Nonlinear Dynamical Systems Research Articles

Research on nonlinear dynamics of planetary gear system of wind turbine gearbox considering wear

Research on nonlinear dynamics of planetary gear system of wind turbine gearbox considering wear

Exploring Non-linear Dynamics: Constructing the Bifurcation Diagram of a Damped Driven Pendulum using Python

In this paper, we present a detailed analysis and construction of the bifurcation diagram for the damped-driven pendulum system. The bifurcation diagram in general presents the qualitative changes of the steady-state behavior for the pendulum. For this purpose, we implement the use of the Python programming language with the inclusion of scientific libraries. This nonlinear dynamical system is an example of a system that exhibits a chaotic regime, which is the sensitivity of its behavior to the initial conditions and the parameters of the system. We investigate the response of the system to a range of drive strengths γ applied. By changing the driving strength, the system reveals patterns of periodicity, quasi-periodic, and chaotic regimes. The critical values where it goes from regular motion to chaotic one are highlighted, offering further understanding of the mechanisms of the transitions. This study presents the use of the Python programming language for the modeling and visualization of non-dynamic systems and contributes to a deeper understanding of nonlinear oscillator dynamics.

A Nonlinear Approach in the Quantification of Numerical Uncertainty by High-Order Methods for Compressible Turbulence with Shocks

This is a comprehensive overview on our research work to link interdisciplinary modeling and simulation techniques to improve the predictability and reliability simulations (PARs) of compressible turbulence with shock waves for general audiences who are not familiar with our nonlinear approach. This focused nonlinear approach is to integrate our “nonlinear dynamical approach” with our “newly developed high order entropy-conserving, momentum-conserving and kinetic energy-preserving methods” in the quantification of numerical uncertainty in highly nonlinear flow simulations. The central issue is that the solution space of discrete genuinely nonlinear systems is much larger than that of the corresponding genuinely nonlinear continuous systems, thus obtaining numerical solutions that might not be solutions of the continuous systems. Traditional uncertainty quantification (UQ) approaches in numerical simulations commonly employ linearized analysis that might not provide the true behavior of genuinely nonlinear physical fluid flows. Due to the rapid development of high-performance computing, the last two decades have been an era when computation is ahead of analysis and when very large-scale practical computations are increasingly used in poorly understood multiscale data-limited complex nonlinear physical problems and non-traditional fields. This is compounded by the fact that the numerical schemes used in production computational fluid dynamics (CFD) computer codes often do not take into consideration the genuinely nonlinear behavior of numerical methods for more realistic modeling and simulations. Often, the numerical methods used might have been developed for weakly nonlinear flow or different flow types other than the flow being investigated. In addition, some of these methods are not discretely physics-preserving (structure-preserving); this includes but is not limited to entropy-conserving, momentum-conserving and kinetic energy-preserving methods. Employing theories of nonlinear dynamics to guide the construction of more appropriate, stable and accurate numerical methods could help, e.g., (a) delineate solutions of the discretized counterparts but not solutions of the governing equations; (b) prevent numerical chaos or numerical “turbulence” leading to FALSE predication of transition to turbulence; (c) provide more reliable numerical simulations of nonlinear fluid dynamical systems, especially by direct numerical simulations (DNS), large eddy simulations (LES) and implicit large eddy simulations (ILES) simulations; and (d) prevent incorrect computed shock speeds for problems containing stiff nonlinear source terms, if present. For computation intensive turbulent flows, the desirable methods should also be efficient and exhibit scalable parallelism for current high-performance computing. Selected numerical examples to illustrate the genuinely nonlinear behavior of numerical methods and our integrated approach to improve PARs are included.

