UK Nonlinear News, May 1996

Seminar Tour by C Grebogi (Maryland)

North Britain: May/June 1996.

Celso Grebogi of the Department of Mathematics, University of Maryland, USA, will tour various UK mathematics departments giving a series of lectures from May 28th to June 6th. The tour is funded by the North British Differential Equations Seminar (a consortium of UK Mathematics Departments) and by the LMS.

See also the web page at:
http://www.ma.hw.ac.uk/nbdes/nbdes.html
The titles of his seminars are:
(1) Chaos, strange attactors, and fractal basin boundaries in nonlinear dynamics,
(2) Chaos: control and communication,
(3) Group properties of shrimps and
(4) Numerical trajectories of chaotic systems.

The dates and venues are:

        28 May      Sheffield               1,2
        29 May      Keele                   2
        30 May      Leeds                   2
        31 May      Edinburgh               1

         3 June     Heriot-Watt             4
         4 June     Dundee                  2
         5 June     St Andrews              1
         6 June     Strathclyde             ?

All seminars are open to everyone interested, but please contact the local mathematics departments for details of the seminar title, time and exact location.

Talks and Abstracts

1. CHAOS, STRANGE ATTACTORS, AND FRACTAL BASIN BOUNDARIES IN NONLINEAR DYNAMICS

The field of chaotic dynamics is undergoing an explosive growth and many applications have been made across a broad spectrum of disciplines. In this lecture, I will introduce the concepts of chaotic and strange attractors and discuss their properties through examples. I will also argue that nonlinear dynamical systems may have more than one attractor. Typically the boundaries separating the basins of attraction of these various attractors are fractal. We will see that as chaos imposes limits in our ability to forecast the future, fractal basin boundaries imposes limits in our ability to predict the time-asymptotic final state of the system.


2. CHAOS: CONTROL AND COMMUNICATION

In this talk, I will review the use of basic properties from dynamical systems to an application. I will discuss the fundamental ideas leading to the control of chaos and the use of chaotic signals for communication. Among the topics to be discussed will be the use of unstable periodic orbits for control, the use of exponential sensitivity of chaos to direct orbits to targets, and the exploitation of symbolic representations of chaotic dynamics for communication. I will also give many relevant experimental applications to the sciences and engineering including applications to biological systems.


3. GROUP PROPERTIES OF SHRIMPS

Universal properties of bifurcation diagrams are some of the more striking phenomena in dynamical systems. In this talk, I will be concerned with a global scaling property for bifurcation diagrams of periodic orbits of smooth scalar maps with both one and two dimensional parameter spaces. Specifically, I will examine "windows" (shrimps) of periodic behavior within chaotic regions of parameter space. In both the one-parameter and two-parameter cases we prove that bifurcations within a periodic window of a given scalar map are well approximated by a linear transformation of the bifurcation diagram of a canonical map.


4. NUMERICAL TRAJECTORIES OF CHAOTIC SYSTEMS

Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. One may therefore question the validity of numerical studies involving long chaotic trajectories. I will report rigorous results in which long numerical trajectories can be approximated by true trajectories of the actual dynamical system. I will also discuss the complications that arise in higher dimensional systems when some of the Lyapunov exponents associated with a given numerical trajectory fluctuates about zero.


Source: Dugald Duncan (dugald@ma.hw.ac.uk)
Heriot-Watt University, Edinburgh

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