Fall 2020
Time: 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
         a) properties of linear systems (1.3, 1.4)
         b) solutions with homogeneous systems with constant coefficients (1.6, 1.7, 1.8)
         c) critical points and linearized stability (1.9, 2.6)
         d) Lyapunov functions and nonlinear stability (2.9)
    3) Hyperbolic Theory
         a) stable and unstable manifolds of dynamical systems (2.7, 2.10)
         b) linearization of hyperbolic systems (2.8)
         c) center manifold and nonlinear stability (2.11, 2.12)
         d) normal forms (2.3)
    4) Periodic and Homoclinic Orbits
         a) Floquet theory and stability of periodic solutions (3.3)
         b) Poincare maps (3.4, 3.5)
         c) Poincare--Bendixon theory (3.6, 3.7, 3.8)
         d) index theory and separatrix orbits (3.12)
         e) structural stability (4.1)

    Primary Reference:

         a) L. Perko, Differential Equations and Dynamical Systems, Third Edition, Springer, (2008), ISBN 0387951164.

    Supplemental References:

         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

    Homework Assignments:

    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.


    Post Date

    Due Date

    HW 1

    Wednesday, September 30

    Wednesday, October 7

    HW 2

    Wednesday, October 21

    Wednesday, October 28

    HW 3

    Wednesday, November 11

    Wednesday, November 18

    HW 4

    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.

    Academic Integrity:

    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 university.

    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 academic dishonesty:
         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.

    Important Notice:

    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.