"One was the face of Nature; if a face:
Rather a rude and indigested mass:
A lifeless lump, unfashion'd, and unfram'd,
Of jarring seeds; and justly Chaos nam'd.
Ovid , Metamorphoses Book 1
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Math 5370:
MTWThF 11:40 - 1:50 PM
in Bell Hall 130A
Introduction to Chaotic Dynamical Systems
Instructor: M. A. Khamsi.
Theory
of nonlinear dynamical systems has applications to a wide
variety of fields, from mathematics, physics, biology, and
chemistry, to engineering, economics, and medicine. This is one
of its most exciting aspects--that it brings researchers from
many disciplines together with a common language. A dynamical
system consists of an abstract phase space or state space, whose
coordinates describe the dynamical state at any instant; and a
dynamical rule which specifies the immediate future trend of all
state variables, given only the present values of those same
state variables. Dynamical systems are "deterministic" if there
is a unique consequent to every state, and "stochastic" or
"random" if there is more than one consequent chosen from some
probability distribution. A dynamical system can have discrete
or continuous time. The discrete case is defined by a map and
the continuous case is defined by a "flow. Nonlinear dynamical
systems have been shown to exhibit surprising and complex
effects that would never be anticipated by a scientist trained
only in linear techniques. Prominent examples of these include
bifurcation, chaos, and solitons.
Target Audience:Senior Undergraduate Math Majors (as Independent Study), Graduate Students in
Mathematics and Statistics, MAT Students, and others.
Course Objectives: After completion of the course the student
will be able to:
·
Determine the qualitative features of a
one-dimensional dynamical problem.
·
Determine the stability of fixed points of a
one-dimensional dynamical problem.
·
Apply one-dimensional dynamics to model
real-world problems.
·
Interpret the qualitative features of a
dynamical problem in the context of a real-world problem.
·
Determine the bifurcation points of a
one-dimensional dynamical problem.
·
Sketch the bifurcation diagram of a
one-dimensional dynamical problem.
·
Determine the qualitative features of a
two-dimensional dynamical problem.
·
Determine the stability of fixed points of a
two-dimensional dynamical problem.
·
Apply two-dimensional dynamics to model
real-world problems.
·
Determine when a limit cycle is present in a
two-dimensional dynamical problem.
·
Determine the qualitative features of a one-dimensional
map.
·
Determine the stability of fixed points and
periodic points of a one-dimensional map.
·
Determine the dimension of a fractal, including
the Cantor set.
Prerequisite: A good foundation in General Topology, Linear Algebra, and Advanced Analysis.
Some notions on Normed and Metric Space theory will be of invaluable help. However, the class
will be essentially self-contained and all mathematical objects (beyond those studied in elementary
undergraduate math classes) needed, will be defined on the way.
Textbook: Robert L. Devaney,
An Introduction to Chaotic Dynamical Systems, Second edition, Addison-Wesley Publ. Co. (1995)
Homework: You will also be assigned written homework problems.
Tests: There will be exams which will be given at the end of every chapter (which count for 50%). The final on
Monday July 7 at 1-3:45 PM is mandatory and comprehensive (which counts for 50%).
Drop Policy: The class schedule lists Friday, June 27 as the last day to drop with
an automatic "W".
Note: The instructor will NOT assign a "W" for students dropping the course after the deadline.
Links of Interest:
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