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Physical Review A

Distribution of subcritical Hopf bifurcations and regular and chaotic attractors in optical bistable systems

Guang Hu, Cun‐Zheng Ning, Hermann Haken

Published Apr 1, 1990• 11 citations
Publisher

Abstract

The distributions, in parameter space, of subcritical Hopf bifurcations and the various attractors of time-dependent motions of optical bistable systems are investigated systematically. The main emphasis is placed on the influence of the variations of the control parameters on the distributions. We have observed the tristability of large-amplitude regular or chaotic motions, periodic pulsations, and the stationary state for the parameters where no instability of the stationary solution exists at all over the complete range of the external field. The problem of how to find the various attractors and how to reveal chaotic motions by adjusting the control parameters is discussed according to the numerical results.

DOI
10.1103/physreva.41.3975

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The distributions, in parameter space, of subcritical Hopf bifurcations and the various attractors of time-dependent motions of optical bistable systems are investigated systematically. The main emphasis is placed on the influence of the variations of the control parameters on the distributions. We have observed the tristability of large-amplitude regular or chaotic motions, periodic pulsations, and the stationary state for the parameters where no instability of the stationary solution exists at all over the complete range of the external field. The problem of how to find the various attractors and how to reveal chaotic motions by adjusting the control parameters is discussed according to the numerical results.
The codimension-two bifurcation set of single-mode optical bistable systems is investigated. The entire set is classified into three classes, a cusp bifurcation set, a degenerate Hopf bifurcation set, and the intersection of a Hopf bifurcation and a saddle-node bifurcation. Based on the specification of degenerate Hopf bifurcation, super- and subcritical Hopf bifurcations can be identified. It is found that, in the subcritical Hopf bifurcation region, an attractor of time-dependent motion may coexist with the stable stationary solution when the cavity is filled by a passive medium. Moreover, the coexistence of three attractors is observed for certain parameter combinations.
We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ``field portrait'' i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.
The theory of dynamic optical bistability and self-pulsations in condensed media is elaborated. The nonlinearity is due to a process in which two excitons are bound into a biexciton by virtue of their Coulomb interaction. A computer experiment is used to study the switching times between the optical bistability branches. The scenario for the transition to the dynamical chaos mode is found. The action of external periodical force upon the system is investigated and the range of amplitude values and frequencies of external harmonic pump were found at which the chaotic regime becomes nonlinear periodic by transformation of strange attractor into a limit cycle. Finally, we discuss the possibility of detecting these phenomena in experiments.
Based on the instability criterion of semi-classical Maxwell -Bloch equations of single-mode optical systems, the global distribution of instability region is depicted in △-θ parameter space (△ detuning of the atoms, θ mistuning of the cavity). In the instability region predicted, various bifurcations and chaos are disclosed. We find that chaos may appear in both upper and lower branches of one bistable curve Moreover, a new time-dependent solution coexisting with the stable stationary solution is found.
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Much attention is being paid of late to the study of cooperative processes in the exciton region of the spectrum. Previously, using generalized Keldysh equations that describe coherent excitons and photons that are weakly nonhomogeneous in space and in time, the theory of optical bistability (OB) was developed in the exciton region of the spectrum, and optical switchover between branches of optical-bistability loop was investigated. The paper presents a theoretical investigation of regular and chaotic self-pulsations in a CuCl crystal with the participation of coherent excitons and biexcitons. We proposed the method of chaotic self-pulsation breakdown in the system of coherent excitons and biexcitons. The method consists in action of external periodical turbulence (Y/spl tilde/=Y+/spl alpha/ sin(/spl omega//spl tau/)) upon a stochastic system. The range of amplitude values and frequencies of an external harmonic pump were found at which the chaotic oscillation regime becomes nonlinear periodic by transformation of strange attractor into a limit cycle.