Math 4FT3/6G03 Dynamical Systems WINTER 2012 Course Outline

Course Homepage: http://www.math.mcmaster.ca/wolkowic/Courses/M4FT3_6G03_2012/Math4FT3_6G03.html

Prerequisite: Registration as a graduate student in the Department of Mathematics and Statistics, or MATH 3F03, or permission of the instructor. M3A03 is recommended.

Brief Course Description: This is an introductory course in Delay Differential Equations with Applications. Delays must be taken into consideration in mathematical models, since they occur naturally in applications, ranging from population biology, epidemiology, economics, and neural networks, to the control of mechanical systems. We will discuss both Distributed Delay (Integro-Differential Equations) and Discrete Delay (Functional Differential Equations), with emphasis on the latter. Ideal prerequisites for this course include a course in the qualitative theory of ordinary differential equations (e.g. M3F03 or M741), calculus of several variables, and some basic analysis and linear algebra. Students will study the basic theory (including existence, uniqueness continuous dependence), study the properties of solutions of linear and nonlinear equations (including exponential solutions, growth estimates, Laplace transforms, stability analysis, and bifurcation analysis). They will thus learn the key tools necessary to understand the applications literature involving delay differential equations and to construct and analyze mathematical models involving (systems of) delay differential equations, both analytically and numerically.

Format: A combination of formal lectures by the instructor, lecturing by the students, and projects. There will also be assignments and computer lab work.

Projects: Individual projects adapted to meet the individual interests of the students comprising of both oral and written components.

Textbook: Hal Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer Texts in Applied Mathematics, Vol 57 (2011)

References:

• R. Bellman and K. Cooke, Differential-Difference Equations, Academic Press.
• J.M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Springer-Verlag.
• R.D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag.
• L.E. El'sgol'ts and S.B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arugments, Academic Press.
• J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag.
• Yang Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering Vol 191, Academic Press.
• Norman MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, Springer-Verlag.
• Selected articles from the literature.

Related Software:

• XPPAUT: For solving many different types of differential equations including Delay Differential Equation is downloadable free on the web at: http://www.math.pitt.edu/~bard/xpp/xpp.html and works for all operating systems. Download the appropriate version for your system.
  • If your operating system is some flavour of Windows, you will need an X-server running. Xming http://sourceforge.net/projects/xming/ is free and easily installed. If your operating system is Unix, Linux, or MacOS you will not need this.
• Matlab
  • Tutorial for solving Delay Differential Equations: http://www.runet.edu/thompson/webddes
  • DDE-Biftool: A Matlab package for the bifurcation analysis of delay differential equations. http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml

Tentative Grading Scheme:

Component	Weight
Assignments (including Computer Labs)	30%
Presentations	20%
Course Participation	25%
Project	25%

Absences and missed work: If you are absent from the university for a minor medical reason, lasting fewer than 5 days, you may report your absence, once per term, without documentation, using the McMaster Student Absence Form. Absences for a longer duration or for other reasons must be reported to your Faculty/Program office, with documentation, and relief from term work may not necessarily be granted. When using the MSAF, report your absence to course_email@mcmaster.ca. You must then contact the instructor immediately (normally within 2 working days) by email (see above for contact information) to learn what relief may be granted for the work you have missed, and relevant details such as revised deadlines, or time and location of a make-up exam. Please note that the MSAF may not be used for term work worth 30% or more, nor can it be used for the final examination.

If you must miss a lecture, it is your responsibility to find out what was covered. The best way to do this is to borrow a classmate’s notes, read them over, and then ask your instructor if there is something that you do not understand.

Late Work: All assignments are due in class on the specified date, prior to class, unless otherwise stated. I reserve the right to penalize late work by 10% per day.

Academic Dishonesty: Academic dishonesty consists of misrepresentation by deception or by other fraudulent means and 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 kinds of a academic dishonesty please refer to the Academic Integrity Policy, specifically Appendix 3, located at http://www.mcmaster.ca/univsec/policy/AcademicIntegrity.pdf

In particular, in this course: