The dynamics and event-collision bifurcations in switched control systems with delayed switching

Abstract

We analyse a switched control system with negative feedback and delayed switching. In particular, we consider the effects of small and arbitrary delays in the switching decision function on the asymptotic dynamics of the system. In the absence of time delay, the phase space contains a set of points which under the action of the system flow are bounded, and trajectories rooted at these points converge to neutrally stable pseudo-equilibria in finite time. This structure is destroyed under the introduction of time delay. For a sufficiently small time delay, the bounded trajectories converge to a unique small scale limit cycle attractor. This is shown by means of the so-called delayed switching lines. For larger delay times, we observe event-collision bifurcations, symmetry breaking bifurcations, homoclinic bifurcations and multistability. For larger time delays, the delayed switching lines play an important role as they may be used to determine the stability properties of limit cycle attractors. By means of the discontinuity mapping, we show why following an event-collision bifurcation the stability of a limit cycle attractor may be radically altered. Our numerical test-bed model we consider here may be used on the macroscopic scale as a model for human neuromuscular control during quite standing or target tracking. It is interesting that much of the complex dynamics we uncover here occurs in the parameter range of delay time of around 150 ms, which is a typical processing time of neurocontrol systems of healthy human subjects during the control of, for example, quite standing.