Nonlinear dynamics, chaos and complex systems
PHYS-460
Program
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Page content
Here you can find the program of the course and the references for each lecture. References appear a few days after the lecture. Unless specified, the chapters refer to S.H. Strogatz, Nonlinear dynamics and chaos, with application to Physics, Biology, Chemistry, and Engineering, Second Edition, Westwiew Press.
| Date | Subject | Reference | Additional material |
|---|---|---|---|
| 18 February |
1. Introduction 1.1 Practical information 1.2 Course structure and Lorenz model 1.3 History 2. Introduction to nonlinear dynamical systems 2.1 Key concepts 2.1.1 Definitions |
Ch. 1 |
Lorenz1963.pdf Saltzmann1962.pdf Théorie du mouvement de la lune, Tome 1, Charles Eugène Delaunay |
| 25 February |
2.1.2 Existence and uniqueness 2.1.3 Continuous dependence of solutions 2.1.4 Equilibria 2.2 First order (or one-dimensional systems) 2.2.1 A simple example and geometrical way of thinking |
2.0, 2.1, 2.2 |
M. W. Hirsh |
| 4 March |
2.2.1 A simple example and geometrical way of thinking (continued) 2.2.2 Transcritical bifurcations 2.2.3 Saddle-node bifurcations 2.2.4 Pitchfork bifurcations 2.2.5 Catastrophes |
2.3, 2.4, 2.6 3.0, 3.1, 3.2, 3.4, 3.6 |
Catastrophes |
18 March |
2.3 Second order (or two-dimensional) systems 2.3.1 Phase portraits 2.3.2 Linear systems |
5.0, 5.1, 5.2, 6.1, 6.2, 6.3 |
|
| 25 March |
2.3.3 Examples of fixed point analysis and phase portraits in non-linear systems
2.3.4 Fixed point stability |
6.4 |
|
| 1 April |
2.3.5 Special types of systems 2.3.6 Limit cycles 2.3.7 Saddle-node, transcritical, and pitchfork bifurcations |
6.5, 7.0, 7.2, 7.3, 8.1, 8.2 |
|
| 8 April |
2.3.8 Hopf bifurcations 2.3.9 An example, chlorine dioxide-iodine-malonic acid reaction 2.3.10 Global bifurcations of cycles 3. Chaos 3.1 The Lorenz system 3.1.1 Elementary properties of Lorenz system |
8.3. 8.4, 9.0, 9.2 |
CMA_reaction.pdf homoclinic_orbits.m infinite_period_bifurcation.m |
| 15 April |
3.1.2 Chaos 3.1.3 Lorenz map | 9.3, 9.4, 9.5 |
|
| 29 April |
3.1.4 Exploring the parameter space 3.2 One-dimensional maps 3.2.1 Fixed points, stability, and cobweb diagrams 3.2.2 Periodic solutions |
10, 10.1, 10.2 |
Lorentz.pdf |
| 6 May |
3.2.3 Bifurcations 3.2.4 Long term dynamics |
10.3, 10.4, 10.5, 10.6 |
1DMaps.pdf 2cycle_example_fig1.pdf 2cycle_example_fig2.pdf |
| 13 May |
3.2.5 The logistic map 3.2.6 Universality |
10.7, 11.0, 11.1 |
Renormalization |
| 20 May |
3.2.7 Renormalisation 3.3 Fractals and strange attractors 3.3.1 Reminder: countable and uncountable sets 3.3.2 A simple example of a fractal: the Cantor set |
||
| 27 May |
3.3.3 Dimension 3.3.4 Strange Attractors 4. Complex systems |
11.2, 11.3, 11.4, 11.5, 12.0, 12.1, 12.2 |
correlation_dimension_vs_r.m correlation_dimension_vs_r_matlab_intrinsic.m |
| Extra |
4. Complex systems 4.1 Introduction 4.1.1 Definitions 4.1.2 Approaches 4.1.3 Features 4.2 Epidemics 4.2.1 Classical models of epidemic spreading 4.2.2. Networks 4.2.3 Epidemic models on networks 4.2.4 Dynamics on networks 4.2.5 Example of dynamics on networks: the SI model |