Duffing System Research Articles

Prediction of Solutions of the Duffing System with Tuned Mass Damper

Abstract In this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

Fast–Slow Dynamics Analysis of a Coupled Duffing System with Periodic Excitation

In this paper, a two-degree-of-freedom nonlinear coupled Duffing equation with an external excitation and two external excitations are studied. For the coupled Duffing system with periodic excitation, the system shows the dynamic behavior on different time scales when the excitation frequency and the inherent frequency of the system are different. Firstly, we discretize the system by using the Euler method, and the discrete equation is obtained. Secondly, the two external excitations are considered as slow variables that are transformed into a slow variable by the Moivre formula, which divides the original system into the fast–slow subsystem. Finally, the oscillation dynamic behavior of the coupled system is discussed by combining fast–slow analysis method and the transformation phase diagram.

Bursting oscillations and mechanism analysis in a non-smooth Duffing system with frequency domain of two time scales

The main purpose of the manuscript is to explore the bursting oscillations in a dynamic system with non-smooth factor. Take the duffing system as a example, by introducing dry friction, and choosing periodic excitation ensure that there exists order gap between the natural frequency and the exited frequency, a non-smooth system with two time scales is established. The whole exciting term can be regarded as a slow-varying parameter, which leads to the generalized autonomous system, and the stability and the associated bifurcation conditions can be derived. By taking suitable parameter values, three typical cases are considered under the three group different parameters which can described as double scrolls, three scrolls and four scrolls. By using the transformed phase portrait and considering the effect of non-smooth factor, the associated bifurcation mechanism of the bursting oscillations is discussed, which may reveal different forms of bursting oscillations and transition mechanism between quiescent states and the spiking states. In addition, aiming at the coexistence of multiple equilibrium states in this non-smooth system, combined with the influence of evolution behavior of the attractor itself, and further reveals that the reason for showing special oscillation phenomena under the influence of non-smooth factor.

The Analysis of Periodic Signal Detection Method Based on Duffing System Chaotic Dynamics

This article presents the analysis of periodic signal detection method based on Duffing system sensitivity to weak influences. The described signal detection method is developed with using of Duffing system that oscillates in chaotic state, without transitions to periodic state. The main advantage of such method is the absence of periodic oscillation modes with low sensitivity. The divergence of Duffing system phase trajectories is investigated with influences of different periodic signals under low signal-to-noise ratio values. The estimation of phase trajectories divergence is performed with using of numeric integration. The signal detection method is analyzed with different forms of input signal: sinusoidal, square, triangle. The analysis shows that a reliable detection of periodic signal can be performed for any of the three presented forms of signal with repeating frequency near the frequency of the driving signal. The obtained results show wide capabilities of Duffing system applications for detection of weak periodic signals.

Tracking Control of Chaotic Systems via Optimized Active Disturbance Rejection Control

For tracking control of chaotic systems, we develop an active disturbance rejection (ADR) control method. Using the first state of the system as the only available state, a time-varying bandwidth extended state observer reconstructs the remaining states and the total disturbance. A time-varying bandwidth feedback controller forces all the states of the system to follow exactly the reference signal and its derivative. The parameters of the ADR controller are optimized using a genetic algorithm. As the objective function, we chose the weighted sum of the integral of the absolute error and the integral of the absolute control signal. Two chaotic systems—the Duffing system and the Genesio–Tesi system—are considered in computer simulation tests. Results of these simulations are presented to demonstrate the effectiveness of the ADRC method in controlling chaotic systems.

The Application of Lagrangian Descriptors to 3D Vector Fields

Since the 1980s, the application of concepts and ideas from dynamical systems theory to analyze phase space structures has provided a fundamental framework to understand long-term evolution of trajectories in many physical systems. In this context, for the study of fluid transport and mixing the development of Lagrangian techniques that can capture the complex and rich dynamics of time-dependent flows has been crucial. Many of these applications have been to atmospheric and oceanic flows in two-dimensional (2D) relevant scenarios. However, the geometrical structures that constitute the phase space structures in time-dependent three-dimensional (3D) flows require further exploration. In this paper we explore the capability of Lagrangian descriptors (LDs), a tool that has been successfully applied to time-dependent 2D vector fields, to reveal phase space geometrical structures in 3D vector fields. In particular, we show how LDs can be used to reveal phase space structures that govern and mediate phase space transport. We especially highlight the identification of normally hyperbolic invariant manifolds (NHIMs) and tori. We do this by applying this methodology to three specific dynamical systems: a 3D extension of the classical linear saddle system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics f-plane approximation model which is described by analytical wave solutions of the 3D Euler equations. We show that LDs successfully identify and recover the template of invariant manifolds that define the dynamics in phase space for these examples.

