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Introduction 37 § 1. The geometry of domains of possible motions 39 §2. Periodic trajectories of natural mechanical systems 46 §3. Periodic trajectories of irreversible systems 55 §4. Asymptotic solutions. Application to the theory of stability of motion 62 References 69

Stochastic and deterministic versions of a discrete dynamical system on a necklace are investigated. This network consists of a sequence of contours NSWE with nodes, i.e. the nodes are common points at W and E. There are two cells and a particle on each contour. Each time instance, the particle occupies a cell and, at every time unit, comes to the next cell in the same direction. The particles of the neighbouring contours move in accordance with rules of stochastic or deterministic type. The behaviour of the model with the rule of the first type is stochastic only at the beginning and after a time interval becomes a pure deterministic system. The system with the second rule comes to a steady mode, which depends on the initial state. The average velocity of particles and characteristics of the system are studied.