Duffing System Research Articles

Effects of overmodulation on the slow progresses of relaxation oscillations in the Duffing system

Effects of overmodulation on the slow progresses of relaxation oscillations in the Duffing system

AI-Lorenz: A physics-data-driven framework for Black-Box and Gray-Box identification of chaotic systems with symbolic regression

Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime, due to their sensitive dependence on initial conditions and the complex non-linear interactions within the system. The challenge is even greater when the physics underlying such systems is not yet understood, and scientific inquiry must solely rely on empirical data. Despite advancements such as the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which has shown success in identifying chaotic systems with reasonable accuracy, challenges remain in dealing with noise, sparse data, and the need for models that generalize well across different chaotic systems. Driven by the need to fill this gap, we develop a framework named AI-Lorenz that learns mathematical expressions modeling complex dynamical behaviors by identifying differential equations from noisy and sparse observable data. We train a physics-informed neural network (PINN) with the eXtreme Theory of Functional Connections (X-TFC) algorithm by using data and known physics (when available) to learn the dynamics of a system, its rate of change in time, and missing model terms, which are used as input for a symbolic regression algorithm (PySR) to autonomously distill the explicit mathematical terms. This, in turn, enables us to predict the future evolution of the dynamical behavior. The performance of this framework is validated by recovering the right-hand sides and unknown terms of certain complex, chaotic systems, such as the well-known Lorenz system, a six-dimensional hyperchaotic system, the non-autonomous Sprott chaotic system, and the slow-fast Duffing system, and comparing them with their known analytical expressions and state-of-the-art regression and system identification methods.

On embedding the Koopmanization into controlled nonlinear systems, its comparison with the Carleman linearisation and concerning results: beyond the feedback linearisation

The feedback linearisation has found its striking applications in many practical situations for control problems that exploit the ideas from the linear system theory. Towards linearisation techniques for controlled nonlinear systems, operator-theoretic frameworks from functional analysis have received attentions as well. The Koopman operator is a linear, but an infinite dimensional operator. Most notably, the novelties of the paper are the following. (i) Sketching a rigorous and comprehensive proof of the Koopmanization of the controlled nonlinear system using the notion of the operator-invariant subspace and unifying the minimiser inequality formula with the Koopmanized system. The proof unfolds that the Koopmanization meets the bilinearisation in control context with the augmented state space. (ii) A formal construction of an observer encompassing the generalised Riccati equation in the Koopman setting of the original controlled nonlinear system. The Carleman version of the generalised Riccati equation is rephrased as well. (iii) Unfolding the usefulness of the Koopman observer. To achieve extensive illustrations of the theory of the paper, we construct two Koopman observers for two practical control problems, i.e. a controlled Duffing system with a cubic nonlinearity and a controlled system with a square nonlinearity. Then, we compare the Koopman observers with the Carleman observers.

Resonance and attraction domain analysis of asymmetric duffing systems with fractional damping in two degrees of freedom

This article establishes a two degree of freedom asymmetric Duffing system model with fractional damping through the study of the Jeffcott rotor system with horizontal support, and analyzes the dynamic behavior of its resonance and attraction domain structures. The second-order approximate solution of the system equation is obtained using a multiscale method, and the slow flow modulation equations of both the amplitudes and phases are extracted. The comparison between approximate and numerical solutions has shown their high consistency and the Routh Hurwitz criterion is utilized to study the stability of the system. Finally, we explore the effect of fractional damping on the vibration behavior of the system using amplitude frequency curves, attractive watersheds, and time trajectory plots, including both negative and positive values of the nonlinear stiffness coefficient. The analysis shows that if the coefficient and order of the fractional damping term are reasonably selected, the horizontal and vertical displacements of the system vibration can be effectively controlled. The importance of this work lies in its application in rotor vibration, which has significant implications for the study of the system under investigation.

A combination of improved spline-kernelled chirplet transform and multi-synchrosqueezing technique for analyzing nonstationary signals and structural instantaneous frequency identification

To improve the accuracy of structural instantaneous frequency (IF) identified by the spline-kernelled chirplet transform, a combination of an improved spline-kernelled chirplet transform and a multi-synchrosqueezing technique is proposed for analyzing nonstationary signals and structural IF identification in this paper. Firstly, an improved spline-kernelled chirplet transform (ISCT) is put forward by introducing a three parameter Gaussian window function for an existing spline-kernelled chirplet transform. These three parameters are successfully obtained through an optimization algorithm and the energy concentration measure (CM). Then, combined with a multi-synchrosqueezing technique, a so-called improved multi-synchrosqueezing spline-kernelled chirplet transform (IMSSSCT) approach is further presented to improve the time–frequency energy concentration. Finally, the accuracy and practicability of the proposed IMSSSCT method are validated by analytical signals, a single-degree-of-freedom Duffing system, a three-layer shear frame, as well as a cantilever beam experiment. The research results indicate that the IMSSSCT technique can effectively and accurately identify the IFs of nonlinear and time-varying structures.