Use of Deep-Learning-Accelerated Gradient Approximation for Reservoir Geological Parameter Estimation

The estimation of space-varying geological parameters is often not computationally affordable for high-dimensional subsurface reservoir modeling systems. The adjoint method is generally regarded as an efficient approach for obtaining analytical gradient and, thus, proceeding with the gradient-based iteration algorithm; however, the infeasible memory requirement and computational demands strictly prohibit its generic implementation, especially for high-dimensional problems. The autoregressive neural network (aNN) model, as a nonlinear surrogate approximation, has gradually received increasing popularity due to significant reduction of computational cost, but one prominent limitation is that the generic application of aNN to large-scale reservoir models inevitably poses challenges in the training procedure, which remains unresolved. To address this issue, model-order reduction could be a promising strategy, which enables us to train the neural network in a very efficient manner. A very popular projection-based linear reduction method, i.e., propel orthogonal decomposition (POD), is adopted to achieve dimensionality reduction. This paper presents an architecture of a projection-based autoregressive neural network that efficiently derives an easy-to-use adjoint model by the use of an auto-differentiation module inside the popular deep learning frameworks. This hybrid neural network proxy, referred to as POD-aNN, is capable of speeding up derivation of reduced-order adjoint models. The performance of POD-aNN is validated through a synthetic 2D subsurface transport model. The use of POD-aNN significantly reduces the computation cost while the accuracy remains. In addition, our proposed POD-aNN can easily obtain multiple posterior realizations for uncertainty evaluation. The developed POD-aNN emulator is a data-driven approach for reduced-order modeling of nonlinear dynamic systems and, thus, should be a very efficient modeling tool to address many engineering applications related to intensive simulation-based optimization.

A ship motion forecasting approach based on Fourier transform, regularized Bi-LSTM and chaotic quantum adaptive WOA

Ship motion forecasting is of great significance to assist ship navigation and ensure ship operation. However, due to the coupling influence of wind, waves, currents and other factors, the ship motion has a strong time variation, which brings great challenges to the improvement of the ship motion forecasting accuracy. This paper proposes a novel approach for ship motion forecasting. Firstly, Fourier Transform (FT) is used for data pre-processing to deal with the data noise. Secondly, the Bi-LSTM is applied to simulate the nonlinear dynamical system of ship motion, and the Dropout mechanism and l2 regularization method are used to improve the generalization performance of the network, and a novel ship motion prediction model (FRB model) is constructed. Thirdly, in view of the inherent defects of Whale Optimization Algorithm (WOA), based on chaotic mapping and quantum computing, a new Chaotic Quantum Adaptive Whale Optimization Algorithm (CQAWOA) is proposed to construct a new ship motion forecasting approach, namely FRB&CQAWOA. Finally, based on the measured pitch and roll data of ship, the feasibility and superiority of the established FRB model and the proposed CQAWOA are tested. The test results indicate that the proposed model have better prediction accuracy than other algorithms selected in this paper.

Numerical optimal control for nonlinear dynamical systems involving mixed-valued inputs and joint probability path-constraints

Numerical optimal control for nonlinear dynamical systems involving mixed-valued inputs and joint probability path-constraints

Decomposition and equivalence of general nonlinear dynamical control systems

AbstractProjection and equivalence concepts have been widely studied in the literature. In the present paper two nonlinear systems with the same number of inputs, but not the same number of state variables, are considered. Under mild assumptions, necessary and sufficient conditions are given for the existence of a submersion such that the higher dimensional system projects locally onto the other one. The solution to this problem has relevant applications, for instance in robotics. It includes as a special case the equivalence of nonlinear systems with no particular structure.

Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator

This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.

Digital-Twin Predictive Control of Nonlinear Systems With Time Delays, Unknown Dynamics, and Communication Delays.

With the advancement of computing technology and big data technology, digital twins have gradually been applied in various fields, such as manufacturing, energy, and healthcare. This article studies the predictive control of nonlinear dynamic systems using digital twins. Based on a digital-twin control system framework, predictive control is discussed for three different nonlinear systems with time delays: 1) known nonlinear systems; 2) unknown nonlinear systems; and 3) unknown nonlinear cyber-physical systems. Both a digital-twin predictive control strategy and a digital-twin control predictor are proposed to compensate for time delays and communication delays actively. With the strategy and predictor, the digital-twin controller of a time-delay nonlinear system can be designed to achieve the desired performance based on the nonlinear system without time delays, which vastly simplifies the controller design procedure. A digital model is constructed using data to deal with unknown nonlinear dynamics. The three different closed-loop digital-twin predictive control systems are analyzed to derive a unified stability criterion. The simulation results show how the proposed digital-twin predictive control method performs well for nonlinear systems with time delays, unknown dynamics, and/or communication delays.

