Nonlinear systems, global expansions and exponential representations

In this paper we consider a nonlinear differential equation and show how to associate with it a set of natural singularities in multidimensional complex spaces. Two special cases are considered: nilpotent and rational systems. In the nilpotent case it is shown that the singularities are all ´at the origin´ or, more precisely, are singular linear manifolds parallel to the complex axes. This directly generalizes the linear case where the poles are all at the origin with multiplicity the order of nilpotency of the system matrix. The rational case gives rise to an even more remarkable generalization; this is that the solutions of such a system can be written as sums of integrals of exponential over singular varieties. Much of the theory of linear systems of equations has been generalized to nonlinear systems. The ideas have been based largely on Volterra series and global linearization. However, apart from bilinear systems, there has been little success in finding a frequency domain theory for nonlinear systems. The authors derive the formal expression of a system using a Lie series type argument and the two special cases are discussed.