Behavior Of Nonlinear Systems Research Articles

Model Optimization and Dynamic Analysis of Inventory Management in Manufacturing Enterprises

This study investigates inventory management systems using a sample of listed manufacturing companies in China from 2019 to 2023. By constructing a static mathematical model, the impact of inventory management on corporate performance was empirically tested. Additionally, based on a classical inventory management dynamical model and considering inventory delay characteristics, a new class of two-dimensional inventory management systems was reconstructed. The system’s periodic and chaotic nonlinear characteristics were verified using 0-1 tests, bifurcation diagrams, Lyapunov exponents, and system eigenvalue plots. Furthermore, MATLAB simulations were employed to examine the effect of resource transfer rates on the nonlinear dynamic behavior of the inventory management system. The results from both static mathematical models and dynamical models provide a theoretical basis for inventory management and safety stock level predictions in the manufacturing industry.

Analysis of Displacement Transmissibility and Bifurcation Behavior in Nonlinear Systems with Friction and Nonlinear Spring

Analysis of Displacement Transmissibility and Bifurcation Behavior in Nonlinear Systems with Friction and Nonlinear Spring

Bifurcation analysis in dynamical systems through integration of machine learning and dynamical systems theory

Abstract Characterizing the nonlinear behavior of dynamical systems near the stability boundary is a critical step toward understanding, designing, and controlling systems prone to stability concerns. Traditional methods for bifurcation analysis in both experimental systems and large-dimensional models are often hindered either by the absence of an accurate model or by the analytical complexity involved. This paper presents a novel approach that combines the theoretical frameworks of nonlinear reduced-order modeling and stability analysis in dynamical systems with advanced machine learning techniques to perform bifurcation analysis in dynamical systems. By focusing on a low-dimensional nonlinear invariant manifold, this work proposes a data-driven methodology that simplifies the process of bifurcation analysis in dynamical systems. The core of our approach lies in utilizing carefully designed neural networks to identify nonlinear transformations that map observation space into reduced manifold coordinates in its normal form where bifurcation analysis can be performed. The unique integration of analytical and data-driven approaches in the proposed method enables the learning of these transformations and the performance of bifurcation analysis with a limited number of trajectories. Therefore, this approach improves bifurcation analysis in model-less experimental systems and cost-sensitive high-fidelity simulations. The effectiveness of this approach is demonstrated across several examples.

Dynamic behavior of a cable-stayed footbridge depending on the calculation accuracy

Footbridges are analyzed in terms of dynamic response for pedestrian comfort. This problem is solved mainly on light and at the same time rigid constructions, which are easily oscillated by pedestrian load. It is often solved on steel footbridges, but with the technological development of UHPC, we are capable nowadays to build relatively light and long span concrete footbridges. Behaviour of a thin concrete cross-section with its geometry is closer to steel structures, but at the same time we have to deal with rheological effects such as creep and shrinkage. There are many influences on the structure that can affect the dynamic behavior of the structure, including the non-linear behavior of the cable system. This paper presents an initial entry into the computational issues of more complex constructions that have more input influences.

Model-Based Sequential Design of Experiments with Machine Learning for Aerospace Systems

Traditional experimental design methods often face challenges in handling complex aerospace systems due to the high dimensionality and nonlinear behavior of such systems, resulting in nonoptimal experimental designs. To address these challenges, machine learning techniques can be used to further increase the application areas of modern Bayesian Optimal Experimental Design (BOED) approaches, enhancing their efficiency and accuracy. The proposed method leverages neural networks as surrogate models to approximate the underlying physical processes, thereby reducing computational costs and allowing for full differentiability. Additionally, the use of reinforcement learning enables the optimization of sequential designs and essential real-time capability. Our framework is validated by optimizing experimental designs that are used for the efficient characterization of turbopumps for liquid propellant rocket engines. The reinforcement learning approach yields superior results in terms of the expected information gain related to a sequence of 15 experiments, exhibiting mean performance increases of 9.07% compared to random designs and 6.47% compared to state-of-the-art approaches. Therefore, the results demonstrate significant improvements in experimental efficiency and accuracy compared to conventional methods. This work provides a robust framework for the application of advanced BOED methods in aerospace testing, with implications for broader engineering applications.

