Dynamical Systems

Lecturer: Thomas Prellberg Semester 1, 2011/2012


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!)

229/92.1One-dimensional ODEs
29/9No Exercise class
36/102.2-2.3Fixed Points and Stability
46/102.4-2.5Linear Stability Analysis; Existence and Uniqueness
6/10Exercise class: 2.2.8,9; 2.3.1-4; 2.4.7,8
513/102.6-2.7Impossibility of Oscillations; Potentials
613/103.1Saddle-Node Bifurcations
13/10Exercise class: 2.7.7; 3.1.2,3,5
720/103.2Transcritical Bifurcation
820/103.4Pitchfork Bifurcation
20/10Exercise class: 3.2.4,6; 3.4.2,16
927/103.6Imperfect Bifurcations
1027/104.1-4.3Flows on the circle
27/10Exercise class: 3.5.8; 3.6.2; 4.3.7
113/114.4Flows on the circle
123/115.1Two-dimensional linear flows
3/11Exercise class: 4.4.4; 5.1.2,9
Reading Week: Insect Outbreak (3.7) and Fireflies (4.5)
1317/115.2Linear flows
1417/116.1Phase plane
17/11Exercise class: 5.1.12; 6.1.3,8
1524/116.2-6.3Phase plane (continued)
1624/116.4Lotka-Volterra model
24/11Exercise class: 6.2.2; 6.3.4,16
171/126.5-6.6Conservative systems; Reversible Systems
181/127.1Limit cycles
1/12Exercise class: 6.5.2; 6.6.3; 7.1.3
198/127.2Ruling out closed orbits
208/127.3Poincare-Bendixson Theorem
8/12Exercise class: 7.2.6,10; 7.3.1
2115/129The Lorenz equations and chaos
2215/129Chaos and strange attractors

Exam Paper

Revision lecture April 27, 11-12, Queens EB4. Exam May 24, 10-1.

Thomas Prellberg
September 2011