Modeling parametric processes—A tutorial review

When a system has a periodically time variable component or parameter it is described by a differential equation with periodically varying coefficients, often of second order. Unfortunately many of these equations are intractable thus precluding convenient analysis of the corresponding "parametric" system; in such cases recourse is generally taken to numerical techniques. In this article a procedure is reviewed whereby intractable equations can be replaced, or modeled, by substitutes which are readily treated analytically, thus permitting both the generation of system responses and the analysis of system stability. It is shown that the principal requirement for this technique is to replace the time variation in the equation by another time function which resembles the former in its lower frequency spectrum and, in addition, leads to a solvable equation. The results presented illustrate how quite crude models of this sort can be used to produce very accurate results, for processes such as parametric amplification, in a fraction of the time required by numerical techniques.