Stability Exponent and Eigenvalue Abscissas by Way of the Imaginary Axis Eigenvalues

Abstract

In this chapler we present a technique for accurate computation of the stability exponent and other eigenvalue abscissas of a matrix delay equation. Previously the author introduced a finite dimensional linear operator, determined by the delay equation coefficients, having spectrum containing all possible imaginary axis eigenvalues of the delay system. Using the eigenvalues of this operator, and introducing a transmision in the equation’s characteristic function, we can make an accurate numerical determination of the system stability or growth exponent and other eigenvalue abscissas. Arter giving the basic theorems for the method, we give an example in which we go over essentials of implementation. Then we explore the method in some special cases, beginning with second order scalar delay equations and an interesting example of positive delay feedback. We proceed to a rather detailed examination of the effect of the delay parameter in some simple first order delay equations, finding that accurate computation of the system abscissa leads us to some interesting and unconventional conclusions on its behavior with respect to this parameter. We then give an example in which the method is adapted to a cenain distributed delay equation. We conclude with some comments on possible future research.