Marks Ruziboev

In this work, we study rates of mixing for small independent and identically distributed random perturbations of contracting Lorenz maps sufficiently close to a Rovella parameter. By using a random Young tower construction, we prove that this random system has exponential decay of correlations.

In this paper, we study linear optimal pursuit evasion games in Banach space of summable sequences with it pth power. We start by proving the existence and uniqueness of mild solutions as well as asymptotic stability for linear differential equations. Further, the dependence of asymptotic stability on the norm of the space is investigated: if the spectrum of the right-hand side lies strictly on the left half-plane then it is asymptotically (in fact, exponentially) stable for all spaces of summable sequences. We also show an example that is asymptotically stable for in the spaces of sequences, which is not asymptotically stable for in the space of essentially bounded sequences. We also study the linear differential game of pursuit and evasion and construct a strategy for the evader that guarantees the possibility of evasion from any initial state on a finite time interval and construct optimal strategy for the pursuer to complete the pursuit game.

An algorithmic method is proposed to generate all cubic polynomials with a critical orbit relation. We generate curves (polynomials of parameters) that correspond to those functions with critical orbit relations. The irreducibility of the polynomials obtained is left as an open problem. Our approach also works to generate critical orbit relations in all families of rational functions with active critical points.

In this article, we consider simple-motion pursuit–evasion differential games in the Hilbert space of square summable sequences. We show that when the players have the same dynamic capabilities, evasion is possible under some assumptions about the initial positions of the players.

We study quenched mixing rates for two classes of random interval maps characterized by the presence of two indifferent fixed points and singular points. Using a random tower construction we prove the existence of an equivariant absolutely continuous probability measures with polynomial decay of correlations.

In this paper, we study the stability, controllability, and differential game of pursuit for an infinite system of linear ODEs in ℓ². The system we consider has a special right-hand side, which is not diagonal and serves as a toy model for controllable system of infinitely many interacting points. We impose integral constraints on the control parameters. We obtain criteria for stability and null controllability of the system. Further, we construct a strategy for the pursuer that guarantees completion of the pursuit problem for the differential game. To prove controllability we use the so called Gramian operators.

Two player pursuit evasion differential game and time optimal zero control problem in ℓ2 are considered. Optimal control for the corresponding zero control problem is found. A strategy for the pursuer that guarantees the solution for the pursuit problem is constructed.

We investigate a differential evasion game with multiple pursuers and an evader for the infinite systems of differential equations in ℓ². The control functions of the players are subject to geometric constraints. The pursuers’ goal is to bring the state of at least one of the controlled systems to the origin of ℓ², while the evader’s goal is to prevent this from happening in a finite interval of time. We derive a sufficient condition for evasion from any initial state and construct an evasion strategy for the evader.

In this article we study random tower maps driven by an ergodic automorphism. We prove quenched exponential correlations decay for tower maps admitting exponential tails. Our technique is based on constructing suitable cones of functions, defined on the random towers, which contract with respect to the Hilbert metric under the action of appropriate transfer operators. We apply our results to obtain quenched exponential correlations decay for several non-iid random dynamical systems including small random perturbations of Lorenz maps and Axiom A attractors.