Damping efficiency of the fractional Duffing system and an assessment of its solution accuracy

Dynamical systems with high damping efficiency in a wide frequency band can be useful on a small scale – for harvesting energy from ambient vibrations, and on a large scale – for damping harmful vibrations of mechanical structures. In this paper we present an assessment of the quality of solutions and damping efficiency of systems with fractional order derivatives. To simulate the fractional system the fourth-order Runge–Kutta method and Grünwald–Letnikov methods are used. We propose a coefficient for assessing the quality of solutions to fractional systems by reference to the quality of the calculated energy balance. As an exemplary system we study the Duffing model with embedded additional fractional-order derivative terms. Based on this coefficient, intervals of key numerical simulation parameters are determined to ensure the appropriate quality for the calculations of energy flows and energy balance. The determined values of these parameters are then used in tests of the damping efficiency of the studied system. Our results show that by modifying the fractional terms it is possible to configure a system that exhibits a “broadband effect”, i.e. a system that is characterized by high-amplitude vibrations and, consequently, high energy efficiency in a wide range of excitation frequencies.

MCGnet with multi-level gated unit transformations: A time series prediction-based model for unknown pattern abnormality detection

The task of anomaly detection in data patterns remains challenging due to the diverse of fault patterns, which often render existing models ineffective when encountered with unknown or unseen data patterns in the training data. Furthermore, insufficient samples, the presence of chaotic behavior, unbalanced anomaly patterns are potential limiting factors in the practical application. While prediction error-based approaches attempt to address these challenges, the discriminability of abnormal patterns in existing scenarios is not yet sufficient, thus limiting the ability to identify abnormal phenomena. To overcome these limitations, we proposed a novel model called MCGnet. The backbone architecture of MCGnet employed a novel design of continuous multi-layer circular convolution transformations. This design enhances the expressive capability of the model by expanding the differences among data patterns. Additionally, we introduced in parallel the newly proposed temporal-frequency gated Gaussian unit, which generates richer feature representations. Subsequently, the obtained feature representations were fed into an explicit causal network to learn complex patterns of data along temporal dependencies. The MCGnet not only solves high-precision time series prediction problems but also significantly improves the discriminability between normal and abnormal patterns. Since the proposed model is accurately modeled using only a small amount of normal data, it bypasses the limiting factors such as insufficient fault samples and sample imbalance, and is able to effectively deal with various unknown data patterns. The accuracy of the proposed method was verified through simulation data (Duffing system) and its potential for engineering applications was demonstrated through real-world data (rotor system).

Frequency switching leads to distinctive fast–slow behaviors in Duffing system

The slowly forced Duffing system has been found to exhibit unique fast–slow behaviors in descending frequency switching scheme, the typical of which is the sliding bursting, which is instructive for understanding the dynamics of ubiquitous frequency switching systems in scientific research and practical engineering. This paper is devoted to further refining the fast–slow dynamics of the slowly forced Duffing system with another commonly encountered frequency switching scheme, i.e., the ascending frequency switching, characterized by switching the frequency synchronously according to the increase or decrease of state variable values. Taking the forcing amplitude as an example, this paper provides a theoretical way to fully summarize the fast–slow behaviors in frequency switching systems with the variation of parameter values based on the proposed superposition analysis of one- and two-parameter bifurcations of subsystems. As a result, two typical bifurcation structures and eight threshold windows with distinct switching vector fields therein are identified, inducing up to ten different bursting patterns. Among them, several novel fast–slow dynamics, such as multiple jumps hysteresis loop formed by boundary equilibrium bifurcations and the switching failure phenomena, are presented and investigated. In particular, the underlying mechanism of threshold modulation, i.e., the evolution of unconventional bifurcations, is also found. These findings contribute to complementing and contrasting the existing studies on the fast–slow dynamics of frequency switching systems.

Combined Impact of Multidelay Feedback Number and Interval: A Novel Mechanism for Controlling the Stability of Stochastic Duffing Systems

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay [Formula: see text] as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay [Formula: see text] of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields.

