Analysis of transcendental nonlinear systems

In the recent literature1¿5 certain aspects of a nonlinear theory were introduced which enable the solving of a class of nonlinear systems whose general dynamic equation is given by one like the following: $Z(D) x (t) + F {x, dot {x}, ldots } = g(t) eqno{hbox{(1)}}$ where the linear part Z(D)x(t), the nonlinear part F{x, x ¿}, and the driving function g(t) are required to satisfy certain conditions given in references 1 and 2, and x is the response function. The dotted letters are time derivatives, so that x = d/dt. The linear integro-differential operator has the form $Z(D) = sum^{N_{2}}_{pi=N_{1}} f_{n}D^{n}$ The coefficients fn are constants for each value of n and the bounds N1 and N2 are positive, negative, or either one is zero, depending on Z(D). Time-varying linear terms are easily added in a similar manner and offer no difficulty either of a theoretical or computational nature. The forcing function g(t), although assumed here to be a deterministic process, may be relaxed to a random process with the aid of the transform ensemble theorem, as explained in reference 6.