### Jonathan Deane: PhD Projects

#### Construction of invariant sets for nonautonomous ODEs

At various points in the proofs contained in the papers below, it was required to construct invariant sets (subsets of the phase plane in which solutions of an ODE remain for all time). Such sets 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 results in an approximation to the actual invariant set that is smaller. 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.
• M.V. Bartuccelli, J.H.B. Deane, G. Gentile and L. Marsh
Invariant sets for the varactor equation
(accepted for Proceedings of the Royal Society of London, Series A, August 2005)
• Michele V. Bartuccelli, Jonathan H.B. Deane and Guido Gentile, Globally and locally attractive solutions for quasi-periodically forced systems (in preparation, 2006).

#### Fast ODE solvers using analytical continuation

At several points in my research I have had need of fast, accurate methods for solving nonlinear ODEs with polynomial nonlinearity. A suitable method, based on numerically-implemented analytical continuation, seems to have been first suggested in Y. F. Chang and G. Corliss, Ratio-like and recurrence relation tests for convergence of series, J. Inst. Maths. Appl., 25: 349--359 (1980).

In brief, the 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 + ½R. 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 something like Runge-Kutta. Full details for the method in this case are to be found in The power series method for fast solution of ODEs with polynomial nonlinearities. See also the conference paper Succinct representation of the Poincaré map for periodically driven differential equations.

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. The project would suit a numerate and computer literate graduate with a good degree in mathematics or physics, with proven computing skills.

I'll be very pleased to hear from you if you are interested in either of the above; click here to e-mail me.