A Hybrid Fault Diagnosis Approach for Rotating Machinery with the Fusion of Entropy-Based Feature Extraction and SVM Optimized by a Chaos Quantum Sine Cosine Algorithm.

As crucial equipment during industrial manufacture, the health status of rotating machinery affects the production efficiency and device safety. Hence, it is of great significance to diagnose rotating machinery faults, which can contribute to guarantee the running stability and plan for maintenance, thus promoting production efficiency and economic benefits. For this purpose, a hybrid fault diagnosis model with entropy-based feature extraction and SVM optimized by a chaos quantum sine cosine algorithm (CQSCA) is developed in this research. Firstly, the state-of-the-art variational mode decomposition (VMD) is utilized to decompose the vibration signals into sets of components, during which process the preset parameter K is confirmed with the central frequency observation method. Subsequently, the permutation entropy values of all components are computed to constitute the feature vectors corresponding to different kind of signals. Later, the newly developed sine cosine algorithm (SCA) is employed and improved with chaotic initialization by a Duffing system and quantum technique to optimize the support vector machine (SVM) model, with which the fault pattern is recognized. Additionally, the availability of the optimized SVM with CQSCA was revealed in pattern recognition experiments. Finally, the proposed hybrid fault diagnosis approach was employed for engineering applications as well as contrastive analysis. The comparative results show that the proposed method achieved the best training accuracy 99.5% and best testing accuracy 97.89%. Furthermore, it can be concluded from the boxplots of different diagnosis methods that the stability and precision of the proposed method is superior to those of others.

Nonlinear multivalued Duffing systems

We consider a multivalued nonlinear Duffing system driven by a nonlinear nonhomogeneous differential operator. We prove existence theorems for both the convex and nonconvex problems (according to whether the multivalued perturbation is convex valued or not). Also, we show that the solutions of the nonconvex problem are dense in those of the convex (relaxation theorem). Our work extends the recent one by Kalita–Kowalski [7].

Stochastic response analysis for nonlinear vibration systems with adjustable stiffness property under random excitation.

Nonlinear vibration systems with adjustable stiffness property have attracted considerable attentions for their prominent broadband performances. In the present manuscript, we consider the stochastic dynamical systems with adjustable stiffness and proposed a numerical method for the random responses analysis of the Gaussian white noise excited systems. A multi-dimensional Fokker-Plank-Kolmogorov equation governing the joint probability density of the mechanical states is derived according to the theory of diffusion processes. We solve the multi-dimensional equation using a splitting method and obtained the stationary probability densities and the mean-square responses directly. Two classical nonlinear vibration systems with adjustable stiffness, including the energy harvesting system and the Duffing system with Dahl friction, are presented as examples. Their comparisons with the results from Monte-Carlo simulations illustrate the effectiveness of the proposed procedure for both monostable and bistable cases, even for cases with strong excitation. In addition, the splitting method is efficient for higher-dimensional problem and has advantages of simple implementation, less storage of intermediate values and so on. Hence, in terms of the application scope, the proposed procedure is superior to the current mainstream methods for the random response evaluation of nonlinear vibration systems with adjustable stiffness.

Adaptive Delayed Feedback Chaos Control of a Class of Uncertain Chaotic Systems Subject to External Disturbance

In this paper, an adaptive nonlinear delayed feedback control is proposed for stabilizing an unstable periodic orbit (UPO) of a class of uncertain chaotic systems using the sliding mode approach. The chaotic system is subjected to external disturbances with unknown bounds. Using the Lyapunov stability theorem and the concept of sliding mode control, an adaptive delayed feedback controller has been designed. The controller and adaptation law guarantee the asymptotic stability of the closed loop system on a periodic trajectory which can be sufficiently close to the UPO of the chaotic system. The capability of the proposed control method in stabilizing the UPO of chaotic systems is established analytically and the effectiveness of the proposed scheme is examined numerically by applying it to the chaotic Duffing and Gyro systems.