Data-driven modeling of multi-stable origami structures: Extracting the global governing equation and exploring the complex dynamics

In recent years, multi-stable origami structures have garnered increasing attention for their applications in dynamic scenarios such as robotic arm motions, impact energy absorption, and spectrum gap regulation. Understanding the intricate working mechanisms and exploring the rich dynamics of these structures necessitate the development of dynamic models. However, existing dynamic modeling methods for origami structures are often cumbersome, and the resulting dynamic models often lack interpretability. To overcome these limitations, we propose a novel data-driven dynamic modeling approach based on B-spline Galerkin method and subset selection strategy. This approach directly captures the dynamics of multi-stable origami structures using measured data, eliminating the need for reliance on empirical or prior knowledge. To validate the effectiveness of the proposed approach, we first evaluate it on the Duffing system, which has explicit expressions, successfully reconstructing the governing equation. Subsequently, we apply this method to the dynamic modeling of the origami ball structure with tri-stability and the multi-cell stacked Miura-origami (SMO) structure with high dimensionality through simulation, showing favorable results. Finally, using experimental data collected from a bi-stable SMO structure prototype, we employ the proposed method to obtain a global model that can accurately predict different dynamic behaviors over a broad range of excitation frequencies, including intra-well periodic vibrations, inter-well periodic vibrations, and inter-well chaotic vibrations. Overall, our method showcases outstanding efficacy in formulating interpretable, manageable, and comprehensive dynamic models. It plays a pivotal role in delving into the intricate dynamics of multi-stable origami structures.

Overmodulation causes a variation in the number of jumps in the Duffing system

Amplitude modulation, involving normal and overmodulation scenarios, is crucial for information transmission. However, the complex dynamics of how modulation phase shift affects relaxation oscillations, especially under overmodulation conditions, has not been fully elucidated. Thus, this paper aims to explore the dynamical mechanism of relaxation oscillations affected by modulation phase shift under overmodulation conditions. The result shows that minor phase adjustments in low overmodulation phases can change the time series of the signal. Notably, at critical modulation phase thresholds, the number of transitions in each period of relaxation oscillations increases, and this phenomenon can be observed across a range of parameter values. However, further increase in phase will lead to the decrease in the number of transitions in relaxation oscillations, which demonstrates a clear correlation between phase adjustments and fold bifurcations affecting oscillation patterns. Based on the tri-parametric and bi-parametric bifurcation analysis, we explore the effect of overmodulation index on the number of transitions, and find that higher indices induce complex variations in it. These findings highlight the intricate interplay between modulation phase and modulation index in determining relaxation oscillation patterns, which are crucial for understanding amplitude modulation diversity and can serve as a reference for future research on other modulation scenarios.

Numerical simulation of fractional-order Duffing system with extended Mittag-Leffler derivatives

In this paper, we studied the dynamics of a nonlinear fractional-order Duffing system combined with Mittag-Leffler derivatives in order to provide dynamic behaviors different from existing ones. The Mittag-Leffler derivative is a generalized version of the exponential kernel derivative. To achieve this goal, we introduced a modified extension to higher-order Mittag-Leffler derivatives to overcome the initialization problem. Moreover, we discussed some properties and relationships of the studied derivatives. Then we presented numerical schemes to handle fractional extensions of the considered oscillatory system including the Mittag-Leffler and the Caputo derivatives. Numerical simulations are carried out and the resulting simulation dynamics of the studied fractional oscillatory system are compared.

Experimental investigation of the chaotification of a Duffing-like electronic oscillator under two-frequency excitation.

Two-frequency excitation has recently emerged as an efficient method to generate strong chaotification of Duffing and Duffing-like dynamical systems with both single- and double-well potentials. For the systems with a double-well potential, large continuous regions with robust chaos (chaotic attractor insensitive to changes in the system parameters) have been predicted to exist when the method is applied. Motivated by these theoretical results, in this work, we investigate experimentally the chaotification under two-frequency excitation of a simple electronic circuit analogous to the double-well Duffing oscillator. The experimental results confirm the theoretical expectations, and a strong chaotification is observed. Evidences are also presented that the chaotic attractor is robust. Therefore, the work establishes experimentally the two-frequency excitation as a simple and reliable method of chaotification. Furthermore, because of its ease of fabrication, the experimental results turn the particular circuit considered in this work into an interesting practical alternative as a reliable source of continuous-time chaotic signals.

