Non Linear Dynamical Systems - #10231 Fall 1998

Course description from the student handbook

Teachers
Morten Brøns (MB), Department of Mathematics, Building 303, room 140, tel. 3067.
E-mail: m.brons@mat.dtu.dk
Tomas Bohr (TB), Department of Physics, Building 309, room 262, tel. 3310.
E-mail: tbohr@fys.dtu.dk

Course material
The lectures will be based on Steven H. Strogatz: Nonlinear Dynamics and Chaos, Addison-Wesley 1994, and supplementary notes available on the web.

Requirements
Four homework assignments are given in the semester. Each assignment will be graded on a scale from 0 to 100. The final grade for the course will be based on the grades of the assignments.

• Homework assignment 1
• Homework assignment 2
• Homework assignment 3
• Homework assignment 4

Supplementary Material

• Using MATLAB to solve differential equations numerically (6 pages)
• Slides on the Hopf bifurcation theorem (2 pages)
• A MAPLE session for the Hopf computations for the Selkov equations (Save as selkov.mws)
• The driven, damped pendulum (7 pages)

Exercises
Selected exercises from the textbook will be discussed. At the Tuesdays following an assignment hand-in, we will go through the problems from the assignments.

Course plan

Sept 1

(MB) Introduction to the course. (Chapter 1)

Sept 4

(MB) Basic properties of differential equations. One-dimensional dynamics. (Chapter 2)

Sept 8

(MB) Bifurcations (Chapter 3)
Exercises: 2.2.2, 2.2.7, 2.2.8, 2.2.11, 2.2.12, 2.5.3, 2.6.1

Sept 11

(MB) Bifurcations (Chapter 3)

Sept 15

(MB) Flows on the circle (Chapter 4)
Exercises: 3.1.2, 3.1.5, 3.2.4, 3.2.5, 3.4.2, 3.4.11, 3.6.5

Sept 18

(TB) Two-dimensional systems (Chapter 5-6)

Sept 22

(TB) Two-dimensional systems (Chapter 5-6).
Exercises: 5.1.9, 5.2.1, 5.2.3, 5.2.6, 5.2.13, 6.1.1, 6.1.2, 6.3.10

Sept 25

(TB) Conservative systems and index theory (Chapter 6)

Sept 29

(MB) Poincaré-Bendixson theorem (Chapter 7)
Exercises: 6.4.4, 6.4.5, 6.4.6, 6.5.19, 6.5.8-10

Oct 2

(MB) Introduction to MATLAB and numerical solution of differential equations (Supplementary notes)
Assignment 1 due

Oct 6

(TB) Non-linear oscillators and averaging (Chapter 7)

Oct 9

(TB) The Poincaré-Lindstedt perturbation method (Chapter 7)

Oct 12 - 16

Fall vacation

Oct 20

(MB) Bifurcations in two-dimensional systems (Chapter 8)
Exercises: 7.1.8, 7.2.13, 7.3.5, 7.6.10, 7.6.13

Oct 23

(MB) Bifurcations in two-dimensional systems (Chapter 8)

Oct 27

(MB) Bifurcations in two-dimensional systems (Chapter 8)
Exercises: 8.1.4, 8.1.5, 8.1.7, 8.1.11, 8.1.12, 8.2.1, 8.2.5

Oct 30

(MB) Bifurcations in two-dimensional systems (Chapter 8)
Assignment 2 due

Nov 3

(TB) The damped driven pendulum (Supplementary notes)

Nov 6

(TB) The damped driven pendulum (Supplementary notes)

Nov 10

(MB) The Lorenz equations (Chapter 9)
Exercises: 8.3.1, 8.4.1, 8.4.2, 8.7.1, 8.7.3

Nov 13

(MB) The Lorenz equations (Chapter 9)

Nov 17

(TB) One-dimensional maps (Chapter 10)
Exercises: 9.2.2, 9.2.4, 9.2.6, 9.4.2

Nov 20

(TB) One-dimensional maps (Chapter 10)
Assignment 3 due

Nov 24

(TB) Fractals (Chapter 11)
Exercises: 10.1.1-10.1.8, 10.1.12, 10.1.13, 10.3.5

Nov 27

(TB) Fractals (Chapter 11).

Dec 1

(TB) Strange attractors (Chapter 12).

Dec 4

(TB) Strange attractors (Chapter 12).

Dec 8

(TB) Hamiltonian Chaos (Suppementary notes).
Assignment 4 due