Analyzing the chaotic and stability behavior of a duffing oscillator excited by a sinusoidal external force

This paper delves into the dynamical, chaotic, and stability aspects of the Duffing oscillator (DO) under sinusoidal external excitation, underscoring its relevance in various scientific and engineering applications. The DO, with its complex behavior, poses a significant challenge to our understanding. A unique perturbation technique, the multiple-scales (MS), is harnessed to tackle this challenge and enrich our knowledge. The analysis entails determining third-order expansions, exploring resonant cases, and factoring in the influence of viscous damping. The accuracy of the analytical solution is cross-validated with numerical results using the Runge–Kutta fourth-order (RK4). The paper also employs effective methods to assess the obtained results qualitatively. As a result, Visual figures, such as bifurcation diagrams and Lyapunov exponents’ spectra (LES), have been presented to illustrate diverse system motions and demonstrate Poincaré diagrams. Moreover, stability analysis has been investigated via resonance curves showing the stable and unstable regions. These aids facilitate our comprehension of the system’s complex behavior and its variations under different conditions. The results are elucidated through displayed curves, offering insights into the dynamics of the DO under sinusoidal excitation and damping effects. The influence of excitation and the quadratic parameter on bifurcation diagrams, LES, and Poincaré maps helps us understand the intricate behavior of nonlinear oscillatory systems, such as DO. The presented oscillator is a compelling example of elucidating the nonlinear behavior observed in numerous engineering and physics phenomena.

Research on rigid-flexible coupling nonlinear dynamics of light commercial vehicle considering frame flexibility

Commercial vehicles play a vital role in the transportation industry. Generally, there will present a complex nonlinear dynamic behavior composed of periodic, quasi-periodic, and even chaotic vibration in vehicle vertical system under the continuous speed bump excitation. To explore the nonlinear dynamic behavior of vehicle system, a new rigid-flexible coupling nonlinear dynamic model for light commercial vehicle has been developed theoretically with considering the frame flexibility, suspension stiffness and damping nonlinearity, and rubber bushing nonlinearity. The dynamic model for frame and suspension have been developed with discrete element method and lumped mass method, respectively. Then, the flexible frame model has been verified through the mode shape and frequency calculated by finite element simulation. And the rigid-flexible coupling nonlinear dynamic model of a light commercial vehicle has been verified with Adams simulation and testing results. Furthermore, the influence of vehicle speed and the height of speed bump, suspension stiffness and damping, as well as cargo weight and distribution on vehicle system dynamics is analyzed with bifurcation diagram, time history, frequency, phase diagram and Poincare section. The results show the significant and complex effects on the nonlinear vibration of vehicle system, especially the nonlinear damping and the flexible frame.

Securing consensus in fractional-order multi-agent systems: Algebraic approaches against Byzantine attacks

This paper investigates the behavior of fractional-order nonlinear multi-agent systems subjected to Byzantine assaults, specifically focusing on the manipulations of both sensors and actuators. We employ weighted graphs, both directed and undirected, to illustrate the system's topology. Our methodology combines algebraic graph theory with fractional-order Lyapunov techniques to develop algebraic requirements for leader-following consensus, providing a robust framework for analyzing consensus dynamics in these complex systems. We present quantitative results demonstrating the effectiveness of our approach, including two numerical examples that validate the proposed requirements for consensus evaluation. Notably, our work highlights the novelty of using fractional-order systems to enhance resilience against adversarial conditions, contributing significantly to the field of multi-agent systems. By clarifying key terms and streamlining our language, we ensure accessibility for a broader audience while emphasizing the implications of our findings for real-world applications.

