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Last updated: January 23, 2002

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. Continuous-time dynamical systems depending upon parameters, structural stability and bifurcations, the geometric point of view: collision of invariant sets.
• Bifurcations of equilibria. Saddle-node bifurcation, transcritical and pitchfork bifurcations, hysteresis and cusp.
• Bifurcations of limit cycles. Hopf and tangent bifurcations, Neimark-Sacker 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 doubly-degenerate (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 frequency-dependent. 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.

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, user-friendly software program to investigate chaotic dynamics. (Exercises and software are available for downloading.)

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, on-off intermittency, together with several variants of it, is a mechanism that produces bursting.  The lecture will discuss the properties of systems undergoing on-off 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 on-off intermittency.