Contraction of orbits in random dynamical systems on the circle

The paper deals with a theoretical justification of the effect, observed in computer experiments, of convergence of orbits (without tending to any particular point) in random dynamical systems on the circle. The corresponding theorem is proved under certain assumptions satisfied, in particular, in some C ¹-open domain in the space of random dynamical systems.It follows from this theorem that the corresponding skew product has two invariant measurable sections, naturally called an attractor and a repeller. Moreover, it turns out that convergence of orbits and the uniqueness of a stationary measure, phenomena that are mutually exclusive in the case of a single map, typically coexist in random dynamical systems.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 36–54, 2004Original Russian Text Copyright © by V. A. Kleptsyn and M. B. NalskiiSupported in part by RFBR grants 02-01-00482 and 02-01-22002 and CRDF grant RM1-2358-MO-02.     

Substitution dynamical systems: Characterization of linear repetitivity and applications

We consider dynamical systems arising from substitutions over a finite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive substitutions to minimal substitutions. This includes applications to random Schrödinger operators and to number theory.     

Growth of periodic orbits of dynamical systems

V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 14–22, October–December, 1991.     

A construction of stable subharmonic orbits in monotone time-periodic dynamical systems

We consider a family of abstract semilinear elliptic-like equationsB(t,u _o (t))=0 for an unknown functionu ₀ (t) parametrized by the time-variablet0 and valued in a Banach spaceX. Suppose that bothB(.,u) andu ₀ areT-periodic in timet, and each Fréchet derivative generates an exponentially decaying, analyticC ₀-semigroup inX. We show that, for every small >0, the abstract parabolic-like evolution equationdu /dt=B(t,u _(t) ),t0, has a linearly stableT-periodic solutionu nearu ₀. Given any integern2, we construct examples ofB andu ₀ such that the minimum periods ofB(.,u) andu ₀, respectively, are=T/n andT. Thenu (t), t0, is alinearly stable subharmonic orbit of minimum periodT for our -periodic evolution equation. The corresponding dynamical systems are strongly monotone.This work was supported in part by, Vanderbilt University Research Council.     

Dynamical systems arising from random substitutions

Random substitutions are a natural generalisation of their classical ‘deterministic’ counterpart, whereby at every step of iterating the substitution, instead of replacing a letter with a predetermined word, every letter is independently replaced by a word from a finite set of possible words according to a probability distribution. We discuss the subshifts associated with such substitutions and explore the dynamical and ergodic properties of these systems in order to establish the groundwork for their systematic study. Among other results, we show under reasonable conditions that such systems are topologically transitive, have either empty or dense sets of periodic points, have dense sets of linearly repetitive elements, are rarely strictly ergodic, and have positive topological entropy.     

Newton's method and complex dynamical systems

This article is devoted to the discussion of Newton's method. Beginning with the old results of A.Cayley and E.Schröder we proceed to the theory of complex dynamical systems on the sphere, which was developed by G.Julia and P.Fatou at the beginning of this century, and continued by several mathematicians in recent years.     

Some dynamical behavior of discrete Nagumo equation

The spatiotemporal dynamics can be effectively studied by continuation from an anti-integrable limit. By using the anti-integrability method we consider the dynamical behavior of discrete Nagumo equation, prove the existence of breather and discuss the spatial disorder of the stationary solutions in the system.     

Tracing of pseudotrajectories of dynamical systems and stability of prolongations of orbits

We investigate properties of dynamical systems associated with the approximation of pseudotrajectories of a dynamical system by its trajectories. According to modern terminology, a property of this sort is called the “property of tracing pseudotrajectories” (also known in the English literature as the “shadowing property”). We prove that dynamical systems given by mappings of a compact set into itself and possessing this property are systems with stable prolongation of orbits. We construct examples of mappings of an interval into itself that prove that the inverse statement is not true, i.e., that dynamical systems with stable prolongation of orbits may not possess the property of tracing pseudotrajectories. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1016–1024, August, 1997.     

Leibniz代数胚上动力系统的轨道范例与图示

首先陈述Leibniz代数胚上的动力系统和在局部坐标系下该动力系统的方程,在此基础上,给出了动力系统轨道的范例和图示.     

Long-run behavior of constrained discrete-time linear dynamical systems

We consider a system of linear equations whose variablesy(t) E ^m describe an economy or process which is observed in succeeding discrete time periodst. With forms given for the equations relating the values of the state vectory(t) in succeeding periods and limiting the set of feasible choices ofy(t) in each period, conditions which determine the nature of feasible sequences of state vectors are given, and the characteristics ofy(t) ast are analyzed. Some of the results are studied in the framework of specified infinite horizon optimization problems.Research for this paper was partially supported by the Office of Naval Research, Contract N00014-69-A-0200-1055 at the Operations Research Center of the University of California, Berkeley.     

Stability analysis of constrained inventory systems with transportation delay

Stability is a fundamental design property of inventory systems. However, the often exploited linearity assumptions in the current literature create a major gap between theory and practice. In this paper the stability of a constrained production and inventory system with a Forbidden Returns constraint (that is, a non-negative order rate) is studied via a piecewise linear model, an eigenvalue analysis and a simulation investigation. The APVIOBPCS (Automatic Pipeline, Variable Inventory and Order Based Production Control System) and EPVIOBPCS (Estimated Pipeline, Variable Inventory and Order Based Production Control System) replenishment policies are adopted. Surprisingly, all kinds of non-linear dynamical behaviours of systems can be observed in these simple models. Exact expressions of the asymptotic stability boundaries and Lyapunovian stability boundaries are derived when actual and perceived transportation lead-time is 1 and 2 periods long respectively. Asymptotically stable regions in the non-linear Forbidden Return systems are identical to the stable regions in its unconstrained counterpart. However, regions of bounded fluctuations that continue forever, including both periodicity and chaos, exist in the parametrical plane outside the asymptotically stable region. Simulation shows a complex and delicate structure in these regions. The results suggest that accurate lead-time information is essential to eliminate inventory drift and instability and that ordering policies have to be designed properly in accordance with the actual lead-time to avoid these fluctuations and divergence.     

Directional complexity of the hypercubic billiard

We consider a minimal rotation on the torus T^d of direction ω. A natural cellular decomposition of the torus is associated to this map. We consider an infinite orbit for this map. We compute the complexity of the associated word. Under some hypothesis on the direction, we obtain an exact formula which shows that the order of magnitude is n^d. This result is related to the billiard map inside a hypercube of R^d+1.     

Numerical optimization of certain dynamical stochastic systems

Two-dimensional dynamical (dynamical stochastic) predator-prey systems are optimized numerically by calculating the variation of the functional of the steady-state solution to the Fokker-Plank equation. Numerical algorithms are used to construct systems with a prescribed response to a certain external action.