PhD projects in Dynamical Systems and Ergodic Theory
Research in this area includes low-dimensional dynamical systems, ergodic theory, iteration of piecewise isometries, mode interactions, perturbed Hamiltonian systems, and thermodynamic formalism. More details, including staff members working in this area, can be found on the dynamical systems and ergodic theory pages.
Below are some examples of PhD projects in dynamical systems and ergodic theory. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Construction of invariant sets for nonautonomous ODEs (Supervisor: Dr Jonathan H.B. Deane)
Invariant sets (subsets of the phase plane in which solutions of an ODE remain for all time) are important because they delineate qualitatively different sorts of behaviour displayed by solutions of an ODE, for instance separating solutions that remain bounded for all time from ones which blow up in finite time. Proving that a given set is invariant generally requires the proof of an inequality on the boundary of the set. Even in the case of a second-order non-autonomous ODE, this essentially planar method yields a subset of the actual invariant set. An investigation into how to optimise this procedure to obtain best possible constructions, and possibly also how to automate the construction using computer algebra or a low-level computer language, is the purpose of this PhD project.
Fast ODE solvers using analytical continuation (Supervisor: Dr Jonathan H.B. Deane)
Fast, accurate methods for solving nonlinear ODEs with polynomial nonlinearity are important in many applications. One suitable method is based on Taylor series, and is also known as the cell-to-cell mapping technique. Roughly speaking, the solution of the ODE is expanded in a power series around a point t=t0, and a suitably modified ratio test applied to the high-order coefficients of the series. The test gives an estimate of the radius of convergence of the series, R (among other things), and so we can compute the solution accurately at, say, t=t0+½ R. This is effectively numerically-implemented analytical continuation. In practice, R can be quite large and so the ODE can be solved in correspondingly large time steps. In some recent work on the varactor equation, an increase in speed by a factor of about 10-50 was obtained using this method compared to, for instance, Runge-Kutta methods. The method appears to be promising but a great deal remains unknown about the assumptions on which it is based, its performance and the circumstances under which it fails.
