Functional Analysis of Systems Characterized by Nonlinear Differential Equations

摘要

An analysis, by functional calculus, of a class of nonlinear systems is presented. The class of nonlinear systems that are analyzed includes all those analytic systems that are characterized by nonlinear differential equations. Applications of this analysis are shown for several actual nonlinear physical systems that are analytic. The precise definition of an analytic system is given. Loosely speaking, an analytic system is any system with these three properties: (i) It is deterministic. (For a given input signal, the system can have one and only one corresponding output signal.) (ii) It is time-invariant. (iii) It is 'smooth.' (The system cannot introduce any abrupt or switchlike changes into its output. All such changes in the output must be caused by the input rather than the system.) Given a nonlinear differential equation, the conditions are shown under which it characterizes an analytic system. Given an analytic system characterized by a nonlinear differential equation, it is shown how that system can be analyzed by an application of functional calculus. Specifically, an inspection technique is developed whereby a Volterra functional power series is obtained for that system's input-output transfer relationship. (Author)