SUMMARYReal-time dynamic substructuring is an experimental technique for testing the dynamic behaviour of complex structures. It involves creating a hybrid model of the entire structure by combining an experimental test piece-the substructure-with a numerical model describing the remainder of the system. The technique is useful when it is impractical to experimentally test the entire structure or complete numerical modelling is insu cient.In this paper, we focus on the in uence of delay in the system, which is generally due to the inherent dynamics of the transfer systems (actuators) used for structural testing. This naturally gives rise to a delay di erential equation (DDE) model of the substructured system. With the case of a substructured system consisting of a single mass-spring oscillator we demonstrate how a DDE model can be used to understand the in uence of the response delay of the actuator. Speciÿcally, we describe a number of methods for identifying the critical time delay above which the system becomes unstable.Because of the low damping in many large structures a typical situation is that a substructuring test would operate in an unstable region if additional techniques were not implemented in practice. We demonstrate with an adaptive delay compensation technique that the substructured mass-spring oscillator system can be stabilized successfully in an experiment. The approach of DDE modelling also allows us to determine the dependence of the critical delay on the parameters of the delay compensation scheme. Using this approach we develop an over-compensation scheme that will help ensure stable experimental testing from initiation to steady state operation. This technique is particularly suited to sti structures or those with very low natural damping as regularly encountered in structural engineering.

We present a control scheme that is able to find and stabilize an unstable chaotic regime in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is noninvasive, however only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effect. We demonstrate the control scheme for so-called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.

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There is currently much interest in examining climatic tipping points, to see if it is feasible to predict them in advance. Using techniques from bifurcation theory, recent work looks for a slowing down of the intrinsic transient responses, which is predicted to occur before an instability is encountered. This is done, for example, by determining the short-term autocorrelation coefficient ARC(1) in a sliding window of the time-series: this stability coefficient should increase to unity at tipping. Such studies have been made both on climatic computer models and on real paleoclimate data preceding ancient tipping events. The latter employ reconstituted time-series provided by ice cores, sediments, etc., and seek to establish whether the actual tipping could have been accurately predicted in advance. One such example is the end of the Younger Dryas event, about 11 500 years ago, when the Arctic warmed by 7• C in 50 yrs. A second gives an excellent prediction for the end of "greenhouse" Earth about 34 million years ago when the climate tipped from a tropical state into an icehouse state, using data from tropical Pacific sediment cores. This prediction science is very young, but some encouraging results are already being obtained. Future analyses will clearly need to embrace both real data from improved monitoring instruments, and simulation data generated from increasingly sophisticated predictive models.

Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity.