Overview

Description

CALENDAR DESCRIPTION:

Stability and bifurcation in vector fields and discrete maps. Centre manifold theory and applications of normal forms. Introduction to chaos, Lyapunov exponents, and normal hyperbolicity.

COURSE DETAILS:

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This course will be delivered online. You are expected to have access to a reliable internet connection. You will need a computer from which you can download course materials and activities and watch live and/or recorded lectures and participate in live tutorials or workshops.

You will need a camera to take photographs of your work. A phone is acceptable.

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Course Outline

Nonlinear Dynamics and Bifurcation
This course is an introduction to the study of dynamical systems. Nonlinear differential equations and iterative maps arise in the mathematical description of numerous systems throughout science and engineering, for instance in physics, chemistry, biology, economics, and elsewhere. Such systems may display complicated and rich dynamical behaviour, and we will develop some linear and nonlinear mathematical tools for their analysis, and consider models in such fields as population biology, ecology, and mechanical and electrical oscillations. Our emphasis throughout will be on the qualitative behaviour of the models, in particular, on the prediction of qualitative change in the nature of the dynamics as a system parameter varies (bifurcation). In this course we will proceed from simpler to more complicated (and more interesting!) systems. We begin with one-dimensional flows, their steady states, stability and bifurcations, and then observe the far more complicated dynamics, including chaos, that may occur in one-dimensional maps. Phaseplane analysis in two dimensions reveals the possibility of oscillations and limit cycles, and we study their bifurcations. As time permits, we will also investigate higher-dimensional dynamical systems, deterministic chaos and strange attractors.

Outline

Introduction to Bifurcation
Fixed points and stability. Saddle node, trans-critical, super-critical and sub-critical pitchfork bifurcation. Linear-systems, Jordan canonical forms, phase plane analysis.

Limit Cycles and Oscillations
Nonlinear centers, index theory, limit cycles, periodic orbits, Poincaré-Bendixon, global attractors, Floquet multipliers.

Normal Forms and Global Bifurcation
Sub- and super-critical Hopf and homoclinic bifurcations, Normal form, center manifold reduction, Poincaré Maps, Normal hyperbolicity.

Chaos and Maps
Maps, chaos, chaotic maps, Lyapunov exponents, area preserving maps renormalization methods.

COURSE DELIVERY

•  Lecture: synchronous- lectures will be held at fixed times, on-line
•  Midterm(s): synchronous; date: TBA
•  Final exam: synchronous; date: TBA

Note: this is a cross-listed course with MATH 767.

Grading

Materials

REQUIRED READING: