear sciences. The efficacy of the modified GREM method in generating a wide range of traveling wave solutions for the coupled Kuralay equation is illustrated in this study.

Cooperative Adaptive Fuzzy Control for the Synchronization of Nonlinear Multi-Agent Systems under Input Saturation

This research explores the synchronization issue of leader–follower systems with multiple nonlinear agents, which operate under input saturation constraints. Each follower operates under a spectrum of unknown dynamic nonlinear systems with non-strict feedback. Additionally, due to the fact that the agents may be geographically dispersed or have different communication capabilities, only a subset of followers has direct communication with the leader. Compared to linear systems, nonlinear systems can provide a more detailed description of real-world physical models. However, input saturation is present in most real systems, due to various factors such as limited system energy and the physical constraints of the actuators. An auxiliary system of Nth order is introduced to counteract the impact of input saturation, which is then employed to create a collaborative controller. Due to the powerful capability of fuzzy logic systems in simulating complex nonlinear relationships, they are deployed to approximate the enigmatic nonlinear functions intrinsic to the systems. A distributed adaptive fuzzy state feedback controller is designed by approximating the derivative of the virtual controller by filters. The proposed controller ensures the synchronization of all follower outputs with the leader output in the communication graph. It is shown that all signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around the origin. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

Simulation of nonlinear system dynamics of calcium and dopamine signaling in neurons

Simulation of nonlinear system dynamics of calcium and dopamine signaling in neurons

Nonlinear dynamical system approximation and adaptive control based on hybrid‐feed‐forward recurrent neural network: Simulation and stability analysis

AbstractWe proposed an online identification and adaptive control framework for the nonlinear dynamical systems using a novel hybrid‐feed‐forward recurrent neural network (HFRNN) model. The HFRNN is a combination of a feed‐forward neural network (FFNN) and a local recurrent neural network (LRNN). We aim to leverage the simplicity of FFNN and the effectiveness of RNN to capture changing dynamics accurately and design an indirect adaptive control scheme. To derive the weights update equations, we have applied the gradient‐descent‐based Back‐Propagation (BP) technique, and the stability of the proposed learning strategy is proven using the Lyapunov stability principles. We also compared the proposed method's results with those of the Jordan network‐based controller (JNC) and the local recurrent network‐based controller (LRNC) in the simulation examples. The results demonstrate that our approach performs satisfactorily, even in the presence of disturbance signals.

Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions

The stick–slip phenomenon is a jerking motion that can occur while two objects slide over each other with friction. There are several situations in which this phenomenon can be observed: between the slabs of the friction dampers used to mitigate vibrations in buildings, as well as between the components of the base isolation systems used for seismic protection. The systems of this kind are usually designed to work in a smooth and flawless manner, but under particular conditions undesired jerking motions may develop, yielding complex dynamic behavior even when only a few degrees of freedom are involved. A simplified approach to the problems of this kind leads to the mechanical model of a rigid block connected elastically to a rigid support and at the same time with friction to a second rigid support, both the supports having a prescribed motion. Despite the apparent simplicity of this model, it is very useful for studying important features of the non-linear dynamics of many physical systems. In this work, after a suitable formulation of the problem, the equations of motion are solved analytically in the sticking and sliding phases, and the influence of the main parameters of the system on its dynamics and limit cycles is investigated and discussed.

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Artificial neural network‐based adaptive control for nonlinear dynamical systems

SummaryThis research article presents an artificial neural network (ANN)‐based indirect adaptive control method for nonlinear dynamical systems. In this article, a modified Elman recurrent neural network (MERNN) is proposed as an identifier and controller for controlling nonlinear systems. The architecture of the proposed controller is a modified form of the existing Elman recurrent neural network. The parameter training of ANN‐based controllers is obtained by using the most popular optimization algorithm which is known as the back‐propagation algorithm. A comparative study includes Elman, Diagonal, Jordan, feed‐forward neural network (FFNN), and radial basis function network (RBFN)‐based controllers to compare with the proposed MERNN controller. To determine the controller's robustness, parameter variations, and disturbance signals have been considered. The performance analysis of the proposed controller is illustrated by two simulation examples. The simulation results reveal that MERNN can not only identify the unknown dynamics of the plant but also adaptively control it compared to the others.

