# MATH44041/64041: Applied Dynamical Systems

### Announcements:

• Announcement before the final exam:
• The final exam will be on Monday 22 January, 2018, from 9:45-12:45. I should be in my office most of the time, except the Christmas and New Year period.
• No lecture on Friday 15 December, and revision notes will be used during the lectures on Monday 11 December.
• The coursework can be collected on Wendesnday 13 December noon, with the full solution released by then.
• Please fill the course unit survey, if you have time to do so.
• Assessed Coursework is released now, and is due Wednesday 6th December 1pm.
• 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) Oct 27
Sec 5.3, 5.4 ⚬ Nonlinear systems and stable manifold Exercise Sheet 5 (Solution) Nov 06
Sec 1.3 ⚬ Maps: fixed points and periodic orbits Exercise Sheet 6 (Solution) Nov 10
Chap 2, 5, 6 ♠ Periodic Orbits Part 3 Exercise Sheet 7 (Solution)
Sec 5.5 ⚬ Poincare-Bendixson theorem for periodic orbtis Nov 13
Sec 2.8 ⚬ Floquet theory for periodic coefficients Nov 17
Chapter 8 ♠ Bifurcation and Centre Manifold Part 4 Exercise Sheet 8 (Solution) Nov 20
Sec 5.6 ⚬ Centre manifold and its approximation Nov 24
Sec 8.1 ⚬ Extended centre manifold Exercise Sheet 9 (Solution) Nov 27
Sec 8.1, 8.4, 8.6, 8.8 ⚬ Bifurcations Dec 01
Chap 1, 21 ♠ Maps and their bifurcation Part 5 Exercise Sheet 10 (Solution) Dec 04
Sec 1.3 ⚬ Stability of fixed points and periodic orbits Dec 08
Sec 21.1, 21.2, 21.3 ⚬ Bifurcation of maps
Sec 1.3 ⚬ Logistic map and two-dimensional maps Dec 11
*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 $\dot{v} = \dot{r}\sin\theta{\color{red}+}r\dot{\theta}\cos\theta = \cdots$.
• Some typos in the solutions to question sheet. Exercise Sheet 1: in Question 5, the imaginary unit $i$ is missing in $\lambda =\pm i \omega$; in Question 6(c), solutions with initial condition $x_0<\mu$ tend to $-2$, not to $\mu$. Exercise Sheet 2: at the end of Question 5, $\dot{\theta}=-1$. Exercise Sheet 3: in Question 2, the factor $a$ should be in the numerator, not in the demominator.
• A little more information is given in Exercise Sheet 4, Question 3(b) on the form of the Lyapunov function you are looking for. The corresponding solution is also updated (the original one has some mistakes).

### "Dynamical" dynamical systems

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