Complex period-1 motions of a periodically forced Duffing oscillator with a time-delay feedback

2016

Int. J. Dynam. Control

Self Cite

In this paper, frequency responses of periodic motions to chaos in a periodically forced, damped, quadratic nonlinear oscillator are investigated through the finite Fourier series analysis of discrete solutions of periodic motions. The discrete solutions of periodic motions are obtained from mapping structures of discrete nodes, and the corresponding stability and bifurcation analysis of periodic motions are completed by the eigenvalue analysis of fixed points in discrete nonlinear dynamical systems. The frequencyamplitude characteristics for bifurcation trees of periodic motions to chaos are discussed, and the quantity levels of harmonic amplitudes for different harmonic orders are illustrated clearly. Numerical results of periodic motions are illustrated to show harmonic frequency responses effects on periodic motions.

Section: Introductionmentioning

confidence: 99%

On frequency responses of period-1 motions to chaos in a periodically forced, time-delayed quadratic nonlinear system

2016

Int. J. Dynam. Control

Self Cite

“…Luo [2013] systematically proposed a methodology for periodic motions in time-delayed, nonlinear dynamical systems. Luo and Jin [2014b] used such a methodology to investigate the bifurcation tree of period-1 motion to chaos in a periodically forced, quadratic nonlinear oscillator with time-delay. To further understand the properties of periodic solutions in time-delayed nonlinear systems, the periodically forced, time-delayed, Duffing oscillator should be investigated analytically, and in [Luo & Jin, 2014b], symmetric and asymmetric period-1 motions of the periodically forced Duffing oscillator with a time-delayed displacement were discussed.…”

Section: Introductionmentioning

confidence: 99%

“…Luo and Jin [2014b] used such a methodology to investigate the bifurcation tree of period-1 motion to chaos in a periodically forced, quadratic nonlinear oscillator with time-delay. To further understand the properties of periodic solutions in time-delayed nonlinear systems, the periodically forced, time-delayed, Duffing oscillator should be investigated analytically, and in [Luo & Jin, 2014b], symmetric and asymmetric period-1 motions of the periodically forced Duffing oscillator with a time-delayed displacement were discussed. Herein, the analytical bifurcation trees of asymmetric period-1 motion to chaos will be investigated to the global behaviors of periodic motions in such a time-delayed oscillator.…”

Section: Introductionmentioning

confidence: 99%

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

Int. J. Bifurcation Chaos

Jin

2014

Self Cite

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 A N/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 timedelay 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.

“…Q 0 and are excitation amplitude and frequency, respectively. In Luo and Jin [1], complex period-1 motions of the periodically forced Duffing oscillator with a time-delayed displacement were investigated, which cannot be obtained from the traditional harmonic balance and perturbation methods. In Luo and Jin [2], the period-m motions of the time-delayed Duffing oscillator was investigated analytically, and complex period-m motions were observed in such a time-delayed Duffing oscillator.…”

mentioning

confidence: 99%

“…Since the time-delayed Duffing oscillator exists in structural vibration control, one is very interested in periodic motions in such a time-delayed Duffing oscillator. In Luo and Jin [1,2], complex period-1 motions and analytical bifurcation trees of period-1 motions to chaos were investigated. To understand complex motions in the time-delayed Duffing oscillator, period-3 motions to chaos will be investigated.…”

mentioning

confidence: 99%

Period-3 motions to chaos in a periodically forced duffing oscillator with a linear time-delay

Jin

2014

Int. J. Dynam. Control

Self Cite

In this paper, bifurcation trees of period-3 motions to chaos in a periodically excited, Duffing oscillator with a linear delay are investigated through the Fourier series. The analytical solutions of period-m motions are presented and the stability and bifurcation of such period-m motions in the bifurcation trees are discussed by eigenvalue analysis. Two independent symmetric period-3 motions were obtained, and the two independent symmetric period-3 motions are not relative to chaos. Two bifurcation trees of period-3 motions to chaos are presented through period-3 to period-6 motion. Numerical illustrations of stable and unstable period-3 and period-6 motions are given by numerical and analytical solutions. The complicated period-3 and period-6 motions exist in the range of low excitation frequency.