Behavior Of Nonlinear Systems Research Articles

Random P-bifurcation in a Duffing-van der Pol vibro-impact system with a Bingham model.

The present work investigates stochastic P-bifurcation phenomena in a Duffing-van der Pol vibro-impact oscillator containing a Bingham model under Gaussian white noise excitation. By employing non-smooth transformations and stochastic averaging techniques, an approximate analytical method is proposed to analyze the stochastic response and bifurcation behavior of nonlinear systems with friction and vibro-impact effects. Using a non-smooth transformation, the stochastically excited vibro-impact oscillator is converted into an approximately equivalent system without velocity discontinuities. Subsequently, the friction term is handled, and stochastic averaging is applied to derive the averaged stochastic Itô equation. The corresponding Fokker-Planck-Kolmogorov equation is then solved to obtain the probability density function of the system's steady-state response. Numerical simulations are conducted to verify the reliability of the proposed method. Based on these results, the critical parameter conditions for stochastic P-bifurcation are derived using singularity theory, considering both the amplitude probability density and the joint probability density of system displacement and velocity. Bifurcation diagrams, extreme value plots, amplitude probability density plots, velocity probability density plots, and joint probability density plots of system displacement and velocity are constructed for different parameter spaces. The findings demonstrate that changes in the viscous damping coefficient of the magnetorheological damper, Coulomb damping force, noise intensity, vibro-impact coefficient, and nonlinear damping coefficient can all induce stochastic P-bifurcations.

Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Abstract In this article, the (3+1)-dimensional Hirota–Satsuma–Ito-like equation is investigated by the modified direct method, from which some interaction solutions among lump, stripe solitons, and Jacobi elliptic function wave solutions are obtained, which are crucial in understanding complex behaviors in nonlinear systems where multiple wave types coexist and interact. The corresponding evolution and dynamics for the interaction solutions under different parameters are discussed. Such interactions are key to modeling realistic systems in which multiple phenomena coexist, such as fluid mechanics, plasma physics, and optical systems, where waves can exchange energy and form stable or unstable patterns. These results reported in this article can reveal the theoretical mechanisms of stability, energy transfer, and pattern formation in nonlinear media and may raise the possibility of related experiments and potential applications in nonlinear science fields, such as oceanography, nonlinear optics, and so on.

Nonequilibrium transport and the fluctuation theorem in the thermodynamic behaviors of nonlinear photonic systems

Nonequilibrium transport and the fluctuation theorem in the thermodynamic behaviors of nonlinear photonic systems

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 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.

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

In this paper, a nonlinear vibration system with friction and linear and nonlinear springs is modeled and analyzed. The analysis examined how the combination of nonlinear variables affects the displacement of the system using the slowly varying amplitude and phase (SVAP) method. The break-loose frequency at which relative motion begins was obtained as a function of the friction ratio, and it was found that the displacement transmissibility differed depending on the change in design parameters. The displacement transmissibility response showed a unique phenomenon in which bifurcation occurred in the front resonant branch before the maximum response point when the linear damping coefficient was small and the friction coefficient was large, and the displacement transfer curve was separated at a specific parameter value. This phenomenon can be divided into three parameter zones considering the bifurcation pattern and stability of the displacement transmissibility curve. In addition, a 3-D spatial zone of dimensionless parameters was presented, which can predict stability during the design process, along with the drawing method and procedure. This can be conveniently utilized in the process of setting the parameters of the isolators considering the stability of the response during the design. In the analysis and design process of vibration isolators with friction damping, this study has important implications for practical 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.

Investigation of Nonlinear Dynamic Behaviors of Vertical Rotor System Supported by Aerostatic Bearings

Abstract A common issue associated with gas bearing–rotor systems is the tendency to generate self-excited vibrations, leading to instability. To address this problem, the fluid–structure coupling model of an aerostatic bearing–rotor system is established in this paper. Then, a hybrid method combining the finite difference method (FDM) and direct integration method is employed to solve the bearing lubrication equation and rotor motion equation simultaneously. Furthermore, based on the orbits of rotor center, frequency spectrum diagrams, Poincaré maps, waterfall diagrams, and bifurcation diagrams, the effects of rotational speed, rotor mass, orifice diameter, and nominal clearance on the nonlinear dynamic behaviors of the bearing–rotor system are investigated. The results indicate that the system exhibits rich nonlinear behaviors with increasing rotational speed and rotor mass, including the occurrence of typical half-speed whirl. However, the nonlinear vibrations of the system can be restricted by selecting appropriate bearing structural parameters, providing theoretical guidance for the design of aerostatic bearing–rotor systems.

Neuromorphic Computing Based on Few-Molecule Vibration Dynamics Achieved By Surface-Enhanced Raman Scattering and Ion-Gating Stimulation

1. ObjectiveArtificial intelligence (AI), which has achieved high-level capabilities in cognition, learning, and inference through technological innovations such as deep learning, has greatly benefited human society, however, problems such as high energy consumption and increased communication traffic have also emerged. As a solution to these problems, physical reservoir computing (PRC), a kind of neuromorphic computing that uses the nonlinear behavior of physical systems to efficiently process information, has been attracting attention.[1-5] Although various methods of PRC have been reported (e.g., memristors, electrochemical cells, optical circuits, soft bodies, and spintronics devices), the molecular computing approach is promising to realize the ultimate compact and integrated PRC. However, the number of molecules in a conventional molecular reservoir system is huge, and the information processing capability of a single molecule to several molecules has not been elucidated.[2,3] In this study, we developed a few-molecule reservoir computing (FM-RC) that utilizes the molecular vibration dynamics of single to few molecules of para-mercaptobenzoic acid (pMBA). The information processing capability of few molecules was evaluated by performing information processing tasks such as blood glucose level prediction on FM-RC. [4]2. ExperimentsFigure 1a shows a schematic diagram of the measurement system of FM-RC. In order to accurately track the nonlinear dynamics of a few molecules, surface-enhanced Raman scattering (SERS) measurements using WOx nanorods/silver nanoparticles (WOx@Ag-NPs) were performed, and molecular vibration reflecting the structural change of pMBA based on the local proton adsorption change due to ion-gating stimulation was observed in the molecular dynamics (Fig. 1b) reflecting the conformational changes of pMBA based on the local proton adsorption changes induced by ion-gating stimuli. The obtained SERS spectra were regarded as independent artificial neurons (nodes) at specific sampling wavenumbers, and these node states were used to perform various information processing tasks (Fig. 1c).3. Results and DiscussionAlthough FM-RC consists of only a few molecules, it achieved a high accuracy of 95.1% to 97.7% in the nonlinear waveform transformation task and 94.3% in the analysis task of second-order nonlinear equations, and achieved the highest computational performance among physical reservoirs in the task of predicting blood glucose levels for type I pediatric diabetes patients (Fig. 1d).[4-6] These high computational performances are attributed to the independent nonlinearity of the nodes distributed in the wavenumber direction, i.e., the high dimensionality achieved by the complex and diverse response characteristics of each vibrational mode to the proton adsorption amount. In this presentation, the origin of the high computational performance of FM-RC will be discussed in terms of nonlinearity and high dimensionality, which are the main properties that determine the performance of PRCs.

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.

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.

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.