Iterations of maps and differential equations; phase portraits,
flows; fixed points, periodic solutions and homoclinic orbits;
stability, attraction, repulsion; Poincaré maps, transition to
chaos. Applications: logistic maps, interacting populations, reaction
kinetics, forced Van der Pol, damped Duffing, and Lorenz equations.
Dynamical systems is a branch of mathematics which
studies processes which change. Such processes occur in all
branches of science, and examples of dynamical systems
include the motion of stars, the change of stock markets,
the variation of the world's weather, the rise and fall of
populations, the reaction of chemicals and the motion of a
simple pendulum. The central goal of the study of
dynamical systems is to predict where the system under
consideration is heading and where it will ultimately go
(for example, one would like to know when the market goes up
or down, whether it will be rainy or sunny, or if
interacting populations become extinct).
The study of dynamical systems originated from differential or
difference equations arising from many applied fields, and has been
one of the most fruitful fields of mathematical research in
this century. Many profound results have been uncovered and applied
to other branches of mathematics as well as to physics, chemistry,
biology and economics.
In this course, we will use maps in low-dimensional Euclidean spaces
to demonstrate the main contents,
methods and applications of dynamical systems. The course will
be structured so that students are gradually introduced to more
and more sophisticated ideas from analysis as the course proceeds.
It starts with only a few elementary notions that can be
explained using graphical methods or simple differential calculus.
Topics (in addition to those listed above from the main
Calendar) also include bifurcations and periodic doubling.
The text is Edward R. Scheinerman, Invitation to Dynamical
Systems, Prentice Hall, 1996.
The final grade may be based on assignments,
one test and a final exam.
Students who have not passed MATH 3210 must obtain permission of the
instructor to enrol.
Prerequisite:AS/SC/MATH 2021.03 or
AS/SC/AK/MATH 2221.03 or
AS/SC/MATH 1025.03; AS/SC/AK/MATH 2270.03.
Exclusion:AS/SC/AK/MATH 3270 3.0
Coordinator: J. Wu