Functional expansions for the response of nonlinear differential systems

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Abstract
This paper is concerned with representing the response of nonlinear differential systems by functional expansions. An abstract theory of variational expansions, similar to that of L. M. Graves (1927), is developed. It leads directly to concrete expressions (multilinear integral operators) for the functionals of the expansions and sets conditions on the differential systems which insure that the expansions give reasonable approximations of the response. Similarly, it is shown that the theory of analytic functions in Banach spaces leads directly to conditions which imply uniform convergence of functional series. The main results on differential systems are summarized in a set of theorems, some of which overlap and extend the recent results of Brockett on Volterra series representations for the response of linear analytic differential systems. Other theorems apply to more general nonlinear differential systems. They provide a rigorous foundation for a large body of previous research on Volterra series expansions. The multilinear integral operators are obtained from systems of differential equations which characterize exactly the variations. These equations are of much lower order than those obtained by the technique of Carleman. A nonlinear feedback system serves as an example of an application of the theory.