Fast cycles detecting in non-linear discrete systems

In the paper below we consider a problem of stabilization of a priori unknown unstable periodic orbits in non-linear autonomous discrete dynamical systems. We suggest a generalization of a non-linear DFC scheme to improve the rate of detecting T-cycles. Some numerical simulations are presented

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 delayed feedback control for stabilizing $T$-cycles of a differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + ... + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + ... + a_N = 1$. Following an approach of Morg\"ul, we construct a map $F: \mathbb{R}^{T+1} \rightarrow \mathbb{R}^{T+1}$ whose fixed points correspond to $T$-cycles of $f$. We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of $F$. We associate to each periodic orbit of $f$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. An example indicating the efficacy of this method is provided

Аналіз пульсових хвиль власних векторів оператора диференціювання в базисі перетворення Уолша-Адамара

The opportunity and prospect of the analysis of signals of a pulse wave is shown in the field of orthogonal transformations, for which transformation are of an own vector of the discrete operator of differentiation.Показана возможность анализа сигналов пульсовой волны в области ортогональных преобразований, для которых трансформантами являются собственные вектора дискретного оператора дифференцирования.Показана можливість і перспективність аналізу сигналів пульсової хвилі в області ортогональних перетворень, для яких трансформантами є власні вектори дискретного оператора диференціювання, а оригіналами - трансформанти Уолша - Адамара

Geometric maximal operators and BMO on product bases

We consider the problem of the boundedness of maximal operators on BMO on shapes in $\mathbb{R}^n$. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO

On differentiation of integrals with respect to bases of convex sets.

Differentiation of integrals of functions from the class $Lip(1,1)(I^2)$ with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in $Lip(1,1)(I^N)$, N ≥ 3, and $H^{ω}_{1}(I^2)$ with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension