J. Comp. Math., Volume 18. The GPL-Stability Of Runge-Kutta Methods Fordelay Differential Systems Biao Yang 1, Lin Qiu 1, Jiao-Xun Kuang 11 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China. Abstract This paper deals with the GPL-stability of the Implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. We focus on the stability behaviour of the Implicit Runge-Kutta(IRK) methods in the solutions of the following test systems with a delay term $$y'(t) = Ly(t) + My(t-\tau), t\ge 0,$$ $$y(t)=\Phi(t), t\le 0,$$ where $L, M$ are $N \times N$ complex matrices, $\tau \gt 0$, $\Phi(t)$ is a given vector function. We shall show that the IRK methods is GPL-stable if and only if it is L-stable, when we use the IRK methods to the test systems above. Key words: Delay differential equation; Implicit Runge-Kutta methods; GPL- stability.