MS19.5: Systems with Time Delay

This contribution gives the formal notion of neutral distributed differential equations, which can be used for Hopf bifurcation calculation using center manifold reduction theorem and normal form calculation. An illustrative simple example is presented to control a robotic arm with nonlinear stiffness characteristics and improve its dynamic behavior by applying acceleration feedback with distributed time delay.

We perform a theoretical study of the Atlantic Meridional Overturning Circulation (AMOC) by means of a conceptual mathematical model in the form of a scalar delay differential equation (DDE) with two time delays. The time delays are associated with the positive temperature feedback between the Equator and the North Pole, and the salinity exchange between the surface and deep water at the Pole, respectively. Studying the interplay between these two delayed feedback mechanisms is a way to gain a conceptual understanding of the long-term dynamics and possible responses of the Atlantic Ocean. Specifically, we perform a numerical bifurcation analysis of the DDE model with the software package DDE-Biftool for Matlab. Our analysis shows rich behaviour organized by homoclinic orbits, as well as a new ’locking’ phenomena between the delays.

High-efficiency and high-accuracy manufacturing relies on theoretical foundation of cutting vibration and its control, which plays a key role in intelligent manufacturing. Based on a new model considering both of regenerative and frictional cutting forces, we improved the prediction accuracy of cutting instability. Moreover, nonlinearity and non-smoothness in regenerative cutting incurs subcritical Hopf bifurcation to introduce large-amplitude chatter into linearly stable region, yielding cutting multi-stability called unsafe cutting (UC) in unsafe zones (UZs). Due to the time delay induced by regenerative effect in chip formation, the cutting multi-stability cannot be analyzed by conventional basin of attraction since delayed terms involve infinite-many dimension. To address this issue, infinite-many dimensional time-delayed states are approximated by a Fourier series aligned on a straight line, and the coefficients of the basis functions and the cutting process are used to construct the statistical basin of attraction. Inside the statistical basin of attraction, there exists a safe basin with no chatter. This findings are instrumental in designing a new state-dependent intermittent control to guide the cutting dynamics towards the safe basins. It is also seen that the state-dependent intermittent control is efficient in improving the cutting safety and shrinking the unsafe zones, even when the targeted basin for the control is larger than the real safe basin.