Analysis of a delay-differential equation in optical bistability

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Abstract
We study the stability boundaries for the steady and periodic solutions of the Ikeda delay-differential equation. In the limit of an infinite delay the differential equation reduces to a nonlinear map. For this map there exist two classes of boundaries: one corresponding to subharmonic cascades and the other either to domains of bistability for steady or periodic solutions or to the emergence of stable periodic solutions from chaos. For a finite delay we show that each of the above boundaries splits into an infinite sequence of secondary boundaries. The crossing of the secondary boundaries associated with the subharmonic sequence results in the progressive squaring of the periodic solution. The crossing of a secondary boundary of the bistability sequence results in the addition of a frequency in the transient oscillatory relaxation.