Geometrically exact post-buckling and post-flutter of standing cantilevered pipe conveying fluid

Although the nonlinear dynamics of hanging cantilevered pipes conveying fluid have been extensively scrutinized, there is limited research on the nonlinear behavior of standing ones. Hence, the objective of this study is to examine the geometrically exact nonlinear static and dynamic responses of cantilevered pipes conveying fluid in a standing position. The geometrically exact rotation-based model, combined with the shooting method and the Galerkin technique, is applied to assess the nonlinear static behavior of system and its stability characteristics. Moreover, to compute the nonlinear dynamics of system, the geometrically exact quaternion-based model, together with the Galerkin technique, is employed. It is revealed that the system may undergo buckling through either a supercritical or subcritical pitchfork bifurcation, depending on the gravity parameter, which may give rise to extremely large-amplitude responses. The system may also experience flutter instability due to a supercritical Hopf bifurcation, which brings about self-excited periodic oscillations. The generic behavior of system for a specific range of the gravity parameter is investigated across four distinct scenarios, which vary based on the gravity parameter and mass ratio. Notably, only one of these scenarios is analogous to the situation that comes to pass for the hanging case.

On the Periodicity Solutions of Five Systems of Rational Systems of Difference Equations of Order Five

The primary aim of this paper is to explore the behavior of several nonlinear systems of difference equations following T_{n+1}=(E_{n}E_{n-4})/(T_{n-3}(\pm 1 \pm E_{n}E_{n-4})), E_{n+1}=(T_{n}T_{n-4})/(E_{n-3}(pm 1 \pm T_{n}T_{n-4})), and obtain solution expressions for them. Moreover, we utilize MATLAB programming to simulare the dynamics and validate our results.

On the study of solitary wave dynamics and interaction phenomena in the ultrasound imaging modelled by the fractional nonlinear system

This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields.

Data-driven modeling of subharmonic forced response due to nonlinear resonance

Complex behavior in nonlinear dynamical systems often arises from resonances, which enable intricate energy transfer mechanisms among modes that otherwise would not interact. Theoretical, numerical and experimental methods are available to study such behavior when the resonance arises among modes of the linearized system. Much less understood are, however, resonances arising from nonlinear modal interactions, which cannot be detected from a classical linear analysis. Academic examples of such phenomena have been known, but no systematic method has been developed to detect and model nonlinear resonant interactions purely from numerical or experimental data. Here, we develop such a data-driven methodology that identifies nonlinear resonant response on low-dimensional spectral submanifolds (SSMs) of the dynamical system. Our approach is generally applicable to nonlinear resonances, but we specifically focus here on one particular behavior: subharmonic response in forced nonlinear systems without any resonance among the linearized frequencies of the unforced system. We first illustrate analytically how such a response is born out of a nonlinear resonance hidden in the conservative limit of the system. We then show how this effect can be identified and modeled purely from data. As our main example, we isolate and model previously unexplained response patterns in fluid sloshing experiments.

Comparative study of LSTM and ANN models for power consumption prediction of variable refrigerant flow (VRF) systems in buildings

The optimized control of variable refrigerant flow (VRF) system requires an accurate time series forecast model for power consumption. Currently, physics-based and black-box models widely used for forecasting power consumption may not be able to capture the dynamic and non-linear behavior of such complex systems. This study presents a long-short-term memory (LSTM), deep learning-based model to accurately predict the power consumption of a VRF system with heat recovery units. The model training used one year of VRF system field test data. The feature selection through the Pearson correlation coefficient was implemented to improve the model's accuracy and computational efficiency. The sensitivity analysis of feature selection was performed by preparing three feature sets, including different levels of relationship with the predicted target. Additionally, the hyperparameters of the models were optimized by Bayesian optimization with the Tree-structured Parzen Estimator algorithm. The deep learning model, LSTM model, was compared to the baseline machine learning model, Artificial Neural Network (ANN) and decision tree. The results show that LSTM-30feat with input time step 4 has the best testing performance of Coefficient of the Variation of the Root Mean Square Error (CvRMSE) 23.3%. The best ANN model is ANN-10feat with input time step 8, which has a CvRMSE of 27.8% in testing and 13,569 trainable parameters. However, LSTM-10feat with input time step 4 has the CvRMSE of 24.8% in testing, and the trainable parameters are 1,809. A higher number of trainable parameters in models might result in increased memory usage on the computer and be computationally expensive.

