Stability analysis of systems with periodic coefficients: An approximate approach

A quantitative stability analysis of the Carson-Cambi equation [(1 + e cos t )(d 2 y /d t 2 ) + py = 0] is carried through, using a new, effective approach. The results are compared with a recent perturbation analysis, and show that this should not be used for e0.4. In the present analysis we go up to e = 0.8, and, in fact, with less effort than the perturbation analysis involves. Detailed stability diagrams are presented.

A Quantitative Stability Analysis of the Solutions to the Carson-Cambi Equation

These notes are neither chapters in a textbook, nor short research papers. They are something in between and they include results known for some years as well as just published results. The six notes are written to be fairly independent without extensive mutual reference and the page limit for each note is set to 10. This page limit means that extensive examples are not included.

Sensitivity Analysis for Dynamic Stability Problems

The initial value problem for a circuit containing a small random capacitance is studied. The randomly varying capacitance is taken to be zero-mean, gaussian processes with prescribed covariance function. The method of first-order smoothing is employed to determine the mean solution. Explicit results are shown for a negative exponential correlation function. The covariance of the solution is also obtained in the form of a double integral over the Green's function and the mean of the solution.

https://iopscience.iop.org/article/10.1088/0022-3727/11/11/006/pdf

The random capacitance equation

Numerical Simulation of Stability and Responses of Dynamic Systems under Parametric Excitation

The periodic differential equation (1+e cos t)y + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |e|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–e plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to e3. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±12, ±32, ±52. Three-timing is employed to extend the solution near k = ±12. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.

Barbara Epstein

Papers

Perturbation solutions of the Carson–Cambi equation