A quantitative study on the stability of quadratic delay difference systems

This paper estimates the size of the stability region around zero for quadratic delay difference systems. When the initial disturbance is in the asymptotic stability region, the solution of the initial value problem of the quadratic delay difference system approaches zero. Examples are given for a three-dimensional system and three one-dimensional equations to demonstrate both stability and instability. Examples 2–4 show that when parameters in the systems do not satisfy the stability conditions, the zero solutions can be unstable. Three evolution features of initial disturbances are shown numerically: decaying to zero, being amplified but bounded, and growing to infinity. Example 3 further shows that the stable zero solution may not be a global attractor. Numerical results confirm the conclusions of the main theorem in this paper and imply that our estimation of the size of the stability region are of reasonable accuracy.