On Some Neutral Functional Differential Equations Occurring in Synchronization

Abstract

The present research started from a synchronization problem going back to Huygens who observed pendula synchronization through the wall they were hanging. The phenomenon has been recently described by R. W. Brockett within the framework of Control Theory. Based on a model describing pendulum hanging on an infinite undamped vibrating string, there was obtained a basic model for two hanging pendula, expressed by neutral functional differential equations. The forcing terms in the equations are determined by the initial conditions on the vibrating string and can be supposed periodic or almost periodic. The model of the synchronization turns to be existence and stability of a forced oscillation. Existence means that there exists a global solution on $\mathbb {R}$ which in the periodic case has the same period as the forcing term; since the dynamics of the two oscillators is contained in the differential part, this means nothing more but synchronization. Stability means that this synchronization is asymptotically noticeable and measurable. However, the basic theorem on forced oscillations for systems like those described here required that the difference operator in the difference subsystem of the model has its spectrum inside the unit disk. But in the case of the mechanical models this spectrum—the eigenvalues of a certain matrix are on the unit circle (being equal to $\pm 1$) what sends to a critical case that must be analyzed separately. An electrical analogue of the aforementioned mechanical model but containing damping at the boundaries satisfies the condition for the difference operator to be strongly stable.