Some results on Liapunov functions and generated dynamical systems

This paper presents several results pertaining to the use of lower semicontinuous Liapunov functions in the analysis of autonomous abstract evolution equations. Such functions can be useful in setting up a nonlinear dynamical system that need not satisfy any exponential estimate, as well as in locating positive invariant sets of the resulting dynamical system. Other results concern the computation of the derivative of a lower semicontinuous Liapunov function, the use of such a function to assure precompactness of positive orbits, and a version of the Invariance Principle that is valid for lower semicontinuous Liapunov functions.     

Integrable dynamical systems generated by quantum models with an adiabatic parameter

Several models solvable in terms of special functions of the Heun class are widely used in quantum mechanics. They are all characterized by the presence of a parameter that can be regarded as an adiabatic variable. An antiquantization procedure applied to such a model generates a dynamical model with properties of the Painlevé equations. The mentioned parameter plays the role of time. We consider examples of such models.     

Topological conjugacy of constant length substitution dynamical systems

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.     

Self-stochasticity phenomenon in dynamical systems generated by difference equations with continuous argument

For dynamical systems generated by the difference equations x(t+1) = f(x(t)) with continuous time (f is a continuous mapping of an interval onto itself), we present a mathematical substantiation of the self-stochasticity phenomenon, according to which an attractor of a deterministic system contains random functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 954–975, July, 2006.     

Pseudorandom vector sequences of maximal period generated by triangular polynomial dynamical systems

In this paper we study the period of vector sequences generated by triangular polynomial systems and we characterize the case when their orbits are of maximal period. Moreover, we estimate multiplicative character sums with these sequences and we obtain better results using a different approach.     

Cyclic solutions in dynamical systems generated by local variational problems with objective-function singularities

The article considers cyclic systems generated by local variational problems with various singularities in the objective function. Conditions for the creation of cyclic solutions are examined. Translated from Nelineinaya Dinamika i Upravlenie, pp. 7–14, 1999.     

Attractors and dynamical systems generated by initial-boundary problems for the equations of motion of viscoelastic fluids

Attractors m and dynamical systems {m; V_t,–相似文献   

Tilings from graph directed iterated function systems

Geometriae Dedicata - A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a...     

Multivalued maps, selections and dynamical systems

Under suitable hypotheses the well known notion of first prolongational set J⁺ gives rise to a multivalued map which is continuous when the upper semifinite topology is considered in the hyperspace of X. Some important dynamical concepts such as stability or attraction can be easily characterized in terms of ψ and moreover, the classical result that an attractor in Rn has the shape of a finite polyhedron can be reinforced under the hypotheses that the mapping ψ is small and has a selection.     

Unilateral and Equitransitive Tilings by Squares

Recent results obtained by Martini et al. [4] facilitate the proof that there are exactly eight unilateral and equitransitive tilings of the plane by squares of three sizes. This refutes a conjecture by Schattschneider that appears in the book Tilings and Patterns by Grünbaum and Shephard [2]. Received January 6, 1999, and in revised form October 2, 1999. Online publication May 16, 2000.     

Planarity, Symmetry and Counting Tilings

We axiomatize the geometrical properties of T-tetromino to what we call generalised k-T-tetromino. Using this set of axioms, we show that the number of tilings of a 2kn × 2km rectangular region is given by T(L _n,m ; 3, 3) if and only if the tile is a k-T-tetromino. This generalizes a result of Korn and Pak (Theor Comp Sci 319:3–27, 2004).     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.