Abstract:
This article presents the results of an analytical and computer study of the collective dynamic properties of a chain
of self-oscillating systems (conditionally — oscillators). It is assumed that the couplings of individual elements of the chain
are non-reciprocal, unidirectional. More precisely, it is assumed that each element of the chain is under the influence of the
previous one, while the reverse reaction is absent (physically insignificant). This is the main feature of the chain. This system
can be interpreted as an active discrete medium with unidirectional transfer, in particular, the transfer of a matter. Such
chains can represent mathematical models of real systems having a lattice structure that occur in various fields of natural
science and technology: physics, chemistry, biology, radio engineering, economics, etc. They can also represent models
of technological and computational processes. Nonlinear self-oscillating systems (conditionally, oscillators) with a wide
“spectrum” of potentially possible individual self-oscillations, from periodic to chaotic, were chosen as the “elements” of
the lattice. This allows one to explore various dynamic modes of the chain from regular to chaotic, changing the parameters
of the elements and not changing the nature of the elements themselves. The joint application of qualitative methods of the
theory of dynamical systems and qualitative-numerical methods allows one to obtain a clear picture of all possible dynamic
regimes of the chain. The conditions for the existence and stability of spatially-homogeneous dynamic regimes (deterministic
and chaotic) of the chain are studied. The analytical results are illustrated by a numerical experiment. The dynamical regimes
of the chain are studied under perturbations of parameters at its boundary. The possibility of controlling the dynamic regimes
of the chain by turning on the necessary perturbation at the boundary is shown. Various cases of the dynamics of chains
comprised of inhomogeneous (different in their parameters) elements are considered. The global chaotic synchronization (of
all oscillators in the chain) is studied analytically and numerically