# MATH44041/64041: Applied Dynamical Systems

Course Unit Specification: MATH44041/MATH64041
Lecturer: Dr. Yanghong Huang (yanghong.huang@manchester.ac.uk)
Office: Alan Turing 1.108
Learning outcomes. On successful completion of this course unit students will:

• solve linear or decoupled system of ODEs or maps, and deduce their long term behaviours of the solutions
• derive qualitative properties of solutions to system of ODEs or maps by semi-group property, conserved quantities or change of variables
• calculate fixed points of system of ODEs, determine their linear types and sketch the phase portrait
• construct Lyapunov function to show the stability of the solutions
• apply the Poincare-Bendixson theorem to show the existence of periodic solution and apply Floquet theory to periodic linear system
• calculate the stable or unstable manifold
• calculate the centre manifold and classify the bifurcation type from the reduced dynamics
• calculate fixed points or periodic orbits of maps, locate and classify bifurcation points
Lecuture Notes: Part 1,Part 2,Part 3,Part 4,Part 5

Meiss's Book Outline Tutorial exercies and solutions Scanned notes
Chap 1 ♠ Introduction 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 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 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 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 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.

### "Dynamical" dynamical systems

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