MATH44041/64041: Applied Dynamical Systems

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.

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

Try the package pplane to get the behaviours of trajectories of ODES on the phase plane. Download the java version (Java Runtime Environment required), as the matlab version does not work for current matlab distribution (MATLAB 2015).
Circular trajectories on the phase portrait for the system \(\dot{x}=xy, \dot{y}=\frac{1-x^2+y^2}{2}\). (Example at the end of Section 1)
Vector field for the system \(\dot{x} = 1-x^2+y,\ \dot{y}=bx\), as the parameter \(b\) changes (Example 3.2).
Dynamics on the phase plane:competing species and mutualistic species (Example 3.4 and 3.5)
Phase portrait for the system of ODEs \( \dot{x} = x, \dot{y}=y^2\) ( Example 3.12).
van der Pol Oscillator on the phase plane (periodic solution that is not on a circle)
Dynamics on the Centre Manifold (Example 5.2)
Bifurcation of system for the system \( \dot{x} =1-y^2,\dot{y}=-x-\mu y - y^2\) (Example 5.7)
Evolution of the Henon Map
Different behaviours of the logistic map \( x_{n+1} = \mu x_n(1-x_n)\) (refinement on a smaller interval of \(\mu\)). (Section 6.3)
Bifurcation of two-dimensional maps for the \(x_{n+1} = \mu y_n + x_n-x_n^2,\ y_{n+1}=x_n\) (Example 6.2)
Intermittency of the logistic map (Section 6.5)
Strange attractors for Lorenz ODEs, Rossler ODEs and various systems proposed by Prott: Case A,Case G, Case I, Case L and Case S.
Julia Set: fractal sets that remain bounded under the iteration \(z\mapsto f(z)=z^2+c\) for some complex number \(c\).