Introduction to Nonlinear Dynamics and Chaos

Topic: 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.

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.

Books required:

• 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

Supporting literature:

• 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
  
  

Topics:

1. "Understanding" Nature's Dynamics
   
   
2. Dynamic Aspects of Modern Science
   
   
3. Different Types of Population Dynamics
   
   
4. Dynamic Models Based on Differential Equations
   
   
5. Interacting Systems Producing (near) Periodicity
   
   
6. Temporal Chaos and Fractal Structures
   
   
7. Periodic Systems Coupled to Periodic Environments
   
   
8. Applications: exotic chemical systems, pattern formation in biology
   
   
9. Oscillation and chaos in social and socioeconomic systems. Controlling chaos
   
   
10. Nonlinear Dynamics and Chaos: where we are now?