Ikeda Hopf bifurcation revisited

A large variety of passive optical systems subject to a delayed feedback have appeared in the literature and are described mathematically by the same class of scalar delay differential equations (DDEs). These equations include Ikeda DDE and their solutions are determined in terms of a control parameter distinct from the delay. We concentrate on the first Hopf bifurcation generated by a fixed delay and determine a general expression for its direction of bifurcation. We then examine our result in the two limits of small and large delays. For small delays, we show that a Hopf bifurcation to nearly sinusoidal oscillations is possible provided that the feedback rate is sufficiently high (bifurcation from infinity). For large delays, we complement the early work by Chow et al. [Proc. Roy. Soc. Edinburgh A 120 (1992) 223–229] and Hale and Huang [J. Diff. Equ. 114 (1994) 1–23] by comparing analytical and numerical bifurcation diagrams as the oscillations progressively change from sine to square-wave.