Nonlinear Oscillatory Systems Research Articles

Discovering dynamics and parameters of nonlinear oscillatory and chaotic systems from partial observations

Despite rapid progress in data acquisition techniques, many complex physical, chemical, and biological systems remain only partially observable, thus posing the challenge to identify valid theoretical models and estimate their parameters from an incomplete set of experimentally accessible time series. Here, we combine sensitivity methods and ranked-choice model selection to construct an automated hidden dynamics inference framework that can discover predictive nonlinear dynamical models for both observable and latent variables from noise-corrupted incomplete data in oscillatory and chaotic systems. After validating the framework for prototypical FitzHugh-Nagumo oscillations, we demonstrate its applicability to experimental data from squid neuron activity measurements and Belousov-Zhabotinsky reactions, as well as to the Lorenz system in the chaotic regime. Published by the American Physical Society 2024

Bifurcation analysis of Van der Pol–Mathieu equation for the dust grain charge with regularized (κ)-distributed electron-ion

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized [Formula: see text] distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced [Formula: see text], as well as waste [Formula: see text] and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor [Formula: see text] of regularized [Formula: see text]-distribution (RKD) distributed ei, the superthermality of ei particles [Formula: see text], the ratio of ion to electron temperature [Formula: see text], and the ratio of dust to electron density [Formula: see text] illustrated. It is noticed that the DAW shows promotion in width and amplitude as the frequency [Formula: see text] increases and it becomes rarefaction as the cut-off parameter [Formula: see text] increases. On the contrary, it becomes compressive under the impact of superthermality [Formula: see text] and [Formula: see text]. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized [Formula: see text] dispersed ei, as well as in the interstellar medium.

Experimental Analysis of Dynamics of a Nonlinear Oscillatory System with Self-Synchronizing Vibration Exciters when Interacting with Bulk Material

Experimental Analysis of Dynamics of a Nonlinear Oscillatory System with Self-Synchronizing Vibration Exciters when Interacting with Bulk Material

A universal application of the general integral transform: New technique for the analysis of nonlinear oscillator systems

The wide applicability of a nonlinear oscillator is the main reason why it has gained so much attention from mathematicians and scientists, while it is difficult to be solved both numerically and theoretically. Variational iteration method is one of the numerical approaches which are used to investigate these nonlinear systems. In this paper, we present a nonlinear oscillator arising in a micro-electromechanical system as an example and derive its analytical solution by He’s frequency formula method and variational iteration method coupled with a new general integral transformation. The Lagrange multiplier is formulated and the new iterative format for the correction functional of the variational iteration method is given by the general integral transformation. The approximate nonlinear frequencies are achieved by this new technique and compared with the numerical ones obtained by the Runge–Kutta method. The new general integral transformation has the same essential qualities as those for Laplace and Fourier transform, and it becomes a promising tool to nonlinear problems when coupled with the variational iteration method.

Two Proper Metrics for Quantifying Remote Synchronization of Identical Oscillator Systems

Remote synchronization (RS) has attracted considerable attention in nonlinear oscillator systems during the past decade. Two metrics, τdirect and τindirect, are proposed in this study to measure RS and identify its associated type in systems of identical oscillators, addressing limitations of traditional indicators used for nonidentical oscillators. We confirm these two metrics' correctness by comparing the identified stable regions with the actual regions of the obtained system through numerical integration, consistent with each other perfectly. Furthermore, these metrics remain valid for systems of nonidentical oscillators and yield conclusions consistent with the traditional Kuramoto order parameters, which are commonly employed to quantify RS. It is expected that the two metrics will enhance the exploration of RS properties in future studies.

Development and Validation of a Mathematical Model for Pyroelectric Temperature Measurement Sensors for Application in Mobile Robotic Systems

A pyroelectric temperature sensor for measuring human body temperature with increased accuracy and speed for application in mobile robotic systems has been developed. This pyroelectric temperature sensor for measuring human body temperature is intended for use in various educational institutions. Its usage will allow for identifying sick or potentially ill people and providing them with preliminary advice and avoid infecting other people. This is particularly important considering the seasonality of dangerous infectious diseases and the emergence of new ones (e.g., COVID-19). It is also advisable to use this pyroelectric sensor in hospitals, where temperature measurement is very crucial for monitoring the course of various diseases. The proposed pyroelectric temperature sensor is based on a nonlinear oscillatory system, which provides high sensitivity and allows for solving the problem of increasing the accuracy of measuring the human body temperature in a non-contact way. Measurement error is ±0.1% in the operating range (32–43) °C, measurement time—1 s, and the frequency instability is 3·10−4.