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

A general method to calculate multi-parameter bifurcation diagram in the parameter space is designed based on top Lyapunov exponent and Floquet multiplier to study the effect of different combinations of system parameters on the system’s dynamics. Bifurcation and chaos of the forced and damped Duffing system in two-parameter plane are investigated by using the method designed in this work. The correlation and matching laws of the Duffing system between dynamic performance and system parameters are analyzed. The effect of different types of bifurcation curves on the bifurcating of coexisting attractors is investigated according to basins of attraction, bifurcation diagrams, top Lyapunov exponent spectrums, phase portraits, Poincare maps, and Floquet multipliers. The evolution of various bifurcation curves and codimension-two bifurcation in the parametric plane is studied as well. Coexisting attractors are found in the parameter plane. The results indicate that the different bifurcating curves are selective for the bifurcation of coexisting attractors. Both the pitchfork bifurcation curve and the period-doubling bifurcation curve just change the stability of some of the coexisting attractors, but have no effect on the stability of the other part of the attractors. The saddle-node bifurcation curve has an effect on the stability of all the coexisting attractors. A series of period-doubling bifurcation curves and codimension-two bifurcation points lead to chaos existence region in two-parameter plane. The special evolution of bifurcation points and bifurcation curves in two-parameter plane with the change of the system parameter is observed. The codimension-two bifurcation points and bifurcation curves play an important role in understanding nonlinear dynamics of the system in the parametric plane. The work in this study emphasizes the importance of the different combinations of system parameters on the system dynamics.

Chaos controls of a duffing system with impacts

The chaos controls of a Duffing system with impacts are investigated using appropriate random phase. Based on the discontinuous map and Wedig’s algorithm, a procedure of calculating top Lyapunov exponent is presented for stochastic Duffing system with impacts. As the random phase changes, two kinds of chaos control are achieved for Duffing system with impacts by the criterion of the top Lyapunov exponent: one is chaos suppression, and the other is chaos generation. In addition, the obtained results are verified by numerical simulations including the phase portrait, Poincare surface of section and time history.

Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations.

This paper aims to report two novel bursting patterns, the turnover-of-pitchfork-hysteresis-induced bursting and the compound pitchfork-hysteresis bursting, demonstrated for the Duffing system with multiple-frequency parametric excitations. Typically, a hysteresis behavior between the origin and non-zero equilibria of the fast subsystem can be observed due to delayed pitchfork bifurcation. Based on numerical analysis, we show that the stable equilibrium branches, related to the non-zero equilibria resulted from the pitchfork bifurcation, may become the ones with twists and turns. Then, the novel bursting pattern turnover-of-pitchfork-hysteresis-induced bursting is revealed accordingly. In particular, we show that additional pitchfork bifurcation points may appear in the fast subsystem under certain parameter conditions. This creates multiple delay-induced hysteresis behavior and helps us to reveal the other novel bursting pattern, the compound pitchfork-hysteresis bursting. Besides, effects of parameters on the bursting patterns are studied to explore the relation of these two novel bursting patterns.

Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

Locating and imaging contact delamination based on chaotic detection of nonlinear Lamb waves

Abstract We propose a novel method for locating and imaging contact delaminations in composite plates, using a network of PZT transducers and the nonlinear Lamb waves. Being immune to noise and sensitive to weak signal, the Duffing chaotic oscillator is utilized to detect the nonlinear harmonics of Lamb waves generated at the contact delamination interfaces. The Lyapunov exponent of the Duffing system output is calculated to quantify the relative intensity of nonlinear second harmonic. Its normalized value is taken as the damage index to locate and image the delaminations by combining with the probabilistic imaging algorithm. The numerical simulation study on the proposed method is performed. Compared with the nonlinearity parameter and the conventional damage index, the Lyapunov damage index (LDI) can image the single delamination or multiple delaminations more effectively, especially under the noise interference. The experiment using the LDI is also conducted on a composite laminated plate with multiple internal delaminations. The results show that the proposed imaging method based on LDI is also valid to the practical delaminations in the composite plate. It has the following advantages: immunity to noise, needlessness of baseline data, high imaging resolution and location accuracy.

