Common periodic trajectories of interval maps

We prove that for any continuous piecewise monotone or smooth interval map f and any subset of the set of periods of periodic trajectories of f, there is another map such that the set of periods of periodic trajectories common for f and , which is denoted by , coincides with . At the same time, for each integer , there exists a continuous map f such that for any map if is an infinite set. Dedicated to Vladimir Igorevich Arnold     

Asymptotic periodicity of recurrent flows in infinite networks

We consider a transport process on an infinite network and, using the corresponding flow semigroup as in Dorn (Semigroup Forum 76:341–356, 2008), investigate its long term behavior. Combining methods from functional analysis, graph theory and stochastics, we are able to characterize the networks for which the flow semigroup converges strongly to a periodic group.     

Asymptotic almost periodicity and motions of semigroups of operators

In this survey we present an overview of recent work in resolving certain questions that arise in a qualitative study of solutions to the abstract Cauchy problem. Focusing on motions of C₀-semigroups of operators and their asymptotic behavior, we address (1) the characterization of asymptotically almost periodic motions and (2) the characterization of those asymptotically almost periodic functions for which this property carries over to their integral. Moreover, we take up the problem raised by Nemytskii and Stepanov as to when the ω-limit set of a motion of a dynamical system is a minimal set of almost periodic motions.     

The order of coexistence of homoclinic trajectories for the maps of an interval

Ukrainian Mathematical Journal - A nonperiodic trajectory of a discrete dynamical system is called n-homoclinic if its α- and ω-limit sets coincide and form the same cycle of period n. We...     

Asymptotic periodicity of the Volterra equation with infinite delay

For some species, hereditary factors have great effects on their population evolution, which can be described by the well-known Volterra model. A model developed is investigated in this article, considering the seasonal variation of the environment, where the diffusive effect of the population is also considered. The main approaches employed here are the upper–lower solution method and the monotone iteration technique. The results show that whether the species dies out or not depends on the relations among the birth rate, the death rate, the competition rate, the diffusivity and the hereditary effects. The evolution of the population may show asymptotic periodicity, provided a certain condition is satisfied for the above factors.     

Asymptotic periodicity of nonlinear discrete Volterra equations and applications

Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

Asymptotic periodicity for some evolution equations in Banach spaces

This work deals with the existence and uniqueness of pseudo-almost periodic and asymptotically ω-periodic mild solutions to some evolution equations in Banach spaces.     

Asymptotic periodicity of a food-limited diffusive population model with time-delay

In this paper, a general reaction-diffusion food-limited population model with time-delay is proposed. Accordingly, the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial-boundary value problem are considered. Finally, the effect of the time-delay on the asymptotic behavior of the solutions is given.     

Asymptotic optimality in sequential interval estimation

The problem of setting a fixed width confidence interval for the mean of a normal distribution with unknown variance is considered. Several procedures are reviewed. An asymptotic lower bound for the expected sample size of any sequential sampling plan with the specified confidence coefficient is obtained as the width of the interval decreases to 0.     

Asymptotic investigation of equations of controlled motion with multivalued trajectories

We consider the possibility of applying the averaging method for optimal control problems with the terminal performance criterion, in which behavior of an object is described by a differential inclusion containing a control.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1409–1412, October, 1990.     

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=G _R with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {u _t} ofG and anyg∈G, the time spent inC by the {u _t}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g ⁻¹u_tg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].     

Asymptotic behavior of trajectories of stationary Banach-valued Gaussian fields. II

The study of the asymptotic behavior of trajectories of stationary Gaussian Banach-valued random fields is continued. Bibliography:3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 220–228     

Asymptotic properties of optimal trajectories of discrete economic and demographic models

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 37–48, November–December, 1988.     

Asymptotic behavior of trajectories for some nonautonomous,almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form $\text{du}\text{dt}\text{+ A (t) u(t) ? f(t)}$, where $\text{?(t)}$ is almost periodic: R → V uniformly convex Banach space and A(t) is a periodic, time-dependent, m-accretive operator in V.     

Asymptotic behavior of the quadratic variation for trajectories of processes with independent increments

One considers a certain characteristic, similar to the quadratic variation, of processes with independent increments. One investigates its almost surely behavior as t.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 78–88, 1983.