An Innovative Approach in Inspecting a Damped Mathieu Cubic–Quintic Duffing Oscillator

Abstract Purpose The objective of the present study is to analyze a damped Mathieu–cubic quintic Duffing oscillator as a parametric nonlinear oscillatory dynamical system. This equation has multiple applications in diverse fields, including optics, quantum physics, and general relativity. There are multiple concerns related to periodic motion and the analysis of boundary-value problems with elliptic symmetries. The current effort aims to determine the frequency amplitude of parametric nonlinear issues. Method The non-perturbative approach (NPA) is employed to transform the nonlinear ordinary differential equation (ODE) into a linear equation. The derivation of the approximate solutions is achieved without relying on typical perturbation approaches, separate from the series expansion. Hence, the objective of this study is to depart from traditional perturbation methods and acquire approximated solutions for minor amplitude parametric components without imposing any limitations. Furthermore, the technique is extended to ascertain optimal solutions for the nonlinear large amplitude of fluctuation. Results The current approach allows for rapid estimation of the frequency-amplitude relationship in order to attain successive approximations of the solutions for parametric nonlinear fluctuations. A validation is obtained for the derived parametric equation, demonstrating a high level of agreement with the original equation. An analysis of stability behavior is conducted in multiple scenarios. In addition, the Floquet theory is used to examine the transition curves. Conclusion The current technique is characterized by its clear principles, making it practical, user-friendly, and capable of achieving exceptionally high numerical precision. The current approach is highly beneficial for addressing nonlinear parametric problems due to its ability to minimize algebraic complexity during implementation.

Improving PID Controller Performance in Nonlinear Oscillatory Automatic Generation Control Systems Using a Multi-objective Marine Predator Algorithm with Enhanced Diversity

Power systems are pivotal in providing sustainable energy across various sectors. However, optimizing their performance to meet modern demands remains a significant challenge. This paper introduces an innovative strategy to improve the optimization of PID controllers within nonlinear oscillatory Automatic Generation Control (AGC) systems, essential for the stability of power systems. Our approach aims to reduce the integrated time squared error, the integrated time absolute error, and the rate of change in deviation, facilitating faster convergence, diminished overshoot, and decreased oscillations. By incorporating the spiral model from the Whale Optimization Algorithm (WOA) into the Multi-Objective Marine Predator Algorithm (MOMPA), our method effectively broadens the diversity of solution sets and finely tunes the balance between exploration and exploitation strategies. Furthermore, the QQSMOMPA framework integrates quasi-oppositional learning and Q-learning to overcome local optima, thereby generating optimal Pareto solutions. When applied to nonlinear AGC systems featuring governor dead zones, the PID controllers optimized by QQSMOMPA not only achieve 14%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} reduction in the frequency settling time but also exhibit robustness against uncertainties in load disturbance inputs.

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

Correlation and collective behaviour in Adler-type locally coupled oscillators at the edge of chaos

Dynamical systems can be analysed as computational devices capable of performing information processing. In coupled oscillators, enlarged capabilities are expected when the set of units is formed by subsets with collective behaviour within them but weak correlation between the subsets. A system of non-linear oscillators weakly coupled in the phase approximation is studied. The informational distance maps show different regimes of collective behaviour, ranging from independent, local, and global, as the control parameters of the system are changed. This rich set of behaviours happens despite the simple nature of the model used. Complex hierarchies between oscillators can be identified near the edge of chaos region in the informational distance dendrograms. We identify the emergence of local collective regimes, further corroborated by the spatiotemporal maps. Correlation diagrams also exhibit non-trivial dependence for the system at the edge of chaos.

