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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 behaviorswill consist only in the contents of harmonics given by the Fourier series attached. [ABSTRACT FROM AUTHOR]

The main purpose of the paper is to give discrete-time counterpart for some strong (robust) stability results concerning periodic linear Hamiltonian systems. In the continuous-time version, these results go back to Liapunov and Žukovskii; their deep generalizations are due to Kreĭn, Gel'fand, and Jakubovič and obtaining the discrete version is not an easy task since not all results migrate mutatis-mutandis from continuous time to discrete time, that is, from ordinary differential to difference equations. Throughout the paper, the theory of the stability zones is performed for scalar (2nd-order) canonical systems. Using the characteristic function, the study of the stability zones is made in connection with the characteristic numbers of the periodic and skew-periodic boundary value problems for the canonical system. The multiplier motion ("traffic") on the unit circle of the complex plane is analyzed and, in the same context, the Liapunov estimate for the central zone is given in the discrete-time case. [ABSTRACT FROM AUTHOR]