Charles Tresser

A new approach to data storage using localized structures

In this paper we describe how to use the bifurcation structure of static localized solutions in one dimension to store information on a medium in such a way that no extrinsic grid is needed to locate the information. We demonstrate that these principles, deduced from the mathematics adapted to describe one-dimensional media, also allow one to store information on two-dimensional media.

The mathematics of halftoning

This paper describes some mathematical aspects of halftoning in digital printing. Halftoning is the technique of rendering a continuous range of colors using only a few discrete ones. There are two major classes of methods: dithering and error diffusion. Some discussion is presented concerning the method of dithering, but the main emphasis is on error diffusion.

Master‐slave synchronization from the point of view of global dynamics

We present a mathematical framework for the theory of a synchronization phenomenon for dynamical systems discovered by Pecora and Carroll [Phys. Rev. Lett. 64, 821-824 (1990)]. From this perspective, we can synchronize, using a single coordinate, an open dense set of linear systems. We use our insights to synchronize nonlinear systems which were not previously recognized as being synchronizable. (c) 1995 American Institute of Physics.

Cyclic renormalization and automorphism groups of rooted trees

Cyclic renormalization.- Itinerary calculus and renormalization.- Spherically transitive automorphisms of rooted trees.- Closed normal subgroups of Aut(X(q)).

The gluing bifurcation: I. Symbolic dynamics of closed curves

The authors study the periodic orbits which can occur in a neighbourhood of a codimension-two gluing bifurcation involving two trajectories bi-asymptotic to the same stationary point. Provided some simple conditions are satisfied they prove that there are either zero, one or two closed curves and that these have a specific symbolic form which, in particular, allows them to associate a rotation number with each of them. Furthermore, pairs of orbits which can coexist are identified: the two rotation numbers must be Farey neighbours.

Master-slave synchronization and the Lorenz equations

Since the seminal remark by Pecora and Carroll [Phys. Rev. Lett. 64, 821 (1990)] that one can synchronize chaotic systems, the main example in the related literature has been the Lorenz equations. Yet this literature contains a mixture of true and false, and of justified and unsubstantiated claims about the synchronization properties of the Lorenz equations. In this note we clarify some of the confusion. (c) 1997 American Institute of Physics.

Topological conjugacy of linear endomorphisms of the 2-torus

We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two-dimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the Latimer-MacDuffee-Taussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classification theory for quadratic forms, we use the rotation number of Poincare.

Red spectra from white and blue noise.

The value of maps of the interval in modelling population dynamics has recently been called into question because temporal variations from such maps have blue or white power spectra, whereas many observations of real populations show time–series with red spectra. One way to deal with this discrepancy is to introduce chaotic or stochastic fluctuations in the parameters of the map. This leads to on–off intermittency and can markedly redden the spectrum produced by a model that does not by itself have a red spectrum. The parameter fluctuations need not themselves have a red spectrum in order to achieve this effect. Because the power spectrum is not invariant under a change of variable, another way to redden the spectrum is by a suitable transformation of the variables used. The question this poses is whether spectra are the best means of characterizing a fluctuating variable.

THE INTERMITTENT SOLAR CYCLE

Abstract A prominent feature of the solar cycle is the rise and fall of the number of sunspots on the surface with a timescale of approximately eleven years. The mathematical description of this behavior is complicated by the interruption of the cycle for 75 years starting around 1650. Similar previous intermissions of this kind are implied by the available data. We explore the possibility of modeling such temporal variations of the sunspot number with a deterministic dynamical system of relatively low order. The system we propose manifests on/off intermittency in which the cyclic variations of the solar activity switch off almost completely for extended periods. We also offer an explanation of the variation of the fluctuating part of the sunspot number over the cycle.

The optimality of the online greedy algorithm in carpool and chairman assignment problems

We study several classes of related scheduling problems including the carpool problem, its generalization to arbitrary inputs and the chairman assignment problem. We derive both lower and upper bounds for online algorithms solving these problems. We show that the greedy algorithm is optimal among online algorithms for the chairman assignment problem and the generalized carpool problem. We also consider geometric versions of these problems and show how the bounds adapt to these cases.