Nonlinear behavior and energy harvesting performance of a new tunable quasi-zero stiffness system.

Energy harvesting from vibrations is a popular research topic. However, it is difficult to change the values of its dynamic stiffness characteristics during the operation of device. This paper analyzes the nonlinear dynamical behavior and energy harvesting performance of an energy harvesting device with a gear unit and an inertial amplifier. The Melnikov method is used to discriminate the chaos behavior of the system and is illustrated by numerical simulations. Chaotic dynamics analysis provides a simplified analytical idea that offers more insight into the performance of this energy harvesting device. In addition, the basins of attraction of the coexistence solution of the system are plotted, and the trajectory and energy harvesting efficiency of the system are analyzed for changes in the initial value. The results show that the device has the convenience to change the form and efficiency of system energy harvesting. The theory of Melnikov functions recognizes the presence of chaos and can provide solutions for the parameterization of the system. The choice of initial value greatly affects the energy harvesting efficiency.

Conversion mechanisms and transformed waves for the (3 + 1)-dimensional nonlinear equation

In this paper, we focus on investigating the (3 + 1)-dimensional nonlinear equation which is used to describe the propagation of waves in the shallow water. The study begins with the application of the Hirota bilinear method to obtain N-soliton solution. Building on this foundation, the research delves into the construction of first-order breather wave by imposing complex conjugate constraints on the parameters of two solitons. Further analysis of the characteristic lines of breathers leads to the derivation of conversion conditions. Under this specific condition, a series of nonlinear transformed waves are presented, including quasi-kink solitons, W-shaped kink solitons, oscillation W-shaped kink solitons, multipeaks solitons, quasi-periodic waves, and line rogue waves. Each of these transformed waves exhibits unique structural and dynamic properties, enriching the understanding of wave behavior in higher-dimensional nonlinear systems. The study also explores the nonlinear superposition mechanism between solitary wave and periodic wave. This mechanism elucidates the formation process of nonlinear waves, explaining how their locality and oscillatory characteristics emerge from the superposition of different wave components. Moreover, the geometric properties of the two characteristic lines of the waves are analyzed to understand the time-varying nature of the transformed waves. This temporal analysis is crucial for predicting the evolution and interaction of these waves over time. Finally, the research extends to the higher-order breather wave and explores the interactions among various waves. These interactions reveal the complex dynamics that may arise in the (3 + 1)-dimensional nonlinear systems and provide deeper insights into the interactions among different wave structures.

Data-driven prediction of vortex solitons and multipole solitons in whispering gallery mode microresonator

Deep learning incorporating physics knowledge has become a powerful tool for studying the dynamic behavior of high-dimensional nonlinear systems. In this paper, the two-stage mini-batch resampling of adaptive physics-informed neural network (TMA-PINN) method is proposed to solve the (2 + 1)-dimensional variable-coefficient Lugiato-Lefever equation (vLLE). The vortex soliton in the WGM microresonator with different external excitation is investigated by TMA-PINN. It is found that external excitation can cause the rotation of vortex solitons. In addition, the effect of topological charge and external excitation on the dynamical characteristics of spatial solitons including vortex solitons and multipole solitons are investigated. The results show that the final shape of the rotation of vortex solitons and the number of azimuth lobes of multipole solitons are controlled by topological charges. Compared with classical PINN, TMA-PINN can better handle the gradient balance of various loss terms in (2 + 1)-dimensional vLLE to reconstruct the dynamic behavior of WGM microresonator solitons, having potential applications in other nonlinear systems.

