Nonlinear Differential Equations and Analytic System Theory

Part I of this paper is a summary of the theory of analytic systems. Analytic systems are nonlinear, time-invariant, deterministic systems, whose input-output relation can be expressed by a Volterra functional power series. Analytic systems include, as special cases, the usual linear, time-invariant, deterministic systems, and also systems whose input-output relation can be expressed by a Taylor series or a power series of convolution integrals. By an application of Volterra’s calculus of functionals, an algebra of analytic systems is developed, which provides a means for the analysis or synthesis of interconnections and combinations of analytic systems. Part II of this paper introduces a solution technique for nonlinear differential equations. This solution technique is an application of the theory summarized in Part I. Specifically, if all the terms in a differential, integral, or integro-differential equation are the results of integration, differentiation, multiplication, and/or addition of signals, then that equation may be synthesized by an analytic system,the decomposition of which results in a simultaneous set of algebraic transform equations. The solutions of these algebraic equations provide a Volterra series solution to the original differential, integral, or integro-differential equation. When a user becomes adept at the solution technique introduced here, then it can be performed by inspection. Several examples are given.