INTRODUCTION to NONLINEAR DYNAMICS and CHAOS

INTRODUCTION to NONLINEAR DYNAMICS and CHAOS

2009 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
TA: Mihály Bányai
email: banmi@digitus.itk.ppke.hu

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 state 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 requested 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. 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 ninth week most likey 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 final written examination. Written and oral reports on a group project are a pre-requirement for making the final examination.

Topics:

1. Dynamical Systems: elementary concepts.

Dynamics everywhere. The early scientific studies of dynamics.
Linear and non-linear dynamics: an example.
Derivation and classification of differential equations.
Ordinary and partial differential equations. Dimension and order of DEs.
Autonomous and non-autonomous systems.
Modeling.
State, change of state, phase space (state space), phase plane.
Variables, parameters, initial values, trajectory, fixed points,
Stability.
Deterministic and stochastic models

Background reading
http://www.scholarpedia.org/article/Dynamical_systems
http://www.scholarpedia.org/article/Phase_space

2. Population Dynamics and related areas
Single variable systems. Liner and exponential growth. Explosion. Logistic growth. Fishery management. Bifurcations.
Difference versus differential equations: preliminary remarks
Two-variables systems. Predator-prey models: the Lotka-Volterra model.

3-4. Stability analysis, attractors, conservative and limit cycle oscillations

Conservative oscillators. Eigenvectors and eigenvalues.

Gradient systems. Hamiltonian systems. Liouville theorem. Linear stability analysis.
Bifurcations: saddle-node, transcritical, pitchfork, subcritical pitchfork.
Limit cycles: van der Pol oscillator. Bendixson negative Criterion. Hopf bifurcation (supercritical, subcritcal, degenerate. Nonlinear oscillators. The Brusselator model.

5-6. Temporal Chaos and Fractal Structures.

Dissipative vs. conservative chaos.
Logistic difference equations. Lorenz model, Rõssler attractor.
The characterization of chaotic systems: Lyapunov exponents, fractal dimension, power spectrum.

Oscillation and chaos in macroeconomics.

7. Spatial patterns: models with partial differential equations.
Reaction – diffusion system. Pattern formation in biology.
Turing structures: how the leopards get their spots? Simple solutions. Traveling waves. Modeling aggregation: a cellular slime molds model.

9. Stochastic models: Markov and non-Markovian process. Diffusion processes, jump processes. The Brownian motion and variations.
Stochastic models of chemical, biological and financial systems

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

Useful books:

Strogatz SH: Non-linear Dynamics and Chaos: With Applications to
Physics, Biology, Chemistry and Engineering. Westview Press 1994 is the most popular introductory (graduate course level) textbook. It is highly recommended.

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