Dynamical Systems

Lecturer: Thomas Prellberg Semester 1, 2012/2013


Lecture times
Exercise classes
Office hours
Module description

A dynamical system is any system which evolves over time according to some pre-determined rule. The goal of dynamical systems theory is to understand this evolution. This module develops the theory of dynamical systems systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations and chaos. Much emphasis is placed on applications.

Module information


Lecture Notes Lecture and Exercise Class Plan (tentative!)

227/92.1One-dimensional ODEs
27/9No Exercise class
34/102.2-2.3Fixed Points and Stability
44/102.4Linear Stability Analysis
4/10Exercise class: 2.2.8,9; 2.3.1,3; 2.4.7,8
511/102.5-2.7Existence and Uniqueness; Impossibility of Oscillations; Potentials
611/103.1Saddle-Node Bifurcations
11/10Exercise class: 2.6.2; 2.7.7; 3.1.2
718/103.2Transcritical Bifurcation
818/103.4Pitchfork Bifurcation
18/10Exercise class: 3.1.5; 3.2.4,6; Sample Exam Question 1
925/103.6Imperfect Bifurcations
1025/104.1-4.3Flows on the circle
25/10Exercise class: 3.4.2,16; 3.5.8; 3.6.2
111/114.4Flows on the circle
121/115.1Two-dimensional flows
1/11Exercise class: 4.3.7; 4.4.4; 5.1.2,9
Reading Week: Insect Outbreak (3.7) and Fireflies (4.5)
1315/115.1Linear flows
1415/115.2Classification of Linear Flows
15/11Exercise class: any of 5.2.3-10; 5.1.12
1529/116.2-6.3Phase plane
1629/116.4Lotka-Volterra model
29/11Exercise class: 6.1.3, 6.2.2; 6.3.4,16
176/126.5-6.6Conservative systems; Reversible Systems
186/127.1Limit cycles
6/12Exercise class: 6.5.2; 6.6.3; 7.1.3
1913/127.2Ruling out closed orbits
2013/127.3Poincare-Bendixson Theorem
13/12Exercise class: 7.2.6,10; 7.3.1

Exam Paper

Revision lecture Monday, April 22, 2-3, Queens EB4a

Thomas Prellberg
September 2012