A Novel Discriminant Approximation of Periodic Differential Equations

TL;DR

This paper introduces a new recursive approximation method for the discriminant of second order periodic differential equations, enabling improved stability analysis through algebraic and integral representations.

Contribution

It presents a novel Walsh function-based approximation of the discriminant that is later expressed as a series of definite integrals, linking to Lyapunov's classical approach.

Findings

The approximation converges to the true discriminant as summation elements increase.

The method simplifies stability boundary calculations for periodic differential equations.

The series form aligns with Lyapunov's discriminant series, providing theoretical validation.

Abstract

A new approximation of the discriminant of a second order periodic differential equation is presented as a recursive summation of the evaluation of its excitation function at different values of time. The new approximation is obtained, at first, by means of Walsh functions and then, by using some algebraic properties the dependence on the Walsh functions is eliminated. This new approximation is then used to calculate the boundaries of stability. We prove that by letting the summation elements number approach to infinite, the discriminant approximation can be rewritten as a summation of definite integrals. Finally we prove that the definite integrals summation is equivalent to the discriminant approximation made by Lyapunov which consists in an alternating series of coefficients defined by multiple definite integrals, that is, a series of the form $A=A_{0}-A_{1}+\ldots +\left( -1\right) ^{n}A_{n}$, where each coefficient $A_{n}$ is defined as an $n-$multiple definite integral.