Examining the performance of Laguerre-based and nonlinear predictive control models for wheeled mobile robots when facing obstacles

This focuses on the problem of controlling the formation of a team of rotating robots that are not homonomic and are not able to interact in a barrier environment. The virtual structure formation control strategy is obtained here to calculate each path of the robot separately while the reference path of the virtual center of the formation is generated by artificial potential fields. In this presentation, we will introduce a new control algorithm that utilizes Model predictive control and nonlinear system dynamics. Three traditional model prediction control controllers (MPC): logger-based MPC, nonlinear MPC, and traditional MPC are used to apply the control algorithm for the nonlinear system. Model predictive control is used to calculate the torques required to track the path by using the dynamic model of the Rotary moving robot. A proposed control rule is the most effective approach to solving the formation and tracking problems. Model Predictive Control will be introduced through the introduction of basic ideas and terms. Traditional MPC, logger-based MPC, and non-linear MPC are among the three MPC methods to be offered. The discrete time display of the power plant under control is taken into account by developing MPC methods for linear or nonlinear systems for practical implementation. The model predictor controller optimizes a cost function to calculate an optimal control sequence at any sampling point. While ignoring the rest, the system only achieves the first control action of this sequence. During subsequent sampling, the optimization problem is resolvable with the use of updated process measurements and a different horizon. The cost function formula is determined by the controller’s purpose. An error function often plays a role in determining the objective by revealing the difference between the desired and actual response. The shape of the formation can be preserved by the law of control by utilizing information from other robots, which can be assured through a new cost function. In the current conditions of disruption, the proposed controllers can also achieve the goal of tracking the path and maintaining formation, according to the results.

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Adaptable reservoir computing: A paradigm for model-free data-driven prediction of critical transitions in nonlinear dynamical systems.

A problem in nonlinear and complex dynamical systems with broad applications is forecasting the occurrence of a critical transition based solely on data without knowledge about the system equations. When such a transition leads to system collapse, as often is the case, all the available data are from the pre-critical regime where the system still functions normally, making the prediction problem challenging. In recent years, a machine-learning based approach tailored to solving this difficult prediction problem, adaptable reservoir computing, has been articulated. This Perspective introduces the basics of this machine-learning scheme and describes representative results. The general setting is that the system dynamics live on a normal attractor with oscillatory dynamics at the present time and, as a bifurcation parameter changes into the future, a critical transition can occur after which the system switches to a completely different attractor, signifying system collapse. To predict a critical transition, it is essential that the reservoir computer not only learns the dynamical "climate" of the system of interest at some specific parameter value but, more importantly, discovers how the system dynamics changes with the bifurcation parameter. It is demonstrated that this capability can be endowed into the machine through a training process with time series from a small number of distinct, pre-critical parameter values, thereby enabling accurate and reliable prediction of the catastrophic critical transition. Three applications are presented: predicting crisis, forecasting amplitude death, and creating digital twins of nonlinear dynamical systems. Limitations and future perspectives are discussed.

Physics-constrained robust learning of open-form partial differential equations from limited and noisy data

Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

Data-driven modeling of interrelated dynamical systems

Non-linear dynamical systems describe numerous real-world phenomena, ranging from the weather, to financial markets and disease progression. Individual systems may share substantial common information, for example patients’ anatomy. Lately, deep-learning has emerged as a leading method for data-driven modeling of non-linear dynamical systems. Yet, despite recent breakthroughs, prior works largely ignored the existence of shared information between different systems. However, such cases are quite common, for example, in medicine: we may wish to have a patient-specific model for some disease, but the data collected from a single patient is usually too small to train a deep-learning model. Hence, we must properly utilize data gathered from other patients. Here, we explicitly consider such cases by jointly modeling multiple systems. We show that the current single-system models consistently fail when trying to learn simultaneously from multiple systems. We suggest a framework for jointly approximating the Koopman operators of multiple systems, while intrinsically exploiting common information. We demonstrate how we can adapt to a new system using order-of-magnitude less new data and show the superiority of our model over competing methods, in terms of both forecasting ability and statistical fidelity, across chaotic, cardiac, and climate systems.

Periodic motions with impact chatters in an impact Duffing oscillator.

The periodic motions of discontinuous nonlinear dynamical systems are very difficult problems to solve in engineering and physics. Until now, except for numerical studies, one cannot find a better way to solve such problems. In fact, one still has difficulty obtaining periodic motions in continuous nonlinear dynamical systems. In this paper, a method is presented systematically for periodic motions in discontinuous nonlinear dynamical systems. The stability and grazing bifurcations of such periodic motions are studied. Such a method is presented through discussion on a periodically forced, impact Duffing oscillator. Thus, periodic motions with impact chatters in a periodically forced Duffing oscillator with one-sidewall constraint are studied. The analytical conditions for motion grazing at the boundary are developed from discontinuous dynamical systems. The impact Duffing oscillator is discretized to generate subimplicit mappings. With impact, the mapping structures are employed to construct specific impact periodic motions for an impact Duffing oscillator. The bifurcation trees of impact chatter periodic motions are achieved semi-analytically. The grazing and period-doubling bifurcations are obtained, and the grazing bifurcations are for the appearing and disappearance for an impact chatter periodic motion. The impact chatter periodic motions with and without grazing are presented for illustration of impact periodic motion complexity in the impact Duffing oscillator.