Two-scale command shaping for feedforward control of nonlinear systems

This paper presents a critical analysis of the two-scale command shaping (TSCS) feedforward control technique as applied to nonlinear Duffing oscillators. TSCS is an approach for tailoring a flexible system’s applied control input to reduce undesirable residual vibrations using multiple problem scales, command shaping of a linear subproblem, and cancelation of a remaining nonlinear subproblem. As shown herein, TSCS proves to be an effective method for feedforward control of nonlinear systems. The strategy outperforms conventional and nonlinearly informed command shaping strategies in traditional and non-traditional Duffing systems (e.g., Duffing systems with quadratic nonlinearity and Coulomb damping). The TSCS approach is also extended herein to nonlinear systems with uncertain parameters through the implementation of robust command shaping strategies and the extended Kalman filtering parameter estimation technique.

Robust backstepping control for uncertain chaotic multi-inputs multi-outputs systems using type 2 fuzzy systems

In this paper, a new backstepping type 2 fuzzy control is developed for a class of uncertain multi-inputs multi-outputs (MIMO) chaotic systems, with a complicate of uncertainties and external disturbances. The system consists of interconnected subsystems that are duffing equation. In the control design, type 2 fuzzy logic systems are used to approximate the unknown functions. Hybrid adaptive robust tracking control schemes that are based upon a combination of bounds of type 2 fuzzy approximation parameters and the backstepping design are developed such that all the states and signals are bounded and the proposed approach alleviate the online computation burden and improves the robustness to dynamic uncertainties and external disturbances. Finally, the control of chaotic duffing systems with unknown parameters is given as example to verify the effectiveness of the proposed approach.

A Method of Changing the Center Position in Duffing Chaotic Systsem

Under the condition of keeping Duffing system in chaotic state, we present a method of changing the center position by adding a non-smooth periodic signal to Duffing system. In this paper, we first make a theoretical analysis which describe the influence of added signal on center point of Duffing system and provide mathematical proofs, and then build a simulation system which can change the center position of Duffing system. Finally, computer simulations are used to verify the analytical results. Simulation results show that the proposed method can effectively work in secure communication applications.

Effects of Second-Order Matched Stochastic Resonance for Weak Signal Detection

Weak signal detection via stochastic resonance (SR) has attracted considerable attention in a wide range of research fields, especially under heavy background noise circumstances. In this paper, a second-order matched stochastic resonance (SMSR) method is proposed to further improve the signal-to-noise ratio of weak period signal. By selecting a proper damping factor in the regime of second-order parameter matched relationship, weak periodic signal, background noise, and nonlinear system can be matched in generating an enhanced output. The matched relationship is deduced in combining noise intensity optimization and signal frequency synchronization with duffing system in a mathematical way, and a normalized scale transformation is further carried out to make it accessible in detecting arbitrary high frequency signals. The numerical analysis and application verification are performed to confirm the validity and effectiveness of theoretical results, which indicate the proposed SMSR method is superior to the first-order parameter matched stochastic resonance in achieving a good band-pass filtering effect with a low-noise output as the driving frequency of received signal is not too small ( $\geqslant 0.1$ Hz). Thus, the proposed method is beneficial to practical engineering weak signal processing and anticipates to be a potential novel technique for ship radiated line-spectrum detection.

An equivalent linearization method for nonlinear systems under nonstationary random excitations using orthogonal functions

Many practical engineering problems are associated with nonlinear systems subjected to nonstationary random excitations. Equivalent linearization methods are commonly used to seek for approximate solutions to this kind of problems. Compared to various approaches developed in the frequency and mixed time-frequency domains, though directly solving the system equation of motion in the time domain would improve computation efficiency, only limited studies are available. Considering the fact that the orthogonal functions have been widely used to effectively improve the accuracy of the approximated responses and reduce the computational cost in various engineering applications, an orthogonal-function-based equivalent linearization method in the time domain has been proposed in the current paper for nonlinear systems subjected to nonstationary random excitations. In the numerical examples, the proposed approach is applied to a SDOF system with a set-up spring and a SDOF Duffing oscillator subjected to stationary and nonstationary excitations. In addition, its applicability to nonlinear MDOF systems is examined by a 3DOF Duffing system subjected to nonstationary excitation. Results show that the proposed method can accurately predict the nonlinear system response and the formulation of the proposed approach allows it to be capable of handling any general type of nonstationary random excitations, such as the seismic load.