Chaos and Optical Bistability: Bifurcation Structure

  • K. Ikeda


The phase of the output light from a bistable optical cavity containing a nonlinear dielectric medium obeys the following differential equation with time delay:
$$dx\left( t \right)/dt = - x\left( t \right) + \pi \mu f\left( {x\left( {t - {t_R}} \right)} \right).$$
Although the equation of this class is familiar in various areas such as ecology, neurobiology, acoustics and study of electric circuit, the behavior of its solution has not been investigated in detail. In this paper we report the results of our recent numerical study of this equation. It is found that, with increase of parameter y, which measures the intensity of the incident light, or delay tR, the solution of Eq. (1) exhibits transition from a stationary state to periodic and chaotic states. In the course of this transition, there appear successive bifurcations of a novel type, which form, so to call it, a hierarchy of coexisting periodic solutions: As y is increased, the stationary solution becomes unstable at the first bifurcation point and breaks into a number of periodic ones. These periodic solutions form a set of higher-harmonics which can coexist with each other. With further increase of y, each of these solutions further bifurcates into a new set of coexisting periodic solutions. Such a bifurcation takes place successively, causing an accelerative accumulation of coexisting periodic states and making the time evolution of the solution more and more complicated, until a chaotic state sets in. In the chaotic regime, the coexisting periodic states in turn coalesce successively into fewer sets and are finally reduced to a single chaotic state with totally complicated time evolution. This type of behavior, which has never been before in any other nonlinear system, appears generic in the class of Eq. (1).


Nonlinear System Periodic Solution Stationary Solution Incident Light Bifurcation Point 
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Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • K. Ikeda
    • 1
  1. 1.Department of PhysicsKyoto UniversityKyoto 606Japan

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