Front dynamics in a delayed-feedback system with external forcing

A scalar nonlinear delay-differential equation of the kind arising in optically bistable systems is reduced, in some physically relevant limit, to a partial differential equation first, and then to a system of ordinary differential equations. These simplified equations are studied analytically. We describe the continuous deformation of periodic solutions from sine waves to square waves away from the oscillation threshold, and determine their stability. We also study the effect of a weak periodic modulation. It is found that a resonant forcing can stabilize a very large number of oscillation modes taking the form of square waves with missing front pairs.

[1]  Pismen Lm,et al.  Mobility of singularities in the dissipative Ginzburg-Landau equation. , 1990 .

[2]  K. Ikeda,et al.  Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity , 1980 .

[3]  Peter Davis,et al.  Oscillation modes of laser diode pumped hybrid bistable system with large delay and application to dynamical memory , 1992 .

[4]  T. Aida,et al.  Oscillation mode selection using bifurcation of chaotic mode transitions in a nonlinear ring resonator , 1994 .

[5]  K. Ikeda,et al.  High-dimensional chaotic behavior in systems with time-delayed feedback , 1987 .

[6]  Meucci,et al.  Two-dimensional representation of a delayed dynamical system. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[7]  Shui-Nee Chow,et al.  Singularly Perturbed Delay-Differential Equations , 1983 .

[8]  Motion of interacting defects in the Ginzburg-Landau model. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[9]  Paul Mandel,et al.  Theoretical Problems in Cavity Nonlinear Optics , 1997 .

[10]  V N Chizhevsky.  Multistability in dynamical systems induced by weak periodic perturbations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  H. Gibbs,et al.  Observation of chaos in optical bistability (A) , 1981 .

[12]  Walter Ledermann,et al.  Handbook of applicable mathematics , 1980 .

[13]  Otsuka.  Winner-takes-all dynamics and antiphase states in modulated multimode lasers. , 1991, Physical review letters.

[14]  P. Fife.  Long time behavior of solutions of bistable nonlinear diffusion equationsn , 1979 .

[15]  Alternate paths to chaos in optical bistability , 1983 .

[16]  Giacomelli,et al.  Relationship between delayed and spatially extended dynamical systems. , 1996, Physical review letters.

[17]  Jin-yue Gao,et al.  The effect of modulation in a bistable system with delay , 1989 .

[18]  Jack Xin,et al.  Front Propagation in Heterogeneous Media , 2000, SIAM Rev..

[19]  L. Schulman,et al.  Route to chaos in a hybrid bistable system with delay , 1983 .

[20]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[21]  Y. Morita,et al.  Collision and collapse of layers in a 1D scalar reaction-diffusion equation , 2000 .

[22]  H. M. Gibbs,et al.  Bifurcation gap in a hybrid optically bistable system , 1982 .

[23]  K. Ikeda.  Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system , 1979 .

[24]  Ikeda,et al.  Cooperative dynamics and functions in a collective nonlinear optical element system. , 1989, Physical review. A, General physics.

[25]  Foss,et al.  Multistability and delayed recurrent loops. , 1996, Physical review letters.

[26]  Alexander N. Pisarchik,et al.  ATTRACTOR SPLITTING INDUCED BY RESONANT PERTURBATIONS , 1997 .

[27]  Antonio Politi,et al.  Multiple scale analysis of delayed dynamical systems , 1998 .

[28]  K Pakdaman,et al.  Metastability for delayed differential equations. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  J. Milton,et al.  NOISE, MULTISTABILITY, AND DELAYED RECURRENT LOOPS , 1996 .

[30]  H. M. Gibbs,et al.  BIFURCATIONS TO CHAOS IN OPTICAL BISTABILITY , 1982 .

[31]  K. Ikeda,et al.  Successive Higher-Harmonic Bifurcations in Systems with Delayed Feedback , 1982 .

[32]  Hiroshi Itoh,et al.  Observation of Bifurcation to Chaos in an All-Optical Bistable System , 1983 .

[33]  Kapral,et al.  Analysis of a delay-differential equation in optical bistability. , 1986, Physical review. A, General physics.

[34]  John Mallet-Paret,et al.  Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation , 1986 .

[35]  J. McLeod,et al.  The approach of solutions of nonlinear diffusion equations to travelling front solutions , 1977 .