P201 (Classical Mechanics I)
42 Lectures + 14 Tutorials
Outcome of the Course
This course teaches the students important concepts and methods in classical nonlinear dynamics, with the aim to build their background for future research work in this area.
- General introduction and motivation: examples of linearity and nonlinearity in physics and the other sciences; modelling systems using iterated maps or differential equations, nonautonomous systems
- General features of dynamical systems : Systems of differential equations with examples; control parameters; fixed points and their stability; phase space; linear stability analysis; numerical methods for nonlinear systems; properties of limit cycles; nonlinear oscillators and their applications; the impossibility of chaos in the phase plane; bifurcations: their classification and physical examples; spatial systems, pattern formation and the Turing mechanism; strange attractors and chaotic behaviour.
- The logistic map: Linear and quadratic maps; graphical analysis of the logistic map; linear stability analysis and the existence of 2-cycles; numerical analysis of the logistic map; chaotic behaviour and the determination of the Lyapunov exponent; universality and the Feigenbaum numbers; other examples of iterated maps.
- Hamiltonian Sytems: Phase space; Constants of motion and integrable Hamiltonians; Nonintegrable systems, the KAM theorem and period-doubling; applications.
- Fractal geometry: dimension of an object, Mandelbrot set, Julia set, iterated function systems.
- Spatio-temporal dynamics: Spatio-temporal chaos.
- Quantum Chaos: Quantum analogies to Chaotic behaviour, Correaltions in wave functions, chaos and Semi-classical approaches to Quantum mechanics.
- Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry and Engineering by S. H. Strogatz
- Chaos and Nonlinear Dynamics by Robert C. Hilborn
- Exploring Chaos: Theory and Experiment by Brian Davies
- An Introduction to Dynamical Systems by K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos
- Chaos in Dynamical Systems by Edward Ott
- Chaos and Integrability in Nonlinear Dynamics: An Introduction by M. Tabor