Synchronization modes of triple flickering buoyant diffusion flames: Experimental identification and model interpretation.

The synchronization modes of a nonlinear oscillator system consisting of three identical flickering buoyant diffusion flames in isosceles triangles were studied experimentally and theoretically. Five synchronization modes, such as the in-phase, flickering death, partially flickering death, partially in-phase, and rotation modes, were experimentally observed and identified by systematically adjusting the flame distance and fuel flow rates. Two toy models were adopted to interpret the experimentally identified dynamical modes: one is the classical Kuramoto model, and the other is a complexified Stuart-Landau model, which was proposed through the introduction of the complex coupling term. The theoretical results show that the Kuramoto model successfully interpreted the dynamical modes except for those associated with amplitude death, and the complexified Stuart-Landau model well interpreted all the dynamical modes identified in our experiment. Remarkably, the proposed complexified Stuart-Landau model breaks a new path in the investigation of globally coupled nonlinear dynamical systems with identical oscillators, especially for the study of amplitude death mode.

An efficient method for generalised Wiener series estimation of nonlinear systems using Gaussian processes

System identification of dynamical systems aims to predict the output of a system for a given input by inferring model details from data. This is particularly challenging for nonlinear vibrating systems with long memory, which often have complex dynamics for which the physics is not fully understood. However, in many engineering applications, it is important to capture as much of the nonlinearity in the response as possible. The Wiener series is a classical data-driven technique that decomposes the system response into a set of orthogonal functionals of increasing order. Unlike standard black-box algorithms, such as neural networks, the series is highly interpretable and can offer insight into the nonlinearities present. However, current methods require vast quantities of data to calculate higher order terms in the Wiener series. In this paper, a novel generalised formulation of the Wiener series is developed by framing the series explicitly as a QR decomposition. Then by placing a Gaussian process prior over the kernels, their inherent structure is exploited to significantly reduce the quantity of data required for inference. Initial investigations were carried out using two systems: first pre-defined kernels and second an idealised nonlinear oscillatory system with unknown kernels. It is demonstrated that the Gaussian Process Generalised Wiener Series (GPGWS) enables the calculation of up to and including the third order kernel for systems with long memory and improves the robustness to noise.

An efficient deep neural network with amplifying sine unit for nonlinear oscillatory systems

Nonlinear differential equations play a pivotal role in modeling various physical phenomena, capturing their oscillatory behaviors. Numerous analytic and numerical methods have been developed to obtain precise or approximate solutions for nonlinear dynamics. This study introduces a novel approach utilizing neural network strategies, known for their computational flexibility. This study proposed a novel architecture for deep neural networks that is intended to simulate the nonlinear oscillations of two systems. It has been modified to include an oscillatory activation function called the amplifying sine unit. This study compares the results from our customized neural networks with those from the conventional stable and reliable numerical technique of the Runge-Kutta method of order four. Remarkably, there is striking agreement between the outcomes of the two methods, highlighting the ability of deep neural networks to identify nonlinear dynamical behaviors without requiring explicit conversion of mathematical models into an equation system.

Stroboscopic control and tracking of periodic states

Numerical continuation tools are nowadays standard methods for the bifurcation analysis of dynamical systems. Unfortunately, the full power of these methods is still unavailable in experiments, in particular, if no underlying mathematical model is at hand. We here aim to narrow this gap by providing control based continuation of periodic states which can be ultimately implemented in real-world experimental set-ups. Taking inspiration from atomic force microscopy, we develop experimentally relevant control and tracking tools for time periodic solutions in driven nonlinear oscillator systems based on stroboscopic maps.

Robust limit cycle control for finite-time generation of sustained oscillations in nonlinear systems with mixed dead-zone and saturation

This paper suggests a novel limit cycle control technique, based on the finite-time stability scheme, for producing sustained oscillations in a class of nonlinear complex systems. The considered systems are constrained to input dead-zone and saturation simultaneously and suffer from non-vanishing perturbations due to model uncertainties, and external disturbances. For this purpose, an innovative sliding manifold is proposed based on the topology of the wanted limit cycle which provides the conditions for developing the finite time stability concept for limit sets. With the aid of the finite-time command filter and the cooperation of the sliding mode control (SMC) method with the backstepping technique, a robust controller is constructed which guarantees the finite-time convergence of phase trajectories of the closed-loop system toward the wanted limit cycle. This leads to the creation of sustained oscillations, with the desirable amplitude and frequency, in the system's output. The stability analysis is proved based on the Lyapunov approach through the set stability concept. Finally, the effectiveness of the suggested methodology is confirmed by a variety of simulations.

