GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK

2004 ◽  
Vol 14 (08) ◽  
pp. 2753-2775 ◽  
Author(s):  
HUAILEI WANG ◽  
HAIYAN HU ◽  
ZAIHUA WANG
Keyword(s):  
Time Delay ◽  
Duffing Oscillator ◽  
Basin Of Attraction ◽  
Poincare Section ◽  
Periodic Motions ◽  
Stability Switches ◽  
Local Bifurcations ◽  
Section Technique ◽  
Trivial Equilibrium ◽  
The Stability

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

On Time-Delay Effects on Period-1 Motions in a Periodically Forced, Time-Delayed, Duffing Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Time Delay ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Delay Systems ◽  
Harmonic Amplitude ◽  
Time Delay Systems ◽  
Periodic Motions ◽  
Delay Effects ◽  
The Stability

In recent decades, nonlinear time-delay systems were often applied in controlling nonlinear systems, and the stability of such time-delay systems was very actively discussed. Recently, one was very interested in periodic motions in nonlinear time-delay systems. Especially, the semi-analytical solutions of periodic motions in time-delay systems are of great interest. From the semi-analytical solutions, the nonlinearity and complexity of periodic motions in the time-delay systems can be discussed. In this paper, time-delay effects on periodic motions of a periodically forced, damped, hardening, Duffing oscillator are analytically discussed through a semi-analytical method. The semi-analytical method is based on discretization of the differential equation of such a Duffing oscillator to obtain the corresponding implicit discrete mappings. Through such implicit mappings and mapping structures of periodic motions, period-1 motions varying with time-delay are discussed, and the corresponding stability and bifurcation analysis of periodic motions are carried out through eigenvalue analysis. Numerical results of periodic motions are illustrated to verify analytical predictions. The corresponding harmonic amplitude spectrums and harmonic phases are presented for a better understanding of periodic motions in such a nonlinear oscillator.

BIFURCATION ANALYSIS OF A DELAYED DYNAMIC SYSTEM VIA METHOD OF MULTIPLE SCALES AND SHOOTING TECHNIQUE

2005 ◽  
Vol 15 (02) ◽  
pp. 425-450 ◽  
Author(s):  
HUAILEI WANG ◽  
HAIYAN HU
Keyword(s):  
Time Delay ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Duffing Oscillator ◽  
Time History ◽  
Pitchfork Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Diagrams ◽  
Shooting Technique ◽  
Local Bifurcations

This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

Period-m Motions to Chaos in a Periodically Forced, Duffing Oscillator with a Time-Delayed Displacement

2014 ◽  
Vol 24 (10) ◽  
pp. 1450126 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Time Delay ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Initial Time ◽  
Periodic Motions ◽  
Delayed System ◽  
The Stability ◽  
Analytical Bifurcation Trees

In this paper, periodic motions in a periodically excited, Duffing oscillator with a time-delayed displacement are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed through eigenvalue analysis. The time-delayed displacement is from the feedback control of displacement. The analytical bifurcation trees of period-1 motions to chaos in the time-delayed Duffing oscillator are presented through asymmetric period-1 to period-4 motions. Stable and unstable periodic motions are illustrated through numerical and analytical solutions. From numerical illustrations, the analytical solutions of stable and unstable period-m motions are relatively accurate with AN/m < 10-6 compared to numerical solutions. From such analytical solutions, any complicated solutions of period-m motions can be obtained for any prescribed accuracy. Because time-delay may cause discontinuity, the appropriate time-delay inputs (or initial conditions) in the initial time-delay interval should satisfy the analytical solution of periodic motions in the time-delayed dynamical systems. Otherwise, periodic motions in such a time-delayed system cannot be obtained directly.

Stability Switches of a Class of Fractional-Delay Systems With Delay-Dependent Coefficients

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
Xinghu Teng ◽  
Zaihua Wang
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Jacobian Determinant ◽  
Stability Switches ◽  
Characteristic Roots ◽  
Fractional Delay ◽  
Analysis Of Stability ◽  
The Stability ◽  
Key Steps ◽  
Delay Dependent

Stability of a dynamical system may change from stable to unstable or vice versa, with the change of some parameter of the system. This is the phenomenon of stability switches, and it has been investigated intensively in the literature for conventional time-delay systems. This paper studies the stability switches of a class of fractional-delay systems whose coefficients depend on the time delay. Two simple formulas in closed-form have been established for determining the crossing direction of the characteristic roots at a given critical point, which is one of the two key steps in the analysis of stability switches. The formulas are expressed in terms of the Jacobian determinant of two auxiliary real-valued functions that are derived directly from the characteristic function, and thus, can be easily implemented. Two examples are given to illustrate the main results and to show an important difference between the fractional-delay systems with delay-dependent coefficients and the ones with delay-free coefficients from the viewpoint of stability switches.

