Case for Support: Efficient Evans Function Calculations via Neumann and Magnus Expansions

significance and objectives. A practical problem when integrating systems of linear ordinary differential equations, is that if we wish to sample the solution for a different value of an inherent parameter, then we must re-integrate. This is particularly inefficient when we want to accurately sample the solution over a continuous widespread set of parameter values. Though continuity methods resolve this issue locally, they still involve some degree of reintegration. A particular application we have in mind is that of evaluating the Evans function for different values of the spectral parameter, which involves repeated integration of the spectral equations. In this project we propose to extensively study a set of new efficient numerical methods based on Neumann and Magnus expansions, that completely avoid the need for reintegration. These methods were recently proposed by Aparicio, Malham & Oliver (2002); they extend and generalize work by Moan (1998). The basic idea is that we expand either the Neumann or Magnus series solution for such systems as a power series in the parameter(s) in question. The coefficients of the series can be precomputed to any required accuracy. Then we evaluate the series for any of the parameter values we wish to sample. This proposal is intended to crystallize and then implement these ideas. The realization of these methods will revolutionize Evans function calculations and the direct construction of the pure-point spectrum of linear operators. Our objectives are: