67-717 Nonlinear Dynamics and Chaos
Summer 2007
Instructor: Zoubir Benzaid
Phone: 424 - 7354
Office: Swart 238
Office Hours: MTWR: 8:30-9:00; MTWR: 12:00-1:00 and by
appointment.
Course Content:
This course is an
introduction to the study of dynamical systems. Nonlinear differential
equations and iterative maps arise in the mathematical description of numerous
systems throughout science and engineering, for instance in physics, chemistry,
biology, economics, and elsewhere. Such systems may display complicated and
rich dynamical behavior, and we will develop some linear and nonlinear
mathematical tools for their analysis, and consider models in such fields as
population biology, ecology, and mechanical and electrical oscillations. Our
emphasis throughout will be on the qualitative behavior of the models, in
particular, on the prediction of qualitative change in the nature of the
dynamics as a system parameter varies (bifurcation). The course will stress
conceptual ideas, geometric intuition and concrete examples and applications. The
mathematical treatment will be friendly and informal but still careful.
In this course we will
proceed from simpler to more complicated (and more interesting!) systems. We
begin with one-dimensional flows, their steady states, stability and
bifurcations, and then observe the far more complicated dynamics, including
chaos, that may occur in one-dimensional maps. Phase-plane analysis in two
dimensions reveals the possibility of oscillations and limit cycles, and we
study their bifurcations. As time permits, we will also investigate
higher-dimensional dynamical systems, deterministic chaos and strange
attractors. By the end of the course you
will understand terms such as bifurcations, limit cycles, Lorenz equations,
chaos, iterated maps, period doubling, fractals and strange attractors. Along the way we will consider interesting
and important applications such as mechanical vibrations, lasers, biological
rhythms, superconducting circuits, insect outbreaks, chemical oscillators,
chaotic waterwheels, and even a technique for using chaos to send secret
messages.
Text: S. Strogatz (1994): Nonlinear Dynamics and Chaos by,
1st edition, Westview Press. ISBN: 0-7382-0453-6.
Topics to be covered:
One-Dimensional Flows:
- One dimensional Flows:
Fixed Points and Stability, Population Growth, Linear Stability Analysis,
Existence and Uniqueness of Solutions, Potentials, Solving Equations on
the Computer.
- Bifurcations:
Saddle-Node Bifurcation, Transcritical Bifurcation, Pitchfork Bifurcation,
Overdamped Bead on a Rotating Hoop, Insect Outbreak.
- Flows on the Circle:
Uniform and Nonuniform Oscillators,
Overdamped Pendulum, Fireflies, Superconducting Josephson Junctions.
Two-Dimensional Flows:
- Linear and Nonlinear
Systems : Classification, Phase Plane, Phase Portrait, Fixed Points and
Linearization, Rabbit versus Sheep, Reversible Systems.
- Limit Cycles:
Poincare-Bendixson Theorem, Lienard Systems, Relaxation oscillators
- Bifurcation:
Saddle-Node, Transcritical and Pitchfork Bifurcations, Hopf Bifurcations,
Oscillating Chemical Reactions, Poincare Maps.
Chaos:
- Lorenz Equations:
Chaotic Waterwheel, Chaos on a Strange Attractor, Lorenz Map.
- One-Dimensional Maps:
Fixed Points and Cobwebs, Analysis and Numerics of the Logistic Map,
Periodic windows, Liapunov Exponents.
- Fractals: Countable and
Uncountable Sets, Cantor Set, Dimension of Self-Similar Fractals, Box
Dimension.
- Strange Attractors: The
Simplest examples, Henon Map, Rossler Map, Chemical Chaos.
Recommended Reading:
Popularizations:
- Gleick, J.
(1987). Chaos, the Making of a New Science.London, Heinemann
- Stewart, I. (1989). Does God Play Dice? Cambridge, Blackwell.
- Devaney, R. L.
(1990). Chaos, Fractals, and Dynamics: Computer Experiments
in Mathematics. Menlo Park, Addison-Wesley
- Lorenz, E.,
(1994) The Essence of Chaos, Univ. of Washington Press
- Schroeder, M.
(1991) Fractals, Chaos, Power: Minutes from an infinite paradise
W. H. Freeman New York:
Intermediate
Texts:
- Devaney, R. L.
(1986). An Introduction to Chaotic Dynamical Systems. Menlo Park, Benjamin/Cummings.
- Kaplan, D. and
L. Glass (1995). Understanding Nonlinear Dynamics, Springer-Verlag New
York.
- Jurgens, H.,
H.-O. Peitgen, et al. (1993). Chaos and Fractals: New Frontiers
of Science. New York, Springer Verlag.
- Alligood, K,
Sauer, T et al (1997). Chaos: An Introduction to Dynamical Systems,
Springer Verlag, New
York,
Introductory
Articles:
- May, R. M.
(1986). "When Two and Two Do Not Make Four."Proc. Royal Soc.
B228: 241.
- May, R.M.
(1976) Simple mathematical models with very complicated dynamics. Nature,
Lond. 261, 459-67 (1976).
