Dmitriy Dmitrishin

On the stability of cycles by delayed feedback control

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a DFC for stabilizing -cycles of a differentiable function of the form where with . Following an approach of Morgül, we construct a map whose fixed points correspond to -cycles of . We then analyse the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrium points of . We associate to each periodic orbit of an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. This polynomial is the characteristic polynomial of a Jacobian matrix that lies in a large class of matrices that encompasses the usual ‘companion matrices’ found in linear algebra; the primary purpose of this paper is to show that this polynomial may be expressed in a surprisingly simple form. An example indicating the efficacy of this method is provided.

Finding Cycles in Nonlinear Autonomous Discrete Dynamical Systems

The goal of this paper is to provide an exposition of recent results of the authors concerning cycle localization and stabilization in nonlinear dynamical systems. Both the general theory and numerical applications to well-known dynamical systems are presented. This paper is a continuation of Dmitrishin et al. (Fejer polynomials and chaos. Springer proceedings in mathematics and statistics, vol 108, pp. 49–75, 2014).

Fejér Polynomials and Chaos

We show that given any μ > 1, an equilibrium x of a dynamic system

From Chaos to Order Through Mixing

\displaystyle{ x_{n+1} = f(x_{n}) }

Fejér and Suffridge Polynomials in the Delayed Feedback Control Theory

(1) can be robustly stabilized by a nonlinear control

Optimal stabilization of a cycle in nonlinear discrete systems

\displaystyle{ u = -\sum _{j=1}^{N-1}\varepsilon _{ j}\left (f\left (x_{n-j+1}\right ) - f\left (x_{n-j}\right )\right ),\,\vert \varepsilon _{j}\vert < 1,\;j = 1,\ldots,N - 1, }