Huygens Synchronization Over Distributed Media—Structure Versus Complex Behavior

This analysis is concerned with Huygens synchronization over distributed media—vibrating strings or LC transmission lines. If e.g., two oscillators with lumped parameters i.e., described by ordinary differential equations, which display self sustained oscillations, are coupled to some distributed medium, they interact in function of the structural properties of the resulting system. If this medium has infinite length then, according to the structural properties of the difference operator describing propagation, either synchronization with the external frequency or some “complex behavior” can occur. If the coupling has a finite length, again the qualitative properties are determined by the structure of the aforementioned difference operator: either the self sustained oscillations are quenched, the system approaching asymptotically a stable equilibrium (opposite of the Turing coupling of two “cells” that is lumped oscillators) or again the aforementioned “complex behavior” can occur. We state finally the conjecture that this “complex behavior” is in fact some almost periodic oscillation and not a chaotic behavior. It is worth mentioning that the two types of qualitative behavior are in connection with the physical nature of the considered systems. Specifically, the difference operator is strongly stable for electrical systems and critically stable for mechanical systems. It is this aspect that explains proneness to standard or “complex” behavior. However, if the aforementioned conjecture will not be disproved, the difference between the corresponding oscillatory behaviors will consist only in the contents of harmonics given by the Fourier series attached.