Squeezed States Generation in a Three-Mode System of Nonlinear Quantum Oscillators

Squeezed States Generation in a Three-Mode System of Nonlinear Quantum Oscillators

Dynamic analysis of magnetic spring-based nonlinear oscillator system

This paper aims to study the dynamics of the single-degree-of-freedom magnetic spring-based oscillator system. The proposed oscillator system contains a nonmagnetic shaft, a floating permanent magnet (PM), and two fixed permanent magnets (PMs). All PMs are placed in such a way that they can repel each other. At first, the proposed system's magnetic properties and magnetic restoring force are studied. Experimental and numerical analyses have been carried out to validate the analytical investigation of the magnetic restoring force. The linear and nonlinear coefficients of the oscillator system are analysed from the magnetic restoring force. Moreover, how the gravitational force affects the equilibrium position is studied by varying the height of the oscillator. The magnetic restoring forces for different oscillator heights are also analysed. In addition, the system dynamics, such as damping ratio, eigenvalues and natural frequencies of the oscillator system, are investigated with and without electromechanical coupling. Finally, the proposed system's energy generation capacity is examined using electromechanical coupling.

Multiple-Frequency Force Estimation of Controlled Vibrating Systems with Generalized Nonlinear Stiffness

An on-line estimation technique of multiple-frequency oscillatory forces combined with the Hilbert–Huang transform for an important class of actively controlled, forced vibrating mechanical systems with nonlinear stiffness forces is proposed. Polynomial parametric nonlinearities are incorporated in the significantly perturbed vibrating system dynamics. This class of nonlinear vibrating systems can exhibit harmful large-amplitude vibrations, which are inadmissible in many engineering applications. Disturbing oscillations can be also provoked due to interactions of the primary mechanical system to be actively protected against dangerous vibrations with other forced uncertain multidegree-of-freedom nonlinear vibrating systems. Taylor’s series expansion to dynamically model uncertain vibrating forces into a small time window for real-time estimation purposes is employed. Intrinsic mode functions of multiple-frequency vibrating forces can be then obtained by the Hilbert-Huang transform. Uncertain instantaneous frequencies and amplitudes of disturbing oscillations can be directly computed in temporal space. An active vibration control scheme for efficient and robust tracking of prescribed motion reference profiles based on multiple frequency force estimation is introduced as well. The presented closed-loop on-line estimation technique can be extended for other classes of nonlinear oscillatory systems. Analytical, experimental and numerical results to prove the estimation effectiveness are presented. Numerical results show reasonable estimation errors of less than 2%.

Nonlinear model reduction to fractional and mixed-mode spectral submanifolds.

A primary spectral submanifold (SSM) is the unique smoothest nonlinear continuation of a nonresonant spectral subspace E of a dynamical system linearized at a fixed point. Passing from the full nonlinear dynamics to the flow on an attracting primary SSM provides a mathematically precise reduction of the full system dynamics to a very low-dimensional, smooth model in polynomial form. A limitation of this model reduction approach has been, however, that the spectral subspace yielding the SSM must be spanned by eigenvectors of the same stability type. A further limitation has been that in some problems, the nonlinear behavior of interest may be far away from the smoothest nonlinear continuation of the invariant subspace E. Here, we remove both of these limitations by constructing a significantly extended class of SSMs that also contains invariant manifolds with mixed internal stability types and of lower smoothness class arising from fractional powers in their parametrization. We show on examples how fractional and mixed-mode SSMs extend the power of data-driven SSM reduction to transitions in shear flows, dynamic buckling of beams, and periodically forced nonlinear oscillatory systems. More generally, our results reveal the general function library that should be used beyond integer-powered polynomials in fitting nonlinear reduced-order models to data.