A New Comparison Method for Stability Theory of Differential Systems with Time-Varying Delays

In this paper, a new comparison method is developed by using increasing and decreasing mechanisms, which are inherent in time-delay systems, to decompose systems. Based on the new method, whose expected performance is compared with the state of the original system, some new conditions are obtained to guarantee that the original system tracks the expected values. The locally exponential convergence rate and the convergence region of the polynomial differential equations with time-varying delays are also investigated. In particular, the comparison method is used to improve the 3/2 stability theorems of differential systems with pure delays. Moreover, the comparison method is applied to identify a threshold, and to consider the disease-free equilibrium points of an HIV endemic model with stages of progress to AIDs and time-varying delay. It is shown that if the threshold is smaller than 1, the equilibrium point of the model is globally, exponentially stable. Another application of the comparison method is to investigate the global, exponential stability of neural networks, and some new theoretical results are obtained. Numerical simulations are presented to verify the theoretical results.

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