IJPAM: Volume 18, No. 2 (2005)

STABILITY OF MULTI-DELAY SYSTEMS OF
DIFFERENTIAL EQUATIONS:
CHARACTERISTIC EQUATIONS AND APPLICATIONS

Terrance J. Quinn$^1$, Sanjay Rai$^2$, Pratik Misra$^3$
$^1$Department of Mathematics
Ohio University Southern
1804 Liberty Avenue, Ironton, Ohio, 45638, USA
e-mail: quinnt@ohio.edu
$^2$Division of Science, Engineering and Mathematics
Montgomery College
Rockville, Maryland, 20850, USA
e-mail: Sanjay.Rai@montgomerycollege.edu
$^3$Department of Chemical Engineering
University of Houston
Houston, Texas, 77204-4004, USA
e-mail: pmisra@mail.uh.edu


Abstract.One approach to stability analysis depends on identifying the signs of the real parts of roots of a characteristic function. Where the characteristic function for an ordinary differential equation typically is a polynomial with real coefficients, the characteristic function for a delay differential equation normally includes exponential terms that involve the delay quantities. These functions, therefore, are called ``exponential polynomials'' [#!1!#], or ``transcendental characteristic functions'' [#!2!#], [#!3!#]. Stability analysis of delay models has, for a main body of work, been on a case by case basis, using a technique for treating exponential polynomials that goes back to [#!4!#]. What is lacking, however, are results that would both (a) apply to the general multi-delay case; and (b) be useful for the clinical scientist. Toward the possibility of a practical general theory, we use a linear algebraic framework. In this context, the traditional technique [#!4!#] is related to a factorization of the general delay equation that is point-wise linear and offers insight into the general structure of the zero set. Necessary and sufficient conditions for roots are obtained that allow for a unified approach to multi-delay equations. Certain classical formulas for the one-delay equation are extended to the multi-delay equation. Results are illustrated with examples and applications from the literature. A general result on Hopf bifurcation in the multi-delay system is given. The paper concludes with an indication of further related lines of enquiry that emerge from the context.

Received: September 30, 2004

AMS Subject Classification: 34K20, 92D25

Key Words and Phrases: delay-differential equation, multi-delay, stability, transcendental characteristic function, exponential polynomial, Hopf bifurcation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 18
Issue: 2