Geometric Completion of Differential Systems
using Numeric-Symbolic Continuation

Greg Reid, Chris Smith, and Jan Verschelde

Abstract:

In previous work symbolic algorithms have been developed which use a finite number of exact differentiations and eliminations to reduce over and under-determined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases the application of exact or numerical integration methods.

However such differential-elimination algorithms, which usually incorporate Groebner bases, have poor complexity and are unsuited for application to systems with approximate coefficients.

A new generation of differential-elimination algorithms is proposed which uses homotopy continuation methods to perform the differential-elimination process on such non-square systems. Examples such as the classic index 3 Pendulum are given to illustrate the new procedure. Our approach uses slicing by random linear subspaces to intersect its jet components in finitely many points. Generation of enough such generic points, enables irreducible jet components of the differential system to be interpolated.

2000 Mathematics Subject Classification : Primary 34M15; Secondary 14Q99, 65H10, 68W30.

keywords : Component of solutions, DAE (Differential-Algebraic Equation), differential elimination, differential index, embedding, geometric completion, interpolation, irreducible component, irreducible decomposition, generic point, homotopy continuation, numerical algebraic geometry, numerical jet geometry, path following, polynomial system.