Xin-Chu Fu

Tangencies in invariant disc packings for certain planar piecewise isometries are rare

For planar piecewise isometries (PWIs) (two-dimensional maps that restrict to isometries on some partition) there is a natural coding given by the itinerary of a trajectory between the pieces (atoms) of the partition on which it is defined. The set of points with the same coding is referred to as a cell and under certain general conditions the periodically coded cells define an invariant set that is a disjoint union of discs. In this paper properties of this invariant disc packing are investigated. For a one-parameter family of PWI on a torus, it is proved that tangencies between discs in this packing are rare. More precisely it is shown, using algebraic constraints on the geometry of the centres of the discs, that tangencies between any two discs can only occur at a finite number of parameter values; hence all tangencies will occur at a set of parameter values that is (at most) countably infinite. If such packings are dense it can be shown that they are maximal in a sense of measure. Examples are provided to show that the packing may not be dense if there is continuity over boundaries in the partition, and also that the absence of tangencies in the packing does not necessarily imply that the complement of the packing has positive Lebesgue measure.

A linear chaotic quantum harmonic oscillator

Abstract We show that a linear quantum harmonic oscillator is chaotic in the sense of Li-Yorke. We also prove that the weighted backward shift map (used as an infinite-dimensional linear chaos model) in a separable Hilbert space is chaotic in the sense of Li-Yorke, in addition to being chaotic in the sense of Devaney.

PROPERTIES OF THE INVARIANT DISK PACKING IN A MODEL BANDPASS SIGMA DELTA MODULATOR

In this paper we discuss the packing properties of invariant disks defined by periodic behavior of a model for a bandpass Σ–Δ modulator. The periodically coded regions form a packing of the forward invariant phase space by invariant disks. For this one-parameter family of PWIs, by introducing codings underlying the map operations we give explicit expressions for the centers of the disks by analytic functions of the parameters, and then show that tangencies between disks in the packings are very rare; more precisely they occur on parameter values that are at most countably infinite. We indicate how similar results can be obtained for other plane maps that are piecewise isometries.

Riddling and invariance for discontinuous maps preserving Lebesgue measure

In this paper we use the mixture of topological and measure-theoretic dynamical approaches to consider riddling of invariant sets for some discontinuous maps of compact regions of the plane that preserve two-dimensional Lebesgue measure. We consider maps that are piecewise continuous and with invertible except on a closed zero measure set. We show that riddling is an invariant property that can be used to characterize invariant sets, and prove results that give a non-trivial decomposion of what we call partially riddled invariant sets into smaller invariant sets. For a particular example, a piecewise isometry that arises in signal processing (the overflow oscillation map), we present evidence that the closure of the set of trajectories that accumulate on the discontinuity is fully riddled. This supports a conjecture that there are typically an infinite number of periodic orbits for this system.

Dynamics of a bandpass sigma-delta modulator as a piecewise isometry

We examine a model for a bandpass /spl Sigma//spl Delta/ modulator introduced by Feely and co-workers (1996). This is shown to have the dynamics of a piecewise isometry of a union of convex polygons on the plane by an appropriate transformation of the linearised parts into normal form. Using these we show that the periodically coded regions form a packing of the phase space by circles and we link this system to a number of similar ones. In particular we conjecture that for typical values of the parameter /spl theta/ there is a positive measure set of points that have aperiodic codings.

Invariant sets for discontinuous parabolic area-preserving torus maps

We analyse a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two-parameter family of maps, the set of non-invertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.

Symbolic representations of iterated maps

This paper presents a general and systematic discussion of nvarious symbolic representations of iterated maps through subshifts.nA unified model for all continuous maps on a metric space is given. nIt is shown that at most the second order nrepresentation is enough for a continuous map. nBy introducing distillations, partial representations nof some general continuous maps are obtained. Finally, partitions and nrepresentations of a class of discontinuous maps and some examples nare discussed.

CHAOTIC PROPERTIES OF SUBSHIFTS GENERATED BY A NONPERIODIC RECURRENT ORBIT

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain nonperiodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a nonperiodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

ON PLANAR PIECEWISE AND TWO-TORUS PARABOLIC MAPS

We investigate properties and examples of iterated planar piecewise parabolic (PWP) maps, including those induced by two-torus parabolic maps and their inverses. PWP maps are area-preserving maps that have constant linearizations but only one eigenvector with eigenvalue one. We obtain necessary and sufficient conditions for a two-torus parabolic map to be invertible. For noninvertible PWP maps, there are a number of questions about the existence and structure of attractors for such maps. We introduce the simplest example of a nontrivial PWP map, defined by a parabolic linear map with a translation that is a different constant on each side of a given line in the plane. For this half plane parabolic map we obtain sufficient conditions for a bounded attractor to exist, discuss their dynamical properties and give some examples of these and related maps.

Time-periodically forced amplitude evolution in spatially extended nonlinear systems

In this paper, we consider the Ginzburg-Landau amplitude evolution model with time-dependent forcing. We show that when the forcing is periodic and its spatial square-integral is bounded in time, the Ginzburg-Landau equation has time-periodic solutions. We use a topological technique from nonlinear global analysis.