Nonlinear performance of aerospace face gear-planetary gear transmission system under multi-factors including the effect of temperature

Face gear-planetary gear transmission system combines the advantages of both planetary and face gears. And has enormous advantages in the aerospace field. The meshing of gear teeth generates heat through friction, resulting in thermal deformation and affects nonlinear dynamic characteristics. Exploring the temperature effect on the nonlinear dynamics of the transmission system of face gear-planetary gear is of great significance. Considering the temperature variations effect and time-varying meshing stiffness, tooth side clearance as well as meshing error, a nonlinear dynamic model of the face gear-planetary gear system is established. The nonlinear dynamic characteristics of the system were described by maximum Lyapunov exponent diagram, phase diagram, bifurcation diagram et al. The findings indicate that the effect of temperature variations on the gear system is related to the values of the time-varying mesh stiffness coefficient and mesh damping ratio. When the mesh damping ratio is small, the influence of temperature changes on the bifurcation characteristics of the system is more obvious. As the time-varying mesh stiffness value increases, the temperature rise gradually stabilizes in the range of 15°C to 42°C; Under certain temperature changes, as the meshing damping ratio and time-varying meshing stiffness coefficient increase, the face gear planetary gear system gradually tends to stabilize.

Modeling and simulation of nonlinear dynamical systems for biosensor sensitivity based on carbon nanocomposites

Nanomaterials have a wide range of applications in various fields of scientific research due to their unique physical and chemical properties. With the rapid development of science and technology, nanocomposites synthesized from various nanoparticles have obtained many excellent properties due to their synergistic effects. In this paper, a series of carbon-based nanomaterials are proposed for in-depth research, and corresponding biosensors are constructed. In this study, ECL biosensors based on a variety of nanocomposite materials will be used to detect and analyze cells, and also detect other cells that are similar to cells. The experimental data show that the relative standard deviation of the detection results of the two methods is within 8%, and the sensor has high sensitivity, excellent stability and fast response speed. The sensor showed excellent performance in the repeatability test, and the relative standard deviation of repeated detection was less than 2%. This result shows that the sensor has highly consistent detection capabilities, providing important support for its reliability in practical applications. By adding the description of repeatability data, the summary more comprehensively reflects the performance advantages of the sensor.

Efficient data-driven nonlinear system identification for structural health monitoring: A proof-of-principle study

Structural health monitoring (SHM) is the process of detecting damage in structures. Typically, the damage-sensitive feature identification is based on fitting some models to the measured system response data. The parameters of these models or the prediction errors associated with these models then become the desired damage-sensitive features. Real-world structures usually exhibit significant nonlinear behaviors. Data-driven nonlinear system identification (NSI) is essential to obtain accurate models of structural dynamics for reliable damage detection. However, on the one hand, the data acquisition of real-world structures for damage detection is usually difficult or scarce. On the other hand, NSI generally needs to solve a nonlinear optimization problem that requires more computational time for convergence than linear system identification. Data efficiency and computational efficiency become two critical challenges in data-driven NSI. To address these challenges, this work presents a proof-of-principle study on developing a meta learning-based method for efficient data-driven NSI. Specifically, the meta learning algorithm leverages a previously collected database of similar but different systems to learn the meta-knowledge about how to identify a new system, enabling a fast identification/modeling with limited data. Numerical simulations are performed to validate the method on fundamental nonlinear dynamical systems widely seen in civil engineering. It is observed that the presented meta learning-based method outperforms the conventional method in terms of both data and computational efficiency. Limitations of this work are further discussed.