Multi-method piecewise low-frequency identification

Inter-area frequency modes of electromechanical oscillations are decisive factors that influence the performance of electrical power systems. However, the accurate estimation of these frequencies remains a challenge due to the non-linear behavior of the response system following a disturbance and the strong coupling among the frequencies. This work describes a curve fitting and electromechanical frequency estimation algorithm, called Multi-method Piecewise Identification (MPI), based on a combination of the Matrix Pencil Method (MPM), the Prony method, and Subspace Identification (SSI), choosing the best approximation of an interval basis according to the model accuracy index (MAI) and the mean square error (MSE). The advantages and limitations of MPI were analyzed by computational analyses that were carried out with Python-based code that estimates the low frequencies for several ringdown signals, including oscillating signals from Australian and Brazilian systems that arise after large disturbances. To ensure robustness to random and noisy conditions, the method was tested with a synthetic signal with Gaussian white noise and time-variant frequencies, and compared to the methods mentioned above, the Hilbert–Huang Transform (HHT), and Eigensystem Realization Algorithm (ERA). The MPI has shown robust results and is capable of obtaining a better fitting for the signal than the other methods.

Self-Triggered Impulsive Control for Lyapunov Stability of Nonlinear Systems in Discrete Time.

This article studies Lyapunov stability for nonlinear systems based on discrete-time self-triggered impulsive control (STIC). With the help of comparison approach, some Lyapunov-based sufficient conditions guaranteeing nonZeno behavior and global asymptotic stability of nonlinear systems under STIC are derived. Different from the existing self-triggered mechanisms using implicit expressions to compute the next impulse instant, which may result in computational burden, we propose a novel discrete-time self-triggered mechanism (STM) by which the next impulse instant can be predicted directly by the information of the comparison system. Moreover, the resulting algorithm is easy to be implemented. It is shown that the designed STIC strategy can not only achieve a tradeoff between computational complexity, communication resource usage and control performance, but also has strong robustness and flexibility. Finally, an illustrative simulation is provided showing the effectiveness of the results.

Exact solutions of the nonlinear space-time fractional Schamel equation

This study focuses on the nonlinear space-time behavior of a plasma system made up of electrons, positive ions, and negative ions using the fractional Schamel (FS) equation. The main goal is to find exact solutions to the nonlinear FS equation by applying the extended hyperbolic function (EHF) method. The study examines how the fractional order affects the phase velocity, amplitude, and wave width of solitary wave solutions. Different exact solutions were found based on various values of the fractional order. Graphical representations are included to show the physical properties of these solutions. Overall, the results demonstrate that the EHF method is effective and reliable for finding exact solutions to the nonlinear FS equation.

Model Predictive Control based on Long-Term Memory neural network model inversion

Long Short-Term Memory (LSTM) neural networks are well suited for representing time series as, compared to other neural networks, their structure avoids vanishing or exploding gradients. LSTM has been embedded into Model Predictive Control algorithms in order to forecast the behavior of nonlinear systems. The new algorithm presented in the paper is of a different nature, as the LSTM network approximates the inverse of the system over a receding horizon and provides a sequence of future inputs as a function of a specified output trajectory. The main advantage of the method appears when the desired output trajectory is generated from a small set of parameters, for example, a convergence rate. The Model Predictive control optimizes its criterion with respect to this small set of variables, and the LSTM supplies the corresponding future control inputs. Eventually, the modeling error of the LSTM can be compensated by feeding the control sequence to the forward model and updating the controller according to the output deviation. The algorithm allows to design Model Predictive controllers for nonlinear systems in a generic way, using a very small number of decision variables even with a long receding horizon.