In this paper, the necessary and sufficient conditions of a system with delayed feedback have been derived. In order to study the bifurcation and chaotic behavior, an asymptotic expansion of the solution in terms of the inverse delay time (TR-1) has been obtained. In long delay time limit, the first order effect in TR-1 is to prolong period of the motion. More interesting phenomena are about the higher order terms. The TR-2 terms lead to some hysteresis processes (some bistability) at each bifurcation point. And the third approximation in TR-1 influences the period-doubling bifurcation, i.e. the ratio of the periods below and at the transition point is not exact two. It is also shown that, some instable windows appear at merging points of the chaotic bands and lead to some mode-locked phenomena.
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