A boundary value approach to dynamical systems estimation

1 Methods for the solution of the parameter estimation problem in ordinary differential equations fall into two categories (a) the simultaneous method in which the objective function (for example the sum of squares of residuals) is minimised subject to the differential system as imposed constraints, and (b) the embedding method in which auxiliary conditions are adjoined to take up the extra degrees of freedom in the differential system. If this is chaotic then the imposition of initial conditions in the embedding method becomes suspect. However,in [1] it is shown that provided system structure and data format can be married appropriately, a non-trivial condition, then a technique known as synchronization can be used successfully to overcome the disconnect between the initial conditions and the chaotic trajectory. This technique has a close similarity to the more familiar continuation methods. However, typically the form of data resembles multi point boundary conditions suggesting that a boundary value context would be more appropriate. Here we illustrate the inherent instability of the IVP formulation in chaotic dynamical systems, then show how to choose appropriate boundary conditions, and finally demonstrate the greater power and flexibility of the boundary value formulation in the estimation problem in this context. ∗Australian Bureau of Statistics, Belconnen, ACT 2616, Australia †MSI, Australian National University, ACT 0200, Australia, mailto:Mike.Osborne@ anu.edu.au This paper is based on the honours thesis of Stephen Broadfoot. His project was directed by M.R. Osborne