Dynamical Systems

Course Aim

• Discriminate the linear from the non-linear behavior of any arbitrary system (physical, biological, even social).
• Know how to approach the nonlinear aspects of the system in a principled manner.
• Recognize chaotic behavior (when present) in a given system and analyze it using the available toolkit.
• Approach any arbitrary (for any quantity) time-series and apply a range of analytical tests to extract quantitative information, including but not limited to, Lyapunov exponents, attractor sets, bifurcations, generalized fractal dimensions, correlations etc.

Course Description

An introduction to chaos theory and related topics in nonlinear dynamics, including the detection and quantification of chaos in experimental data, fractals, and complex systems. Most of the important elementary concepts in nonlinear dynamics are discussed, with emphasis on the physical concepts and useful results rather than mathematical proofs and derivations: there are several other resources for the latter. Courses in Chaos & Nonlinear Dynamics tend to be either purely qualitative or highly mathematical; this course attempts to fill the middle ground by giving the essential equations, but in their simplest possible form.

Course Contents

Week 1 One-dimensional Maps Reading Assignment
Computer Project: Logistic Equation
Week 2 Nonchaotic multi-dimensional flows Assignment
Computer Project: Bifurcation diagrams
Week 3 Dynamical Systems Theory Assignment
Computer Project: Lorenz Attractor
Week 4 Lyapunov Exponents Assignment
Computer Project: Lyapunov exponent
Week 5 Strange Attractors Assignment
Computer Project: Hénon Map
Week 6 Bifurcations Assignment.
Computer Project: Poincaré sections
Week 7 Hamiltonian Chaos Assignment
Computer Project: Chirikov Map
Week 8 Time-series properties Assignment
Computer Project: Autocorrelation function
Week 9 Nonlinear prediction & noise reduction Assignment
Computer Project: Nonlinear prediction
Week 10 Fractals Assignment
Computer Project: State-space reconstruction
Week 11 Calculation of fractal dimension Assignment
Computer Project: Correlation dimension
Week 12 Non-chaotic fractal sets Assignment
Computer Project: Iterated function systems
Time permitting Nonchaotic fractal sets
Spatiotemporal chaos and complexity

Assessment

weekly assignment and a weekly computer project

Prerequisites or Prior Knowledge

Classical mechanics, e.g. B10

Reference Books

Chaos and Nonlinear Dynamics: An introduction to scientists and engineers by R. C. Hilborn (2edn) Oxford, 2000
Chaos in Dynamical Systems by Edward Ott. Cambridge University Press, 2002

Notes

Alternate years course, AY2023

Research Specialties