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The scheme of bifurcations in systems described by a certain class of delay-differential equations is investigated in detail. It is shown that higher-harmonic oscillating states appear successively in the course of transition to developed chaos. First-order transitions between these states account for the frequency-locked anomaly recently observed by Hopf et al. in a hybrid optical bistable device. PACS numbers: 05.40.+J, 42.65.BpAmong dynamical systems capable of displaying chaotic behavior, systems with delayed feedback are of interest since their time evolution is determined by a concurrence of a discrete step which tends to induce chaos and a continuous step which tends to smear it. A physical example of such systems is an optical cavity filled with a nonlinear dielectric medium and irradiated with a laser light of constant intensity. Several authors investigated the phenomenon that the transmitted light from such a cavity, a part of which is fed back to the medium by mirrors, displays bistable behavior 1 * 2 and chaotic behavior 3 " 5 under suitable conditions. Other examples are found in neurobiology 6 and ecology. 7 Recently, Hopf and others observed, in a system equivalent to a nonlinear optical cavity, a novel phenomenon they named frequency-locked anomaly 8 : In the course of transition from periodic states to developed chaos, there appear chaos with traces of periodic structure and its higher harmonics successively, being accompanied by discontinuous transitions between them. In this Letter we investigate in detail the scheme of bifurcations on a delay-differential equation which models the dynamics of the system and point out that the above phenomenon is of universal character in a certain class of systems with delayed nonlinear feedback.The delay-differential equation we investigate is of the formwhere n is the bifurcation parameter. The second term on the right side represents nonlinear feedback with delay t R . For an optical cavity, f(x; ii) in Eq. (1) takes the special form 3 f{x;where \i is proportional to the power of the incident light, and B (<1) represents the dissipation of the electromagnetic field in the cavity. The delay t R is the time required for light to make a round trip in the cavity.

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