Search for cycles of nonlinear periodic discrete systems using the method of average predictive control

The dynamics of even the simplest nonlinear stationary discrete systems is very complex. It includes both periodic motions and quasiperiodic or recurrent ones. Such systems almost always have chaotic attractors the nature of which has been currently studied quite well, at least for a wide class of model stationary equations. In nonstationary systems such dynamics becomes even more complex. In many cases chaotic attractors can be modeled using periodic motions with large periods, i.e. construct the so-called skeleton of an attractor. The search for both attractors and minimal invariant sets on them is an important problem of applied mathematics. The solutions are used in the physical, chemical, economic sciences, coding theory, signal transmission, etc. One of the approaches to solving search and verification problems of cycles is based on the application of methods for these cycles stabilization. These methods can be divided into two groups: Delayed control using the knowledge of the system previous states and the predictive control using future values of the system state in the absence of control. The main result of this work is the representation of Jacobian matrix of the cycle of the system with control via the corresponding Jacobian matrix of the system without control. From this representation control gain coefficients are immediately obtained if the cycle multipliplicators are known. If they are unknown, the method for estimating the gain coefficients using the approximate values of Lyapunov exponents is proposed. Methods for verification of the found points of the cycle are proposed in the form of three necessary conditions for the point cyclicity: Checking the residual smallness, periodicity and the local asymptotic stability of the cycle. The algorithm operation is demonstrated on the model examples.