Welcome to MATH 741 - APPLIED MATHEMATICS I
11:00-12:30 on Tuesdays and Fridays
Place: virtually via Zoom (access details are available on Avenue)
Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: by appointment
The final exam will be posted no later than 9am on Wednesday, December 16, and solutions will be due electronically by 9am on Friday, December 18.
The class on October 30 is moved to 9:30am.
The first class will take place on Tuesday, September 8.
All lectures will be recorded and will be available for later viewing via links provided on Avenue.
Information about remote access to library computers with Maple can be found here.
Outline of the Course:
The main goal of this course is to offer an introduction to classical
methods of applied mathematics. We will focus on the qualitative
theory of systems of ordinary differential equations (ODEs). Following
a review of standard results concerning existence and uniqueness of
solutions and their continuous dependence on parameters, we will study
linear system, stability theory, invariant manifolds, ending with a
survey of periodic and homoclinic solutions. A second objective of
this course is to introduce students to modern methods of symbolic and
numerical computing useful in quantitative analysis. We will use the
software environment MAPLE to illustrate a number of problems
discussed in the course. In the optimistic variant, the specific
topics to be discussed will include (number is parentheses correspond
to sections in the textbook by L. Perko):
1) Elements of
the ODE Theory a) existence of
solutions (2.1, 2.2), b) uniqueness
of solutions (2.2) c) dependence on
parameters (2.3) d) flows defined by
differential equations (2.5)
2) Linear Systems and Stability
properties of linear systems (1.3, 1.4)
solutions with homogeneous systems with constant coefficients (1.6, 1.7, 1.8)
critical points and linearized stability (1.9, 2.6)
Lyapunov functions and nonlinear stability (2.9)
3) Hyperbolic Theory
stable and unstable manifolds of dynamical systems (2.7, 2.10)
linearization of hyperbolic systems (2.8)
center manifold and nonlinear stability (2.11, 2.12)
normal forms (2.3)
4) Periodic and Homoclinic Orbits
Floquet theory and stability of periodic solutions (3.3)
Poincare maps (3.4, 3.5)
Poincare--Bendixon theory (3.6, 3.7, 3.8)
index theory and separatrix orbits (3.12)
structural stability (4.1)
L. Perko, Differential Equations and Dynamical Systems, Third
Edition, Springer, (2008), ISBN 0387951164.
b) S. Lynch, Dynamical Systems with Applications Using MAPLE,
Second Edition, Birkhauser, (2010). ISBN 978-0-8176-4389-8.
c) R. K. Miller and A. N. Michel, Ordinary Differential Equations,
Academic Press, (1982). ISBN 0-12-497280-2.
In addition to the above references, example MAPLE codes will be made
available to students on the course webpage.
Real analysis and basic differential equations; no programming skills
in MAPLE are required
There will be four homework assignment which may involve some elements
of MAPLE programming. The tentative post and due dates are
indicated in the table below. Submissions are due electronically at
11:59pm on the due date.
Wednesday, September 30
Wednesday, October 7
Wednesday, October 21
Wednesday, October 28
Wednesday, November 11
Wednesday, November 18
Wednesday, November 25
Wednesday, December 2
The final grades will be based on
a) four homework assignments (4 x 15% = 60%),
b) take-home final exam (40%).
The instructor reserves the right to alter your final grade, in which
case, however, the grade may only be increased.
You are expected to exhibit honesty and use ethical behaviour in all
aspects of the learning process. Academic credentials you earn are
rooted in principles of honesty and academic integrity.
Academic dishonesty is to knowingly act or fail to act in a way that
results or could result in unearned academic credit or advantage. This
behaviour can result in serious consequences, e.g., the grade of zero
on an assignment, loss of credit with a notation on the transcript
(notation reads: "Grade of F assigned for
academic dishonesty"), and/or suspension or expulsion from the
It is your responsibility to understand what constitutes academic
dishonesty. For information on the various types of academic
dishonesty please refer to the Academic Integrity
Policy,. The following illustrates only three forms of
1) Plagiarism, e.g., the submission of work that is not one's own or for
which other credit has been obtained.
2) Improper collaboration in group work.
3) Copying or using unauthorized aids in tests and examinations.
The instructor and university reserve the right to modify elements of
the course during the term. The university may change the dates and
deadlines for any or all courses in extreme circumstances. If either
type of modification becomes necessary, reasonable notice and
communication with the students will be given with explanation and the
opportunity to comment on changes. It is the responsibility of the
student to check their McMaster email and course websites weekly
during the term and to note any changes.