General introduction about linear versus nonlinear dynamics.
Dynamical systems with one variable.
Bifurcations in one variable systems: saddle-node, cusp, transcritical and imperfect bifurcations.
Bifurcations on the circle, synchronisation.
Linear dynamics with two variables: classification of the fixed points (saddle, node, center, degenerate).
Nonlinear dynamics with two variables: phase space analysis, reversibility, Lyapunov function, theory of the index.
Limit cycles: relaxation oscillations, singular perturbation.
Chaos: Lorentz model and analysis.
One dimensional maps: bifurcations, period doubling and intermittency route to chaos, universality.
Fractals: self-similarity, fractal dimension.
Strange attractors: stretching and folding, baker’s map, Henon map.
Pattern formation.
ALGEMENE COMPETENTIES
The overall objective of this course is to be able to analyze dynamical systems using geometrical methods on the phase space. This includes carrying out linear stability, bifurcation and phase plane analyses. We will first focus on one and two dimensional systems. Chaotic phenomena in physical systems will be described with two classical examples: the Lorentz strange attractor and the logistic map. Solving problems and reading literature related to the course material is also foreseen.
