INTRODUCTION to NONLINEAR DYNAMICS and CHAOS

2007 Winter

M-W 4.10-6.00pm, Olds/Upton Library

Instructor: Péter Érdi Henry R. Luce Professor of Complex Systems Studies

Office:Olds/Upton 208B

Phone: (269)337-520

email: perdi@kzoo.edu

Topics:

Dynamical systems are mathematical objects used to model phenomena of natural and social phenomena whose state changes over time. Nonlinear dynamical systems are able to show complicated temporal, spatial and spatio-temporal behavior. They include oscillatory and chaotic behaviors and spatial structures including fractals. Students will learn the basic mathematical concepts and methods used to describe dynamical systems. Applications will cover many scientific disciplines, including physics, chemistry, biology, economics, and other social sciences.

Goal:

The first goal is to teach WHY nonlinear dynamics and chaos theory is important in understanding complicated behaviors. The second goal is to give an introductory overview about HOW the basic methods of nonlinear dynamic works. The course teaches the fundamental mathematical concepts of dynamical systems, such as phase space, attractors, stability analysis, bifurcations etc. The course is designed for physics and math students, but other (even social) science majors interested in mathematical modeling might take the class.

Prerequisite: MATH 113 or permission.

Course Structure:

Each week a topic will be discussed. Students are asked to learn how to use xppaut, a software tool to simulate and study sets of equations that arise in a variety of applications. For free downloading of the source code go to http://www.math.pitt.edu/~bard/xpp/xpp.html. It is required to make simulations each week. During the term it will be possible to attend demonstrations and give reports on readings.

In addition, small groups will be formed to work on specific projects. They should collect data and run simulations.

Special excuse:

In the tenth week I will attend the conference Dynamic Brain Forum in Japan. A long evening class will be held (date to be discussed) during the term, as a compensation.

Exams:

There will be a sixty minutes long midterm and final written examination. Written and oral reports on a group project are a pre-requirement for making the final examination. Active participation in events organized by the Center for Complex System Studies will also be considered in assigning the final grade.

Topics:

1. Dynamical Systems: elementary concepts (state, change of state, variables, parameters, initial values. Deterministic and stochastic models

2. Population Dynamics: single variable systems. Two-variables systems. Difference versus differential equations: preliminary remarks

3-4. Stability analysis, attractors, conservative and limit cycle oscillations. Newton equations and Hamiltion equations. Lotka-Volterra model. Brusselator model. Hopf bifurcation.

5-6. Temporal Chaos and Fractal Structures.

Dissipative vs. conservative chaos.

Logisitc difference equations. Lorenz model, Rõssler attractor.

The characterization of chaotic systems: Lyapunov exponents, fractal dimension, power spectrum.

7. Spatial patterns.

Reaction – diffusion system. Pattern formation in biology.

Turing structures: how the leopards get their spots? Travelling waves.

9. Stochastic models: Markov and non-Markovian process. Diffusion processes, jump processes. The Brownian motion and variations.

Stochastic models of chemical and biological systems

10. Nonlinear Dynamics and Chaos: where we are now?

Useful books:

E. Atlee Jackson: Exploring Nature's Dynamics, John Wiley, 2001 ISBN: 0-471-19146-9

Bard Ermentrout: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM Philadelphia 2002.

Thompson, J. M. T and Stewart HB: Nonlinear dynamics and chaos.

Wiley, 2002

Strogatz SH: Non-linear Dynamics and Chaos: With Applications to

Physics, Biology, Chemistry and Engineering. Westview Press 1994

Epstein Y: Nonlinear Dynamics, Mathematical Biology and Social

Science (Santa Fe Institute Series, Lecture Notes, Vol 4

Gleick J: Chaos: Making a New Science. Penguin 1988

Kaplan D, Glass L: Understanding nonlinear dynamics. Springer-Verlag, 1995.

Ellner SP, Guckenheimer J: Dynamic Models in Biology. Princeton University Press, 2006