Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals

All-optical clock recovery using multi-section distributed-feedback lasers

Recovering the frequency of incoming data sequences in optical transmission lines is important for signal processing. It has been suggested to use all-optical devices, for instance lasers diodes, for this purpose. Recently, self-pulsations have experimentally been discovered in multi-section distributed-feedback lasers. If a self-pulsating laser is exposed to an external data signal, it is expected that the frequency of the self-pulsation locks to the frequency of the data signal, and clock recovery would be obtained. Mathematically, this problem amounts to investigating frequency locking of periodic solutions, that is self-pulsating laser states, on invariant tori under external forcing which represents the external data sequence. In this article, Melnikov functions are derived for periodic forcing, and results on frequency locking under aperiodic forcing are given. The results are applied to a model describing the multi-section distributed-feedback lasers which have been used in experiments. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(356)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Forced frequency locking in S"1-equivariant differential equations

The aim of this paper is to present a simple analytic strategy for predicting, or engineering, two frequency locking phenomena for S"1-equivariant ordinary differential equations. First we consider the forced frequency locking of a rotating wave solution of the unforced equation with a forcing of 'rotating wave type', and we describe the creation of modulated wave solutions which is connected with this locking phenomenon. And second, we consider the forced frequency locking of amodulated wave solution with a forcing of 'modulated wave type'. Especially, we describe the sets of all control parameters and of all forcings such that frequency locking occures, the dynamic stability and the asymptotic behavior (for the forcing intensity tending to zero) of the locked solutions and the structural stability of all the phenomena. This paper is essentially founded on results from our previous work concerning abstract forced symmetry breaking. The equations considered in the present paper are finite dimensional prototypes of certain infinite dimensional models describing the behavior of continuous wave operated or self-pulsating multisection DFB lasers under continuous or pulsating light injection, respectively. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(257)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Abstract forced symmetry breaking

We consider abstract forced symmetry breaking problems of the type F(x, #lambda#)=y, x#approx#O(x_0), #lambda##approx##lambda#_0, y#approx#0. It is supposed that for all #lambda# the maps F (x, #lambda#) are equivariant with respect to representations of a given compact Lie group, that F(x_0, #lambda#_0)=0 and, hence, that F(x, #lambda#_0)=0 for all elements x of the group orbit O(x_0) of x_0. We look for solutions x which bifurcate from the solution family O(x_0) as #lambda# and y move away from #lambda#_0 and zero, respectively. Especially, we describe the number of different solutions x (for fixed control parameters #lambda# and y), their dynamic stability, their asymptotic behavior for y tending to zero and the structural stability of all these results. Further, generalizations are given to problems of the type F(x, #lambda#)=y(x, #lambda#), x#approx#O(x_0), #lambda##approx##lambda#_0, y(x, #lambda#)#approx#0. This work is a generalization of results of J.K. HALE, P. TABOAS, A. VAN-DERBAUWHEDE and E. DANCER to such extend that the conclusions are applicable to forced frequency locking problems for rotating and modulated wave solutions of certain S"1-equivariant evolution equations which arise in laser modeling. (orig.)Available from TIB Hannover: RR 5549(256)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders

In appliclations, solitary-wave solutions of semilinear elliptic equations #DELTA#u+g(u, #nabla#u)=0 (x,y) element of IR x #OMEGA# in infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems in x has proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov-Schmidt reduction near solitary waves can be applied. As an example existence of shift dynamics near solitary waves is shown if a perturbation #mu#h(x, u, #nabla#u) periodic in x is added. (orig.)Available from TIB Hannover: RR 5549(266)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman