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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.

For intuitionistic fuzzy values (IFVs), there are more or less some drawbacks in the existing comparison methods, so it is necessary for us to develop a more proper technique for comparing or ranking IFVs in this paper. To do so, we first formalize an IFV as a fuzzy subset in order to analyze the fuzzy meaning of an IFV, and then according to the basic properties of the fuzzy subset, we determine the dominance relation (order relation) between two IFVs by defining a dominance degree. In order to explain the feasibility of the dominance relations, we validate the monotonicity of intuitionistic fuzzy operational laws, and additionally, we improve and prove the monotonicity of several intuitionistic fuzzy aggregation operators on the basis of the dominance relations. Because it is of importance for some practical problems (e.g., intuitionistic fuzzy multi-attribute decision making) to rank IFVs, we finally develop a method for ranking IFVs by constructing a dominance matrix based on the dominance degrees. A simple example is taken to illustrate the validity of our ranking method.