Systematic solution for multi-model control approaches in nonlinear dynamic systems based on the s-gap metric

This paper proposes a systematic approach for optimizing the distribution of local models in multi-model control systems (MMCS) to enhance overall robustness. While existing literature discusses this method for linear parameter varying (LPV) and uncertain linear time-invariant (LTI) systems, significant limitations persist in addressing nonlinear dynamic systems. Robust control tools like the gap metric and generalized stability margin (GSM) have limited effectiveness in analyzing the robustness of nonlinear feedback systems. To address these challenges, novel concepts of the gap metric and GSM are introduced to determine central operating points (COPs) within local operating areas (LOAs) across the total operating area (TOA). These COPs guide the extraction of affine disturbance local models (ADLMs). Additionally, an optimization problem based on the s-gap metric and GSM is presented to optimize COPs placement and LOAs boundaries. Challenges such as non-monotonic behavior of the cost function and complexity arising from the s-gap metric formulation necessitate novel solution methods. To address these, constraints are applied to the cost function, and a novel discrete optimization approach is introduced. Finally, theoretical findings are applied to the Duffing system, pH neutralization process, and continuous stirred tank reactor (CSTR) plant to evaluate the proposed method's effectiveness. This comprehensive validation across different systems underscores the versatility and practical utility of the proposed approach.

Weak Signal Detection Application Based on Incommensurate Fractional-Order Duffing System

Weak Signal Detection Application Based on Incommensurate Fractional-Order Duffing System

Guided wave localization of small defects based on stochastic resonance characteristics of Duffing systems

To improve the location accuracy of small defects in anti-corrosion pipelines by using ultrasonic guided wave, this study developed a guided wave location method for small defects based on stochastic resonance (SR) characteristics of a time-moving Duffing system. Firstly, it solved the shortcomings of the selection of the key parameters of the system, which depended on human subjectivity and lacked interaction correlation in the previous research. By using the stochastic resonance characteristics of the Duffing system, the best fit parameter group of the system was determined synchronously and coordinately. Secondly, with the conjoint analysis of the key parameters of the system and the numerical simulation verification method, found that part of the energy of the noise signal would be transferred to the system output response, leading to the system output response was strengthened. Finally, A time-moving window function was introduced in the experimental verification, and the enhanced system output response in the window was analyzed using power spectral density (PSD), which could realize the location of guided waves with small defects. The experimental result showed that the small defect guided wave with a cross-section loss rate of at least 6 % could be effectively located.

Bursting dynamic analysis of generalized Duffing systems under two periodic excitations

Bursting dynamic analysis of generalized Duffing systems under two periodic excitations

Rail Flaw Detection via Kolmogorov Entropy of Chaotic Oscillator Based on Ultrasonic Guided Waves.

Ultrasonic guided wave (UGW) inspection is an emerging non-destructive testing(NDT) technique for rail flaw detection, where weak UGW signals under strong noise backgrounds are difficult to detect. In this study, a UGW signal identification model based on a chaotic oscillator is established. The approach integrates the UGW response into the critical state of the Duffing system to serve as a disturbance control variable. By evaluating the system's motion state before and after introducing the UGW response, identification of UGW signals can be realized. Thus, the parameters defining the critical state of Duffing oscillators are determined by Ke. Moreover, an electromagnetic transducer was specifically devised to enable unidirectional excitation for UGWs targeted at both the rail base and rail head. Experimental studies showed that the proposed methodology effectively detected and located a 0.46 mm notch at the rail base and a 1.78 mm notch at the rail head. Furthermore, Ke was directly proportional to the notch size, which could be used as a quantitative index to characterize the rail flaw.

Resonance study of fractional-order strongly nonlinear duffing systems

Resonance study of fractional-order strongly nonlinear duffing systems

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Relaxation oscillations are pervasive in diverse areas of natural sciences and engineering, and exploring the dynamical mechanisms of relaxation oscillations is one of the most significant issues. Typical relaxation oscillations can be observed in the Duffing system. Recently, amplitude modulation has emerged as a novel control mechanism for investigating the behavior of fast-slow dynamics in systemic tension oscillations. It has demonstrated the ability to prolong the quasi-static slow process of the system and increase the number of bifurcation points. However, the exploration of the mechanistic aspects of amplitude modulation is still in its early stages, with many unreported dynamical mechanisms. Among these, investigating the modes of relaxation oscillations induced by amplitude modulation is one of the most important issues. Therefore, this manuscript focuses on studying the effect of amplitude modulation on relaxation oscillations, using the classical forced Duffing system as a representative model. Significantly, we report an intriguing finding for the first time, revealing a new amplitude-modulated mechanism by which the disappearance of relaxation oscillations can be induced. By employing the fast-slow analysis, we have examined the underlying dynamical mechanisms, revealing a strong correlation with the modulation index of amplitude modulation. Notably, when the system operates under low amplitude modulation, an extension of the quasi-static process is observed, manifesting as a prolonged slow process. Conversely, under high amplitude modulation, relaxation oscillations suddenly disappear. Our results serve to enrich the potential mechanisms of amplitude modulation, and our analysis provides a reference for investigating the dynamical behavior induced by amplitude modulation in other dynamical systems.