This is an introductory course on dynamical systems mainly concerned with linear systems and one and two dimensional nonlinear systems of differential equations. We will investigate how to determine the qualitative behaviour of the solutions of these differential equations, without having to determine the actual solutions explicitly (most interesting equations do not have neat closed form solutions). This is an applied mathematics course, and the main focus of the course will be on understanding and explaining the behaviour of solutions to differential equations, as opposed to a pure mathematics course where the focus might be more on stating and proving theorems. The honours version of the course will contain more analytical questions.
The material in this course complements the material in math263/math315/math325. In those courses you learn solve the few differential equations for which neat closed form solutions can be written down, but you do not learn a formula for solving all differential equations, because no such formula exists. In this course we will give an introduction to the techniques for investigating the behaviour of the solutions of differential equations in the cases where neat formulae for the solutions cannot be written down (that is most of the time, in the real world). The final chapter of the recommended text for math315/math325 gives an introduction to the material in math326/376, so if you are planning on taking both math326/376 and math263/math315/math325 I would suggest taking math263/math315/math325 first or both first in the same semester, if your schedule allows. However, formally math263/math315/math325 is not a pre- or co-requisite for this course, and it is quite possible to take this course without the other one.
Linear systems of differential equations, linear stability theory. Nonlinear systems: existence and uniqueness, numerical methods, one and two dimensional flows, phase space, limit cycles, Poincare-Bendixson theorem, bifurcations, Hopf bifurcation, the Lorenz equations and chaos.
The course is intended for all students with an interest in nonlinear dynamics, and sufficient mathematical grounding. In the past students have been drawn from across science and engineering as well as mathematics. To that end the prerequisites are
The Textbook for this course is
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