The course begins with a look back to the discovery of chaos in the 1960s and why it took so long to become known. Topics studied are as follows:
· Chaos in Iterated Maps:
Three properties which characterise chaos are considered and are proved to hold for the simple Doubling Map. These results are extended to other maps using the ideas of topological conjugacy.
· Chaos in Differential Equations:
It is shown that systems of first order differential equations must be of dimension 3 or higher in order to find chaos. Poincare maps are introduced as a means of reducing the study of differential equations to the study of iterated maps. Some examples are considered.
· Fractals:
The concept of a non-integer fractal dimension is introduced and applied to some simple fractals. It is shown that basin boundaries can be fractals. Julia sets and the Mandelbrot set are considered briefly. Chaotic attractors of some iterated maps are also shown to be fractals.
· Statistical Properties of Chaos:
The probability distribution function which describes where chaotic orbits spend most time is considered and the Ergodic Theorem is explored which allows time averages to be replaced with integrals weighted by the probability distribution function.
· Lyapunov Exponents:
The property of sensitive dependence on initial conditions can be characterised in terms of Lyapunov exponents which are considered for both iterated maps and differential equations. It is shown that systems of autonomous differential equations always have one Lyapunov exponent which is zero.
· Shadowing:
The shadowing properties of computed orbits of the Doubling Map are considered.
· Routes to Chaos:
The period-doubling route to chaos including a description of Feigenbaum’s universal numbers is considered together with the intermittency route to chaos associated with turning points.
Applications:
Two applications of chaos are considered, namely control of chaos using small parameter perturbations and synchronisation of two coupled chaotic maps.
