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CO903 Complexity and Chaos in Dynamical Systems

Module Leader: Dr Yulia Timofeeva (Computer Science and Complexity)

Online Course Materials

Taken by students from:

Code Degree Title Year of study core or option credits
P-F3P4 Complexity Science MSc

1

core

12 CATS

P-F3P5 Complexity Science MSc+PhD

1

core

12 CATS

P-F3P6/7 Erasmus Mundus Masters in Complex Systems

1

optional core

6 ECTS

Context: This module forms of the Complexity DTC taught programme.

Module Aims:

The module aims to introduce some of the techniques used in the modern theory of dynamical systems and the concepts of chaos and strange attractors, and to illustrate a range of applications to problems in the physical, biological and engineering sciences.

Dynamical systems are represented by mathematical models that describe different phenomena whose state (or instantaneous description) changes over time. Examples are mechanics in physics, population dynamics in biology and chemical kinetics in chemistry. One basic goal of the mathematical theory of dynamical systems is to determine or characterise the long-term behaviour of the system using methods for analysing differential equations and iterated mappings. Interestingly, simple deterministic nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behaviour, which might seem to be random. This behaviour of systems is known as deterministic chaos.

link to Learning Outcomes

Syllabus:

1. Introduction to Dynamical Models
Differential eq’ns (continuous time) and difference eq’ns (discrete time)
- First-order systems
Existence and uniqueness, Bifurcations, Flow on the circle
- Second-order systems
- Linear systems in two and higher dimensions
- Linearization and stability, Liapunov functions

2. Nonlinear Oscillations
- Limit cycles in two dimensions, Poincaré -Bendixson theorem
- Linear stability of limit cycle, Floquet theory
- Poincaré sections, circle-maps and mode-locking
- Relaxation & Coupled oscillators
- Perturbation methods

3. Introduction to Chaos
- Lorenz equations
- Tests for chaos: Liapunov exponents
- Strange and chaotic attractors, fractal boundaries
- Logistic map

4. Bifurcations and Routes to Chaos
- Local bifurcations: saddle, transcritical, pitchfork; Hopf
- Global bifurcations, homoclinic and heteroclinic connections

- Routes to chaos
- State space reconstruction

Illustrative Bibliography:

1. Strogatz, S.H. (2000). Nonlinear dynamics and chaos. (Westview Press).
A very clear exposition of nonlinear dynamics with many applications in the applied sciences.

2. Jordan, D.W. and Smith P. (1999). Nonlinear ordinary differential equations (Oxford University Press, 3rd ed.).
A standard text in the qualitative theory of differential equations. Particularly good for perturbation theory.

3. Glendinning, P. (1994). Stability, instability and chaos. (Cambridge University Press).
Examines qualitative methods for studying nonlinear differential equations, bifurcation theory and chaos.

4. Alligood, K.T., Sauer, T.D. and Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. (Springer).
A gentle introduction to the science and application of chaos theory with many suggested problems.

5. Ott, E. (2002). Chaos in dynamical systems. (Cambridge University Press).
Provides a broad discussion of chaotic dynamics that emphasises fundamental concepts rather than technical details.

6. Boccara, N. (2004) Modeling complex systems (Springer).

Teaching:

Lectures per week

2 x 2 hours

Classwork sessions per week

2 x 2 hours

Module duration

5 weeks

Total contact hours

40

Private study and group working

60

Assessment Info 2013:

Assessment

Issue date

Deadline

% weighting

Assignment 1

1 Oct
10 Oct (10am)
25

Assignment 2

14 Oct
25 Oct (10am)
25

Oral Examination

28, 29 Oct
50