Stability analysis of nonlinear dynamic systems with slowly and periodically varying delay

On the basis of the geometric singular perturbation theory and the theory of delayed Hopf bifurcation in slow–fast systems with delay, the stability of nonlinear systems with slowly and periodically varying delay is investigated in this paper. Sufficient conditions ensuring asymptotic stability of those systems are obtained. Especially, though a time-varying delay usually increases complexity in the analysis of system dynamics and it usually deteriorates system stability as well, the study indicates that under certain conditions, the stability of the systems with a time-invariant delay only can be improved by incorporating a slowly and periodically varying part into the constant delay. Two illustrative examples are given to validate the analytical results.

► Sufficient conditions ensuring asymptotic stability are obtained in a compact form.
► A simple condition is given to ensure the improvement of stability.
► Our observation makes it possible to use time-varying delay positively.