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.

Modeling parametric processes&#8212;A tutorial review

On the 12th of August 2007 Tony Pritchard, professor of mathematics at the University of Warwick, died suddenly at the age of 70 from a heart attack while on holiday with his family on the island of Lanzarote.  Both of the authors have had the privilege of working with him for many years. We have lost both a colleague and a friend. The following notes are dedicated to his memory.

Obituary to A. J. Pritchard

This paper presents an electronic circuit that exhibits parametric resonance, namely a parametric RLC oscillator. The circuit exploits parametric resonance in an analog application, namely, a gain-frequency dependent switch. In such circuit, when the amplitude and frequency of the parametric excitation varies then a transition curves, in the parameter space, are computed by using an algorithm that calculates the symbolic monodromy matrix. Such curves are interpreted to implement an analog application.

On the implementation of a parametric oscillator in analog applications

We show how differential moments can be used to construct Lyapunov functions for general, autonomous and nonautonomous, secondand third-degree ordinary differential equations. In certain instances, one can extend classical results to sequences of Lyapunov functions.

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Lyapunov stability via differential moments

https://ieeexplore.ieee.org/iel5/9/24104/01099043.pdf

P. Parks

Papers

Stability of the damped Mathieu equation