Marachkov type stability results for functional differential equations

This paper is concerned with systems of functional differential equations with either finite or infinite delay. We give conditions on the system and on a Liapunov function to ensure that the zero solution is asymptotically stable. The main result of this paper is that the assumption on boundedness in Marachkov type stability results may be replaced (in both the finite and the infinite delay case) with the condition that $|f(t,\varphi)|\le F(t)$ such that $\int^{\infty} 1/F(t) dt=\infty$

Continuity, compactness, fixed points, and integral equations

An integral equation, $x(t)=a(t)-\int^t_{-\infty} D(t,s)g(x(s))ds$ with $a(t)$ bounded, is studied by means of a Liapunov functional. There results an a priori bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer type. It is a very flexible fixed point theorem which enables us to show that the solution inherits properties of $a(t)$, including periodic or almost periodic solutions in a Banach space

Attractors and basins of dynamical systems

There are several programs for studying dynamical systems, but none of them is very useful for investigating basins and attractors of higher dimensional systems. Our goal in this paper is to show a new algorithm for finding even chaotic attractors and their basins for these systems. We present an implementation and examples for the use of this program

On some possible extensions of Massera's theorem

In this paper we consider ordinary and functional differential equations with $T$-periodic right hand side and look at some conjectures on proving the existence of a periodic solution. One of the ''best'' conditions to obtain a periodic solution is the existence of a bounded solution, because it would be then not only a sufficient but also a neccessary condition. We take one step in the direction to prove this conjecture

Linear time ordering of bins using a conveyor system

A local food wholesaler company is using an automated commissioning system, which brings the bins containing the appropriate product to the commissioning counter, where the worker picks the needed amounts to 12 bins corresponding to the same number of orders. To minimize the number of bins to pick from, they pick for several different spreading tours, so the order of bins containing the picked products coming from the commissioning counter can be considered random in this sense. Recently, the number of bins containing the picked orders increased over the available storage space, and it was necessary to find a new way of storing and ordering the bins to spreading tours. We developed a conveyor system which (after a preprocessing step) can order the bins in linear space and time

The structure of pairing strategies for k-in-a-row type games

In Maker-Breaker positional games two players, Maker and Breaker, play on a finite or infinite board with the goal of claiming or preventing the opponent from getting a finite winning set, respectively. For different games there are several winning strategies for Maker or Breaker. One class of winning strategies is the so-called pairing (paving) strategies. Here, we describe all possible pairing strategies for the 9-in-a-row game. Furthermore, we define a graph of the pairings, containing 194,543 vertices and 532,107 edges, in order to give them a structure. A complete characterization of the graph is also given