Homoclinic Orbits of a Buckled Beam Subjected to Transverse Uniform Harmonic Excitation

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

Delay-Induced Bogdanov–Takens Bifurcation and Dynamical Classifications in a Slow-Fast Flexible Joint System

A slow-fast delay-coupled flexible joint system is investigated in this paper. To understand the effects of time delay on the stability and oscillation of the manipulator, the geometric singular perturbation method is extended in dealing with delay differential equations. Bogdanov–Takens (BT) bifurcation of the fast subsystem is obtained, which leads to the existence of homoclinic orbits and is proved to be related to the formation of spiking. After the break of homoclinic orbits, Melnikov theory is introduced to predict the threshold curve indicating the occurrence of chaos. Numerical results show that with the increase of time delay, the stability of the system gets worse, and complicated oscillations including bursting, chaotic-bursting and complete chaos turn up. Besides, it is briefly summarized that the effect of the small parameter in the slow-fast system is to influence the convergence rate of solution trajectories, which is widely neglected in previous works.

Oscillation Suppression in a Delay-Coupled Flexible-Joint System

A delay-coupled flexible-joint system is investigated in this paper. Because of the different time scales, the flexible-joint system could be transformed into a fast subsystem and a slow subsystem. The geometric singular perturbation method is used to obtain the slow manifold defining as the equilibrium of the fast subsystem. The eigenvalue analysis of the fast subsystem reveals a relation between the stability of the slow manifold and the time delay. The analysis results provide an idea of suppressing the small amplitude periodic oscillation via adjusting the time delay. Numerical simulations are performed to display the effectiveness of this method.