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.

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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.

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. This paper investigates the possibility of improving aircraft landing gear touch-down performance by adding an inerter alongside a linear passive shock strut. The inerter is a novel mechanical element with the property that the applied force is proportional to the relative acceleration between its terminals. A simplied landing gear model is presented and the baseline performance of a conventional oleo-pneumatic shock absorber is established. Candidate layouts with linear mechanical components including inerters are considered using three objective functions: the strut eciency, the maximum strut load and the maximum stroke. It is demonstrated that improved touch-down performance can be achieved with a linear inerter-based conguration. However it is also observed that the potential energy stored in the gear at the end of the rst compression stroke exceeds that of the baseline nonlinear system. This suggests a poorer elongation stage might be observed. To address this, an additional constraint on energy dissipation is then considered. To achieve a reduced potential energy, a double-stage compression spring is introduced. With this, inerter-based congurations that provide improvements for the performance indices of interest are identied and presented.

This paper presents a detailed analysis on the dynamics of a two-dimensional delayed small-world network under delayed state feedback control. On the basis of stability switch criteria, the equilibrium is studied, and the stability conditions are determined. This study shows that with properly chosen delay and gain in the delayed feedback path, the controlled small-world delayed network may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or the multistability solutions via three types of codimension two bifurcations. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are determined by using the normal form theory and center manifold theorem. In addition, the study shows that the controlled model exhibits period-doubling bifurcations which lead eventually to chaos; and the chaos can also directly occur via the bifurcations from the quasi-periodic solutions. The results show that the delayed feedback is an effective approach in order to generate or annihilate complex behaviors in practical applications.