# MATH44041/64041: Applied Dynamical Systems

### Announcements:

• The test on Monday Oct 30th will cover all materials up to and inclding Section 3.3 of the notes (the first three exercise sheets). The format is similar to last year's questions (and the reference solution).

Course Unit Specification: MATH44041/MATH64041
Lecturer: Dr. Yanghong Huang (yanghong.huang@manchester.ac.uk)
Office: Alan Turing 1.108
Office Hour: Thursday 11:00am-12:00pm or appointment or drop by the office (check weekly grid).
Lectures times and locations:

• Monday 13:00 - 15:00 in Ellen Wilkinson A2.6 (NO. 77 on Campus Map). Most of the second hour will be used for additional examples and tutorial problems.
• Friday 10:00 - 11:00 in Beyer Building TH (NO. 49 on Campus Map)

Important Dates: (For a few of you, the Teaching and Learning Office make special arrangements for you to sit your test in a separate location. If this is the case please ignore this general information and go to the location given to you by the Teaching and Learning office.)

• In Class Test (%15 of the final grade), Monday 30th October 2pm-3pm in Ellen Wilkinson A2.6 (no other lectures during this reading week)
• Assessed Coursework (%10) Due Wednesday 6th December 1pm, to the reception desk.
• Final Exam (%75), during the exam week 15–26 January 2018 (will be finalised in early Decmeber).

Meiss's Book Outline Lecture Notes Tutorial exercies and solutions Scanned notes
Chap 1 ♠ Introduction Part 1 Exercise Sheet 1 (Solution)
Chap 1 ♠ Notation and Basic Concepts Sep 29
Sec 1.2, 4.1, 4.2 ⚬ ODEs: trajectories, phase portrait and flow Oct 02
Sec 4.1 ⚬ Fixed points, perioidic orbits, invariant sets Exercise Sheet 2 (Solution) Oct 06
Sec 3.3 ⚬ Existence and uniqueness
Chap 2,4,5,6 ♠ Linearisation and Equilibria Part 2 Exercise Sheet 3 (Solution) Oct 13
Sec 2.1, 2.3, 2.5 ⚬ Linear systems Oct 20
Sec 6.1, 6.2 ⚬ Planar ODEs
Sec 4.5, 4.6 ⚬ Stability and Lyapunov functions Exercise Sheet 4 (Solution)
Sec 5.3, 5.4 ⚬ Nonlinear systems and stable manifold Exercise Sheet 5 (Solution)
Sec 1.3 ⚬ Maps: fixed points and periodic orbits Exercise Sheet 6 (Solution)
Chap 2, 5, 6 ♠ Periodic Orbits Part 3 Exercise Sheet 7 (Solution)
Sec 5.5 ⚬ Poincare-Bendixson theorem for periodic orbtis
Sec 2.8 ⚬ Floquet theory for periodic coefficients
Chapter 8 ♠ Bifurcation and Centre Manifold Part 4 Exercise Sheet 8 (Solution)
Sec 5.6 ⚬ Centre manifold and its approximation
Sec 8.1 ⚬ Extended centre manifold Exercise Sheet 9 (Solution)
Sec 8.1, 8.4, 8.6, 8.8 ⚬ Bifurcations
Chap 1, 21 ♠ Maps and their bifurcation Part 5 Exercise Sheet 10 (Solution)
Sec 1.3 ⚬ Stability of fixed points and periodic orbits
Sec 21.1, 21.2, 21.3 ⚬ Bifurcation of maps
Sec 1.3 ⚬ Logistic map and two-dimensional maps
*Here blue chapters/sections about maps are referring to Wiggins's book Introduction to applied nonlinear dynamical systems and chaos.
Typos in the online lecture notes (corrected and updated in red):
• In Example 2.19, the equality "=" should be inequality "" in the definition of sub-level sets
• In Example 3.3, the characteristic equation should be (s+1)^2+4=0 (the minus sign should be plus). The origin is a stable focus (not a stable node as in the notes).
• In the transformation of coordinates from $(u,v)$ to $(r,\theta)$ right before Equation (3.5), the minus sign in the expression for $\dot{v}$ should be plus sign. That is $\dot{v} = \dot{r}\sin\theta{\color{red}+}r\dot{\theta}\cos\theta = \cdots$.

### "Dynamical" dynamical systems

Below is a list of programs (in matlab)/animations that help you understand the material better.