MTH744U / MTH744P - Dynamical Systems - 2023/24

The complete list of chapters for the module  are:

Chapter 1. Dynamical systems on the line R 

Chapter 2. Bifurcations of dynamical systems on  R

Chapter 3. Dynamical systems on the circle S¹

Chapter 4. Linear dynamical systems on the plane R² 

Chapter 5. Nonlinear dynamical systems on  R²

STROGATZ p142

5.1.2, 5.1.3, 5.1.5,  5.1.7, 5.1.9, 5.1.10 ((a),(c),(e))

You need access to this file  for the phase portraits required for QUIZ 5. 

Strogatz p.182  (Exercises for Chapter 6 of Strogatz)

Ex 6.3.1 

Ex 6.3.15

Ex 6.3.16 (a).  You can try part (b) by using StreamPlot or equivalent. 

Ex 6.4.1

Ex 6.5.2  Write the second order equation as a first order system in 2-variables x and y.  The "homoclinic orbit" is the trajectory that leaves a saddle point  as an unstable manifold (with increasing time) and returns to the fixed point as a stable manifold (see notes for an example). 

Ex. Investigate the following two systems described in polar coordinates, and sketch their  phase portraits:

(a) r˙=r(1−r),θ˙=1$\stackrel{}{}\mathrm{}\stackrel{}{}\mathrm{}$

(b) r˙=r(1−r),θ˙=1−cos(θ)$\stackrel{}{}\mathrm{}\stackrel{}{}\mathrm{}$.

Introduction  to nonlinear systems. 

This video discusses fixed points of nonlinear systems and their linearisations. 

Use of nullclines in sketching phase portraits. 

Conservative and Gradient Systems