The Analysis and Dynamical Systems group at Manchester is very active, and covers a wide range of areas from the more pure to specific applications. Several members of the group work on both pure and applied aspects, and there are many opportunities for postgraduate study within the group. We also have close interactions with many of the other researchers in the school. We run a regular seminar, together with a series of collaborative meetings as part of two nationwide networks. The group played a major role in the CICADA project.
At its most basic, a dynamical system consists of two things: a set (the 'state space'), and a rule, or 'dynamic' that specifies how each element evolves to new elements as time advances. On this foundation a wide variety of mathematical structures can be built, according to the particular properties of the state space and its evolution rule. Dynamical systems are often classified roughly into two types: in the first, time is a continuously variable quantity, and the dynamical system usually takes the form of a differential equation; in the second, time advances in discrete steps, and the dynamical system is a difference equation or discrete-time map.
Dynamical systems is closely intertwined with most major areas of pure mathematics, including topology, differential hyperbolic and symplectic geometry, functional analysis, number theory, geometric group theory, fractal geometry and measure theory.
On the other hand, the interdisciplinary nature of the subject means there is a wide variety of research activities in addition to the proving of theorems, including the computer simulation of model systems and the analysis of experimental data. Some researchers focus on one particular activity, others prefer to get involved in several. There are also strong links with other areas: applications to mathematical biology are particularly active at the moment, and interactions with the areas of complex systems and computer science are growing fast.
- Applications to biological and social systems
- Arithmetic dynamics
- Bifurcation theory
- Communications and signal processing
- Coupled dynamical systems
- Ergodic theory and thermodymanic formalism
- Fractal Geometry and geometric measure theory
- Geometry of differential equations and integrable systems
- Hamiltonian formalism and geometric mechanics
- Iterated Function Systems
- Stochastic dynamical systems
- Symmetry in dynamical systems