- Berry,
M. V. (1981). "Regularity and Chaos in Classical
Mechanics,Illustrated by Three Deformations of a Circular Billiard." Eur.
J. Phys. 2: 91-102.
- Crawford, J.
D. (1991). "Introduction to Bifurcation Theory."Reviews of
Modern Physics 63(4): 991-1038.
- Shinbrot, T.,
C. Grebogi, et al. (1992). "Chaos in a Double Pendulum." Am. J.
Phys 60: 491-499.
- David Ruelle.
(1980). "Strange Attractors," The Mathematical Intelligencer 2:
126-37.
Advanced
Texts:
- Arrowsmith, D.
K. and C. M. Place (1990), An Introduction to Dynamical Systems.Cambridge,
Cambridge University Press.
- Guckenheimer,
J. and P. Holmes (1983), Nonlinear Oscillations, Dynamical Systems, and
Bifurcation of Vector Fields, Springer-Verlag New
York.
- Ott, E.
(1993). Chaos in Dynamical Systems. Cambridge University Press,
- Kattok et al.,
Introduction to Modern Dynamical Systems. Cambridge University Press.
Software:
UWO has a full site license
for the Computer Algebra System Maple 10.
This software can be
accessed using any PC or Mac at any computer lab on campus.
Maple 10 is extremely user
friendly and I expect you will be using it quite heavily to
complete your homework and Maple assignments.
I will give a general introduction to
this software the first week of classes and shorter
presentations on specific topics dealing
with the extensive differential equations package
included in Maple.
Website: I will to maintain a website for this course at http://www.uwosh.edu/faculty_staff/benzaid. The site will contain the syllabus, homework
assignments, solutions to tests, solutions to homework problems, Maple 10
worksheets, miscellaneous lecture notes and links to other interesting Nonlinear
Dynamics sites.
Exams and Grading:
Your
grade will be based on two take home exams, 4 homework and Maple assignments, and
2 class presentations.
Exams: 45%
Homework
and Maple Assignments: 40%
Presentations: 15%
Homework:
Homework problems from the book and Maple assignments will be assigned every week; they will be posted on the web. You are encouraged to work together and
discuss problems with each other, but solutions must be worked out and
submitted individually; you are responsible for your own homework. Please work neatly and clearly and explain
your reasoning, and produce neat and clearly labeled graphs when
appropriate.
Presentations:
The class will be divided into groups of 2 students.
Each group will be responsible for giving two short presentations on an
appropriate topic related to dynamical systems.
Some possible presentation topics are:
- The
mathematics and computation of fractals
- Further
exploration of iterative maps: rigorous definitions and detailed proofs of
chaos in one-dimensional maps, Sarkovskii's ordering and "period
three implies chaos", universality; or numerical explorations of
iterative maps, possibly in higher dimensions.
- Biological
oscillations, models of population dynamics and ecology, epidemiology or
immunology, models of HIV/AIDS dynamics
- Chemical or
biochemical oscillations
- Physical
applications, such as models for lasers
- Topics in
classical mechanics
- Celestial
mechanics, planetary motion, ...
- Pattern
formation, such as convection in fluids or biological patterns
- Chaos and
cryptography; controlling chaos
- Philosophical
aspects: the implications of chaos for chance and determinism
- Other topics
related to dynamics, bifurcation, chaos, complexity, ...
Course Objectives:
Upon
successful completion of the course a student is expected to
- Analyze one and two dimensional
flows graphically, analytically and numerically.
- Understand phase
portraits
- Understand and compute
fixed points and equilibrium solutions.
- Classify linear systems
- Study the stability of
fixed points for nonlinear systems using linearization.
- Understand the concepts
of stable and unstable manifolds.
- Analyze some one dimensional and two dimensional models arising in
applications such as population growth, population dynamics such as
predator-prey problems, nonlinear oscillators, linear and nonlinear circuits
and have some appreciation of the
range of physical and biological problems to which this theory is
applicable.
- Understand the
concept of a bifurcation and bifurcation diagrams and be familiar with the
most common types of bifurcations.
- Understand the
concept of limit cycle.
- Understand and
be able to use the Poincare-Bendixson Theorem in simple cases.
- Use Liapunov
functions in the study of stability and closed orbits.
- Understand and
analyze some simple three dimensional flows graphically, analytically and
numerically.
- Understand properties of important chaotic systems
derived from nonlinear differential equations such as the Lorenz and
Rossler systems.
- Gain a descriptive knowledge of strange attractors
- understand how chaotic attractors can be characterized
by the Lyapunov exponent and by various types of dimension
- Gain a working knowledge of discrete chaotic
dynamical systems.
- Understand
periodic points and their role in chaotic systems.
- Be able to
give the characteristic properties of discrete chaotic systems.
- Understand the
mathematics of bifurcation.
- Understand the
mathematics involved in the period doubling route to chaos.
- Able to
describe the genealogy of periodic points.
- Understand the
difference between chaotic behavior and random behavior.
- Use the computer
algebra system Maple to compute solutions of dynamical systems, graph
phase portraits and solutions, simulate nonlinear dynamical systems
behavior, study and compute bifurcations points.