Optimal strategies and transitions between them in Arnold’s model

An optimal strategy which exists among periodic rotations or stationery strategies in Arnold's model is described. The study of periodic phenomena of various natures is reduced to analyze the averaged profit of such processes. The infinite horizon averaged profit can be maximized by a level motion or a suitable stationary point for a continuous control system and a continuous profit density on the circle. This makes it possible to complete classification of these singularities by dividing them into three groups that include maximum averaged profit singularities for optimal level cycles, for optimal stationary strategy, and for transitions between these two type of motions. The maximum averaged profits for generic family of pairs are mapped into each other by smooth equivalence close to identity. It is necessary to study them for the stationary domain and prove their stability with respect to all small perturbations in a generic family of systems, to classify the generic singularities