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A Very Simple Criterion for Characterizing the Crossing Direction of Time-Delay Systems with Delay-Dependent parameters

In this paper, a very simple criterion has been established for checking the crossing direction of a characteristic root crossing the imaginary axis for a class of time-delay systems with delay-dependent parameters, an important topic both in stability analysis and Hopf bifurcation. The criterion uses two easy-to-obtain functions with clear physical meaning from the critical stability condition and it generalizes a popular result in the literature.