# MATH44041/64041: Applied Dynamical Systems

Course Unit Specification: MATH44041/MATH64041
Lecturer: Dr. Yanghong Huang (yanghong.huang@manchester.ac.uk)
Office: Alan Turing 1.108
Lectures location:

Tutorial Class: Some of the Wednesday at 10.00 slots (others will be lectures).
Office hour: Wednesday 11.00-12:00 in 1.108 Alan Turing Building, or appointment or drop by the office (check weekly grid).
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
Lecture Notes: Part 1,Part 2,Part 3,Part 4,Part 5 .

Meiss's Book Outline Tutorial exercies and solutions Scanned notes
Chap 1 ♠ Introduction
Chap 1 ♠ Notation and Basic Concepts
Sec 1.2, 4.1, 4.2 ⚬ ODEs: trajectories, phase portrait and flow
Sec 4.1 ⚬ Fixed points, periodic orbits, invariant sets
Sec 3.3 ⚬ Existence and uniqueness
Chap 2,4,5,6 ♠ Linearisation and Equilibria
Sec 2.1, 2.3, 2.5 ⚬ Linear systems
Sec 6.1, 6.2 ⚬ Planar ODEs
Sec 4.5, 4.6 ⚬ Stability and Lyapunov functions
Sec 5.3, 5.4 ⚬ Nonlinear systems and stable manifold
Sec 1.3 ⚬ Maps: fixed points and periodic orbits
Chap 2, 5, 6 ♠ Periodic Orbits
Sec 5.5 ⚬ Poincare-Bendixson theorem for periodic orbtis
Sec 2.8 ⚬ Floquet theory for periodic coefficients
Chapter 8 ♠ Bifurcation and Centre Manifold
Sec 5.6 ⚬ Centre manifold and its approximation
Sec 8.1 ⚬ Extended centre manifold
Sec 8.1, 8.4, 8.6, 8.8 ⚬ Bifurcations
Chap 1, 21 ♠ Maps and their bifurcation
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.

### "Dynamical" dynamical systems

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

### Recommended Reading:

(for library ebooks, you have to use VPN for off-Campus connection). You can also check the official reading list of this module.
• Meiss, James D. Differential dynamical systems. Vol. 14. Siam, 2007. Ebook link
• Strogatz, Steven H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press, 2014. Ebook link
• Wiggins, Stephen. Introduction to applied nonlinear dynamical systems and chaos. Vol. 2. Springer Science & Business Media, 2003.
• Hirsch, Morris W., Stephen Smale, and Robert L. Devaney. Differential equations, dynamical systems, and an introduction to chaos. Academic press, 2012. Ebook link