Global Dynamics of a Duffing Oscillator With Delayed Displacement Feedback

Abstract

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.

Control of Bistability in a Delayed Duffing Oscillator

“…In the region III (antidashed zone), quasiperiodic oscillations resulting from a secondary Hopf bifurcation take place when crossing from region II to region III. It is worthy to notice that a similar equation to (7) was studied numerically, and it was shown that as the delay gain is increased, the system undergoes a secondary Hopf bifurcations [12,13,17]. Figure 3(c) indicates that by increasing the frequency Ω, the Hopf and the secondary Hopf bifurcation curves overlap giving birth to regions (dashed region IV in Figure 3(c)) on which a stable trivial steady state and a stable limit cycle coexist.…”

“…As it can be seen from this figure, the bifurcation value of this limit cycle is τ = π/2. Instead of employing the system (9a) and (9b), to obtain the relation between the amplitude of the periodic solution, R, and time delay, τ, as done in [12], we will take advantage from this modulation system to determine the region of existence of this periodic motion born by Hopf bifurcation.…”

“…The bifurcation curves of the nontrivial steady states given by (12) and (14), corresponding to the secondary Hopf bifurcation, are illustrated in Figure 3 when crossing from region I to region II. In the region III (antidashed zone), quasiperiodic oscillations resulting from a secondary Hopf bifurcation take place when crossing from region II to region III.…”

“…In [1,[11][12][13], attention has been paid to the linear stability analysis of the trivial steady state of delayed oscillators of type (1). While the investigation of stability of nontrivial solutions in (1) has received little attention from analytical view point, the influence of high-frequency excitation on nontrivial steady state has not been tackled.…”

“…While the investigation of stability of nontrivial solutions in (1) has received little attention from analytical view point, the influence of high-frequency excitation on nontrivial steady state has not been tackled. In [12], for instance, a detailed and systematic study on the dynamic of a delayed Duffing oscillator was confined to the linear stability analysis.…”