Machine learning model‐based optimal tracking control of nonlinear affine systems with safety constraints

AbstractThis work focuses on the development of a machine learning (ML) model‐based framework for safe optimal tracking control of a class of nonlinear control‐affine systems to ensure simultaneous closed‐loop stability and safety. Specifically, a novel multilayer feedforward neural network (FNN) with a control‐affine architecture is designed to model nonlinear dynamic systems. Subsequently, a model‐based reinforcement learning (RL) framework is presented, utilizing a novel cost function with Control Lyapunov‐Barrier Function (CLBF) properties, to learn both the control policy and the optimal value function for an infinite‐horizon optimal tracking control problem for nonlinear systems with safety constraints. The efficacy of the proposed methodology is demonstrated through simulations of a one‐link robot manipulator and a chemical process example.

Investigation of the nonlinear dynamics of thin sandwich shells composed of functionally graded materials with double curvature in thermal environments

Double curvature structures play a crucial role as load-bearing components across various engineering fields. Functionally graded materials, blending metals and ceramics, boast superior material characteristics, fueling their expanding applications. This study pioneers the non-linear dynamic analysis of sandwich shells that feature functional gradient compositions in thermal settings. We assume that both the upper and lower shells of these double curvature structures are crafted from functionally graded materials, with a ceramic core. This variation in material exhibits a ceramic layer on the inner surface and a metallic layer on the outer surface, showing properties that vary along the shell thickness in a power-law gradient. Non-linear dynamic equations are derived using third-order shear theory, encompassing geometric non-linearity and shear deformation. Employing the Galerkin method, we discretize the equations of motion into a non-linear dynamic system with five degrees of freedom, and subsequently give an analytical expression for the nonlinear naturalfrequency by means of multiscale analysis. Our discussion examines the impacts of structural parameters, porosity volume fraction, volume fraction index, and temperature differences on the non-linear/linear frequency ratio of doubly curved sandwich shells. Calculations reveal a sharp rise followed by a rapid decline in the non-linear/linear frequency ratio with increasing structural aspect ratio b/a. Conversely, it decreases with increasing thickness-to-length and radius-to-length ratios, with temperature differences initially reducing and later increasing it. These findings offer practical insights for designing functional gradient double curvature sandwich shells.

Adaptive control with lower-order dynamics for nonlinear systems with unknown control coefficients

While adaptive control has been extensively studied for a variety of uncertain nonlinear systems, overparametrization still exists in many works. By overparametrization, it means that the order of the adaptive controller is (much) higher than the number of unknown parameters of the system, which is a defect from a theoretical perspective and indicates high cost in controller's implementation. This paper investigates adaptive control with lower-order dynamics for nonlinear systems with unknown control coefficients, managing to significantly reduce or eliminate overparametrization and showing the differences in controllers' dynamic orders of different control strategies. The uncertain nonlinear system in question has n unknown control coefficients and a p-dimension unknown vector in the system nonlinearities. We consider two cases of unknown control coefficients: case (a): their signs are known; case (b): their signs are unknown, i.e. unknown control directions. For case (a), first by compensating for unknown control coefficients and their reciprocals, as well as the unknown vector, we design an adaptive controller with ( 2 n + p − 1 ) -order dynamics, where no inequality estimate is used to avoid conservatism of the controller. Second, for case (a), via bounding the system uncertainties by a single unknown constant, we design an adaptive controller with only 1-order dynamics, where inequality estimates are used and hence the controller suffers conservatism. For case (b), by adopting multiple Nussbaum functions, we design a ( 2 n + p − 1 ) -order adaptive controller, where n-order dynamic compensation is employed to overcome n non-identical unknown control directions. As in case (a), because no inequality estimate is used, conservatism of the controller is avoided. Regardless of conservatism, for case (b), by bounding the uncertainties in the destabilizing terms, we design n-order adaptive controller, where, of course, inequality estimates have to be used.