Local Bifurcations and Optimal Theory in a Delayed Predator–Prey Model with Threshold Prey Harvesting

2015 ◽  
Vol 25 (07) ◽  
pp. 1540015 ◽  
Author(s):  
Israel Tankam ◽  
Plaire Tchinda Mouofo ◽  
Abdoulaye Mendy ◽  
Mountaga Lam ◽  
Jean Jules Tewa ◽  
...  
Keyword(s):  
Time Delay ◽  
Piecewise Linear ◽  
Optimal Harvesting ◽  
Threshold Policy ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Bifurcations ◽  
Prey Model ◽  
Linear Threshold ◽  
The Stability

We investigate the effects of time delay and piecewise-linear threshold policy harvesting for a delayed predator–prey model. It is the first time that Holling response function of type III and the present threshold policy harvesting are associated with time delay. The trajectories of our delayed system are bounded; the stability of each equilibrium is analyzed with and without delay; there are local bifurcations as saddle-node bifurcation and Hopf bifurcation; optimal harvesting is also investigated. Numerical simulations are provided in order to illustrate each result.

scholarly journals Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Mikhail Guskov ◽  
Fabrice Thouverez
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Response ◽  
Duffing Oscillator ◽  
Point Of View ◽  
Periodic Motions ◽  
Computational Costs ◽  
Nonlinear Responses ◽  
Time Marching ◽  
The Stability ◽  
Good Agreement

Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach.

Multistability in the Centrifugal Governor System Under a Time-Delay Control Strategy

2019 ◽  
Vol 14 (11) ◽  
Author(s):  
Shuning Deng ◽  
Jinchen Ji ◽  
Shan Yin ◽  
Guilin Wen
Keyword(s):  
Time Delay ◽  
Control Strategy ◽  
Large Amplitude ◽  
Stability Boundary ◽  
Physical Parameters ◽  
Time Delay Control ◽  
Delay Control ◽  
Delayed Feedback Controller ◽  
Trivial Equilibrium ◽  
The Stability

Abstract The centrifugal governor system plays an indispensable role in maintaining the near-constant speed of engines. Although different arrangements have been developed, the governor systems are still applied in many machines for its simple mechanical structure. Therefore, the large-amplitude vibrations of the governor system which can lead to the function failure of the system should be attenuated to guarantee reliable operation. This paper adopts a time-delay control strategy to suppress the undesirable large-amplitude motions in the centrifugal governor system, which can be regarded as the practical application of the delayed feedback controller in this system. The stability region of the trivial equilibrium of the controlled system is determined by investigating the characteristic equation and generic Hopf bifurcations. It is found that the dynamic behavior of multistability can be induced by the Bautin bifurcation, arising on the stability boundary of the trivial equilibrium with a constant delay. More specifically, a coexistence of two desirable stable motions, i.e., an equilibrium or a small-amplitude periodic motion, can be observed in the controlled centrifugal governor system without changing the physical parameters. This is a new feature of the motion control in the centrifugal governor systems, which has not yet been reported in the existing studies. Finally, the results of theoretical analyses are verified by numerical simulations.

Nonlinear analysis of electrostatic micro-electro-mechanical systems resonators subject to delayed proportional–derivative controller

2020 ◽  
pp. 107754632092562 ◽  
Author(s):  
Ulrich Gaël Ngouabo ◽  
Peguy Roussel Nwagoum Tuwa ◽  
Samuel Noubissie ◽  
Paul Woafo
Keyword(s):  
Time Delay ◽  
Nonlinear Analysis ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Mechanical Systems ◽  
Micro Electro Mechanical Systems ◽  
Regular Motion ◽  
The Stability ◽  
Proportional Derivative Controller ◽  
Delay Dependent

The present study deals with the nonlinear analysis of electrostatic micro-electro-mechanical systems resonators with two symmetric electrodes and subjected to delayed proportional–derivative controller. After a brief description of the model, the stability analysis of the linearized system is presented to depict the stability charts in the parameter space of proportional gain and time delay. The bifurcation diagram is used to confirm the existence of the delay-dependent and delay-independent regions and to analyze the effect of proportional–derivative gains and time delay on the dynamics of the system. Using Melnikov’s theorem, the criterion for the appearance of horseshoe chaos from homoclinic and heteroclinic bifurcations is presented. Melnikov’s predictions are confirmed by using the numerical simulations based on the basin of attraction of initial conditions. It is found that the increase in proportional gain contributes to increase the region of regular motion in both bifurcations. However, the increase in derivative gain contributes rather to reduce the region of regular motion for homoclinic bifurcation, although it increases rather this region in the case of heteroclinic bifurcation. Moreover, it is also observed, depending on proportional–derivative gains, the existence of a critical value of the delay where before it, the region of regular motion increases and after it, decreases rather.

Analytical Solutions for Periodic Motions in a Hardening Mathieu-Duffing Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis M. O’Connor
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Series Solutions ◽  
Periodic Motions ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

Analytical solutions for period-m motions in a hardening Mathieu-Duffing oscillator are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the hardening Mathieu-Duffing oscillator are presented. Period-1 asymmetric and period-2 symmetric motions are illustrated.

Period-1 Motions in a Time-Delayed Duffing Oscillitor With Periodic Excitation

2014 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Fourier Series ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Amplitude Characteristics

In this paper, analytical solutions of period-1 motions in a time-delayed Duffing oscillator with a periodic excitation are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The symmetric and asymmetric period-1 motions in such time-delayed Duffing oscillator are obtained analytically, and the frequency-amplitude characteristics of period-1 motions in such a time-delayed Duffing oscillator are investigated. Numerical illustrations of period-1 motions are given by numerical and analytical solutions.

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