Abstract
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.
Highlights
In 1979, Kensuke Ikeda considered a nonlinear absorbing medium containing two-level atoms placed in a ring cavity and subject to a constant input of light
For large delays (ε → 0), we find from Eq (22) that the first Hopf bifurcation exhibits the frequency ω0 ≡ ω(0) = π, (27)
We find that stable and unstable Hopf bifurcation points are possible and investigate the first stable Hopf bifurcation in detail
Summary
In 1979, Kensuke Ikeda considered a nonlinear absorbing medium containing two-level atoms placed in a ring cavity and subject to a constant input of light (see Fig. 1 ). Efforts to develop an experimental device that is accurately modeled by a simple DDE like Eq (3) immediately followed the early work of Ikeda. We briefly review these studies and emphasize quantitative comparisons between experiments and theory. The dynamical variable is the optical-path difference in a coherent modulator driven electrically by a nonlinear delayed feedback loop [29]. Mackey [32] studied the case of an autoimmune disease that causes periodic crashes in the production of red blood cells He formulated an equation exhibiting a delayed negative feedback which can be rewritten as ελy′ = −y + λ[1 + yp(t − 1)]−1.
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