Research of a nonlinear dynamic system describing the mathematical model of the bipolar world

Research of a nonlinear dynamic system describing the mathematical model of the bipolar world

Enhancing trajectory tracking accuracy in three-wheeled mobile robots using backstepping fuzzy sliding mode control

The rise in robotics technology has increased interest in ThreeWheeled Mobile Robots (TWMRs) due to their agility and adaptability across various applications. However, effectively controlling TWMRs presents a significant challenge owing to their inherent nonholonomic constraints, which restrict independent movement in all directions. Factors like sensor noise, nonlinear system dynamics, and uncertain system parameters also add to the complexity of controlling TWMRs. This research endeavors to enhance the precision of trajectory tracking in TWMRs. Specifically, it employs Backstepping Fuzzy Sliding Mode Control (BFSMC) with parameters optimized through Particle Swarm Optimization (PSO), coupled with the Extended Kalman Filter (EKF) for state estimation. The study conducts a comprehensive performance comparison between Backstepping Sliding Mode Control (BSMC) and Backstepping Fuzzy Sliding Mode Control(BFSMC) across various trajectory patterns, revealing substantial improvements in trajectory tracking accuracy with BFSMC. BFSMC demonstrates improvements in performance across various trajectory types when considering the integral time absolute error (IAE). Specifically, it achieves a 51.97% improvement for circular trajectories, an 82.09% improvement for infinity trajectories, and an 84.073% improvement for spiral trajectories. Moreover, BFSMC demonstrates superior robustness in the presence of disturbances, noise, parameter variations, and unmodeled dynamics compared to BSMC. Integrating the Extended Kalman Filter further improves accuracy, particularly in noisy conditions.

Type-3 fuzzy neural networks for dynamic system control

This paper presents a Type-3 Fuzzy Neural Network (T3FNN) for dynamic system control. The structure of the T3FNN controller, which is based on the compositional rule of inference of interval type-3 fuzzy systems, is proposed. TSK-type fuzzy rules, employing three-dimensional membership functions (MFs) in the antecedent part and linear functions with coefficients represented as interval sets in the consequent part, are utilized to design T3FNN. Gaussian functions with uncertain centers are employed to describe the type-3 membership functions. By employing three-dimensional MFs, type-3 fuzzy logic rules can accurately model a high degree of uncertainty and effectively handle uncertain information. The design of the T3FNN was accomplished by integrating the type-3 fuzzy logic system (T3FLS) with neural networks. The learning of the T3FNN was conducted using the genetic algorithms and gradient descent algorithm. As a result, the type-3 rule base of the system was effectively developed. The learning algorithms are presented and the proposed T3FNN structure is employed for the development of dynamic system control. The simulations of the T3FNN system for the control of the nonlinear and time-varying dynamic systems were carried out. The results obtained from the simulations indicate the suitability of using the T3FNN in controlling dynamic systems.

A modified neural network method for computing the Lyapunov exponent spectrum in the nonlinear analysis of dynamical systems

This study presents a modified neural network method for computing the Lyapunov exponent spectrum in non-linear dynamical systems. Its mathematical description is an introduction. The proposed modified neural network method allows the addition of bias and constant neurons to the neural network topology and the use of different numbers of activation functions to suit different cases. Various algorithms for computing Lyapunov exponents, such as the Benettin method, the Wolf method, the Rosenstein method, the Kantz method, the Synchronisation method, the Sano-Sawada algorithm and the proposed modification of the neural network method, are used for classical problems in nonlinear dynamics. These problems include the generalised Hénon map, the chaotic attractor of the Baier-Klein map, and the vibrations of mechanical systems such as the flexible Bernoulli-Euler beam and the flexible functionally graded porous closed cylindrical shell under alternating load. The comparative analyses presented in this study are aimed at validating the accuracy and effectiveness of the methods, and at identifying the most relevant approaches for different types of systems and classes of problems. The proposed method is demonstrated to be superior to existing methods based on time-series evaluation in terms of sample size and accuracy. Furthermore, it does not require the initial system equations.

Stability analysis for viscoelastic fluid model with thermorheological effects: Linear and nonlinear approaches

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D2-Chebyshev-τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes.