International School
Topics in Nonlinear Dynamics
Venice (Italy),
January 30  February 1,
2002
Download the miniposter
of the school (pdf)
Technical support by DEI
 PdM
Mail to the webmaster
Last updated: January
23, 2002 
International School
Topics in Nonlinear Dynamics
Venice International University,
Venice (Italy),
January 30  February 1,
2002
PROGRAM
Wednesday, January 30

Thursday, January 31

Friday, February 1

9:00  registration
9:45  Welcome address
Ignazio Musu
Dean of Venice International University
10:00  OPENING LECTURE
Mathematicians and scientists were
the last people to learn about chaos
James A. Yorke
Coffee break
GEOMETRIC APPROACH TO BIFURCATIONS
11:30  Introduction to bifurcations
of dynamical systems
Sergio Rinaldi

NUMERICAL BIFURCATION ANALYSIS
9:30  Numerical detection, analysis,
and continuation of bifurcations
Yuri A. Kuznetsov
Coffee break
11:30  Bordered matrix techniques in
bifurcation analysis
Willy Govaerts

CHAOTIC DYNAMICS: THEORY
AND NUMERICAL SIMULATIONS
9:30  Nonlinear dynamics and chaos:
The ergodic approach (part I)
Sergio Invernizzi
10:30  Nonlinear dynamics and chaos:
The ergodic approach (part II)
Alfredo Medio
Coffee break
11:30 – Dynamical model crunching: A
tutorial
Marji Lines

14:30  Bifurcations of equilibria
Carlo Piccardi
15:30  Bifurcations of limit cycles
Sergio Rinaldi
Coffee break
17:00  Homoclinic and heteroclinic
bifurcations
Carlo Piccardi

EVOLUTIONARY DYNAMICS: THEORY AND
APPLICATIONS
14:30  Introduction to adaptive dynamics
theory
Ulf Dieckmann
Coffee break
16:00  Adaptive dynamics and technological
change
Michael Obersteiner
16:30  Evolutionary bifurcations
Ulf Dieckmann
17:00  Attractor switching and evolutionary
cycles
Fabio Dercole

ANALYSIS OF CHAOTIC TIME SERIES
14:30  Chaotic data analysis: An introduction
Eric Kostelich
15:30  Viewing chaotic dynamics in
experiments
James A. Yorke
Coffee break
17:00  Mechanisms of bursting
Antonello Provenzale

ABSTRACTS
OPENING LECTURE, James A. Yorke

Mathematicians and scientists were the
last people to learn about chaos. A number of illustrations of chaotic
dynamics will be presented.
GEOMETRIC APPROACH TO BIFURCATIONS,
Sergio Rinaldi, Carlo Piccardi

Introduction to bifurcations of dynamical
systems. Continuoustime dynamical systems depending upon parameters,
structural stability and bifurcations, the geometric point of view: collision
of invariant sets.

Bifurcations of equilibria. Saddlenode
bifurcation, transcritical and pitchfork bifurcations, hysteresis and cusp.

Bifurcations of limit cycles. Hopf
and tangent bifurcations, NeimarkSacker bifurcation and frequency locking,
flip bifurcation and Feigenbaum's cascade.

Homoclinic and heteroclinic bifurcations.
Homoclinic and heteroclinic orbits and their bifurcations, Andronov's and
Shilnikov's results.
NUMERICAL BIFURCATION ANALYSIS, Yuri
A. Kuznetsov, Willy Govaerts

Numerical detection, analysis, and continuation
of bifurcations. The lecture is an overview of modern methods for
numerical bifurcation analysis of nonlinear ordinary differential equations,
including continuation and numerical normal form computation for doublydegenerate
(codimension 2) equilibria.

Bordered matrix techniques in bifurcation
analysis. Rank deficiency of a big matrix can be expressed as that
of a smaller matrix that can be obtained by solving an auxiliary bordered
linear system. Various applications of this idea to bifurcation problems
will be discussed.
EVOLUTIONARY DYNAMICS: THEORY AND APPLICATIONS,
Ulf Dieckmann, Michael Obersteiner, Fabio Dercole

Introduction to adaptive dynamics theory.
Whenever
the evolutionary success of an adaptive strategy depends on which other
strategies it competes with, selection is called frequencydependent. Adaptive
dynamics theory is designed to describe such evolutionary processes and
establishes macroscopic evolutionary laws from the microscropic ecological
interactions between individuals.

Adaptive dynamics and technological change.
Technological
change and its interactions with the market are studied through the canonical
equation of adaptive dynamics.

Evolutionary bifurcations. Attractors
of adaptive dynamics driven by slow mutation and fast selection possess
two independent properties: evolutionary stability and convergence stability.
Adaptive dynamics bifurcations are therefore considerably richer than those
of ordinary differential equations.

Attractor switching and evolutionary cycles.
When
the number of attractors of a population model depends upon an adaptive
trait, evolutionary cycles can occur and entrain recursive switches between
the attractors.
Download
Ulf Dieckmann's lecture (pdf 900k)
CHAOTIC DYNAMICS: THEORY AND NUMERICAL
SIMULATIONS, Sergio Invernizzi, Alfredo Medio, Marji Lines

Nonlinear dynamics and chaos: The ergodic
approach. In these lectures, we discuss the statistical properties
of ensembles of orbits generated by deterministic dynamical systems. Some
elementary notions of measure theory are provided together with examples
of applications to certain canonical models. The question of predictability
is also discussed.

Dynamical model crunching: A tutorial.
Numerical
simulation plays a fundamental role in the study of dynamical systems.
This tutorial makes use of a specifically designed, userfriendly software
program to investigate chaotic dynamics. (Exercises and software are available
for downloading.)
Preview
Sergio Invernizzi's lecture.
Preview
Alfredo Medio's and Marji Lines' lectures.
ANALYSIS OF CHAOTIC TIME SERIES, Eric Kostelich,
James A. Yorke, Antonello Provenzale

Chaotic data analysis: An introduction.Much
effort has been undertaken over the past two decades to elucidate the dynamical
properties of chaotic processes from time series measurements. The
lecture surveys some of the basic ideas: the time delay embedding method,
attractor reconstruction, and the estimation of dynamical information like
dimension and Lyapunov exponents. While the methods work very well in many
circumstances, they are subject to large uncertainties in some cases.
The lecture will outline one approach to quantifying the uncertainty in
Lyapunov exponent calculations.

Viewing chaotic dynamics in experiments.
The
lecture will address the question of when a projection of a dynamical system
(from a high dimension to a lower dimension) preserves the essential aspects
of the original system. This technique of projection is often used by scientists
studying experiments having chaotic attractors. It is important to know
if and when it is valid.

Mechanisms of bursting. As its name
suggests, bursting is a process in which episodes of high activity are
alternated with periods of inactivity. In dynamical systems, onoff intermittency,
together with several variants of it, is a mechanism that produces bursting.
The lecture will discuss the properties of systems undergoing onoff intermittency
and will attempt to see whether one can actually tell from the output of
a signal when an observed bursting behavior is caused by the presence of
onoff intermittency.
