SEARCH FOR CYCLES OF NONLINEAR PERIODIC DISCRETE SYSTEMS USING THE METHOD OF AVERAGE PREDICTIVE CONTROL

Abstract

The dynamics of even the simplest nonlinear stationary discrete systems is very complex. It includes both periodic motions and quasiperiodic or recurrent ones. In such systems, chaotic attractors are almost always present, the nature of which has been studied quite well today, at least for a wide class of model stationary equations. In non-stationary systems such dynamics becomes even more complex. In many cases chaotic attractors can be modeled using periodic motions with large periods, i.e. build the so-called skeleton of an attractor. The search for both attractors themselves and minimal invariant sets on them is an important task of applied mathematics — solutions are used in the physical, chemical, economic sciences, in the theory of coding, signal transmission, etc. One of the approaches to solving search and verification of cycle problems is based on the application methods for stabilizing these cycles. These methods can be divided into two groups: delayed control, using knowledge of the previous states of the system, and predictive control, using future values of the state of the system in the absence of control. The main result of this work is the representation of the Jacobi matrix of a system cycle with control through the corresponding Jacobi matrix of a system without control. From this representation control gains are immediately obtained if the cycle multipliers are known. If they are not known, then a method is proposed for estimating the gain using approximate values of Lyapunov exponents. Verification methods for the found points of the cycle are proposed in the form of three necessary conditions for the cyclicity of the point: checking the smallness of the residual, checking the periodicity, and checking the local asymptotic stability of the cycle. The operation of the algorithm is demonstrated on model examples.