Huygens' synchronization of dynamical systems : beyond pendulum clocks

Synchronization is one of the most deeply rooted and pervasive behaviours in nature. It extends from human beings to unconscious entities. Some familiar examples include the fascinating motion of schools of fish, the simultaneous flashing of fireflies, a couple dancing in synchrony with the rhythm of the music, the synchronous firing of neurons and pacemaker cells, and the synchronized motion of pendulum clocks. In a first glimpse to these examples, the existence of selfsynchronization in nature may seem almost miraculous. However, the main "secret" behind this phenomenon is that there exists a communication channel, called coupling, such that the entities/systems can influence each other. This coupling can be, for instance, in the form of a physical interconnection or a certain chemical process. Although synchronization is a ubiquitous phenomenon among coupled oscillatory systems, its onset is not always obvious. Consequently, the following questions arise: How exactly do coupled oscillators synchronize themselves, and under what conditions? In some cases, obtaining answers for these questions is extremely challenging. Consider for instance, the famous example of Christiaan Huygens of two pendulum clocks exhibiting anti-phase or in-phase synchronized motion. Huygens did observe that there is a "medium" responsible for the synchronized motion, namely the bar to which the pendula are attached. However, despite the remarkably correct observation of Huygens, even today a complete rigorous mathematical explanation of this phenomenon, using proper models for pendula and flexible coupling bar, is still missing. The purpose of this thesis is to further pursue the nature of the synchronized motion occurring in coupled oscillators. The first part, addresses the problem of natural synchronization of arbitrary self-sustained oscillators with Huygens coupling. This means that in the analysis, the original setup of Huygens’ clocks is slightly modified in the sense that each pendulum clock is replaced by an arbitrary second order nonlinear oscillator and instead of the flexible wooden bar (called here Huygens’ coupling), a rigid bar of one degree of freedom is considered. Each oscillator is provided with a control input in order to guarantee steady-state oscillations. This requirement of having a control input to sustain the oscillations can be linked to Huygens’ pendulum clocks, where each pendulum is equipped with an escapement mechanism, which provides an impulsive force to the pendulum in order to keep the clocks running. Then, it is shown that the synchronized motion in the oscillators is independent of the kind of controller used to maintain the oscillations. Rather, the coupling bar, i.e. Huygens’ coupling is considered as the key element in the occurrence of synchronization. In particular, it is shown that the mass of the coupling bar determines the eventual synchronized behaviour in the oscillators, namely in-phase and anti-phase synchronization. The Poincaré method is used in order to determine the existence and stability of these synchronous motions. This is feasible since in the system there exists a natural small parameter, namely the coupling strength, which value is determined by the mass of the coupling bar. Next, the synchronization problem is addressed from a control point of view. First, the synchronization problem of two chaotic oscillators with Huygens’ coupling is discussed. It is shown that by driving the coupling bar with an external periodic excitation, it is possible to trigger the onset of chaos in the oscillators. The mass of the coupling bar is considered as the bifurcation parameter. When the oscillators are in a chaotic state, the synchronization phenomenon will not occur naturally. Consequently, it is demonstrated that by using a master-slave configuration or a mutual synchronization scheme, it is possible to achieve (controlled) synchronization. Secondly, the effect of time delay in the synchronized motion of oscillators with Huygens’ coupling is investigated. In this case, the wooden bar, is replaced by a representative dynamical system. This dynamical system generates a suitable control input for the oscillators such that in closed loop the system resembles a pair of oscillators with Huygens’ coupling. Under this approach, the oscillators do not need to be at the same location and moreover, the dynamical system generating the control input should be implemented separately, using for instance a computer. Consequently, the possibility of having communication time-delays (either in the oscillators or in the applied control input) comes into play. Then, the onset of in-phase and anti-phase synchronization in the coupled/controlled oscillators is studied as a function of the coupling strength and the time delay. In addition to computer simulations, the (natural and controlled) synchronized motion of the oscillators is validated by means of experiments. These experiments are performed in an experimental platform consisting of an elastically supported (controllable) rigid bar (in Huygens’ example the wooden bar) and two (controllable) mass-spring-damper oscillators (the pendulum clocks in Huygens’ case). A key feature of this platform is that its dynamical behaviour can be adjusted. This is possible due to the fact that the oscillators and the coupling bar can be actuated independently, then by using feedback, specific desirable oscillators’ dynamics are enforced and likewise the behaviour of the coupling bar is modified. This feature is exploited in order to experimentally study synchronous behaviour in a wide variety of dynamical systems. Another question considered in this thesis is related to the modeling of the real Huygens experiment. The models used in the first part of this thesis and the ones reported in the literature are simplifications of the real model: the coupling bar has been considered as a rigid body of one degree of freedom. However, in the real Huygens experiment, the bar to which the clocks are attached is indeed an infinite dimensional system for which a rigorous study of the in-phase and antiphase synchronized motion of the two pendula is, as far as is known, still never addressed in the literature. The second part of the thesis focuses on this. A Finite Element Modelling technique is used in order to derive a model consisting of a (finite) set of ordinary differential equations. Numerical results illustrating all the possible stationary solutions of the "true" infinite dimensional Huygens problem are included. In summary, the results contained in the thesis in fact reveal that the synchronized motion observed by Huygens extends beyond pendulum clocks