Pattern Formation in Systems with Multiple Delayed Feedbacks

Abstract

Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities. In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behaviour in the case of two delays is shown to "encode" two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the attainment of the long-delay regime is introduced. We also demonstrate this phenomenon for a semiconductor laser with two delayed optical feedbacks.Systems with time delays are common in many fields, ranging from optics (e.g. laser with feedback [1-4]), vehicle systems [5], to neural networks [6], information processing [7], and many others [8]. A finite propagation velocity of the information introduces in such systems a new relevant scale, which is comparable or higher than the intrinsic timescales. It has been shown that the complexity of such systems, e.g. the dimension of attractors, is finite and it grows linearly with time delay [9]; moreover, the spectrum of Lyapunov exponents approaches a continuous limit for long delay [10][11][12]. As a result, in this case essentially high-dimensional phenomena can occur such as spatio-temporal chaos [13], square waves [8], Eckhaus destabilization [14], or coarsening [3]. In the above mentioned situations, the system involves one long delay, which can be interpreted as the size of a onedimensional, spatially extended system [13,15]. This approach has proven to be instrumental in explaining new phenomena in systems with time delays [16,17].In this Letter, we show that many new challenging problems arise when a system is subject to several delayed feedbacks acting on different scales. In contrast to the single delay situation, essentially new phenomena occur, related to higher spatial dimensions involved in the dynamics, such as spirals or defect turbulence. As an illustration, we consider a specific physical system, namely, a model of a semiconductor laser with two optical feedbacks.A simple paradigmatic setup for the multiple delays case is the following systeṁEq.(1) describes a very general situation: the interplay of the oscillatory instability (Hopf bifurcation) and two delayed feedbacks z τi = z(t − τ i ), that we consider acting on different timescales 1 τ 1 τ 2 . The variable z(t) is complex, and the parameters a, b, and c determine the instantaneous, τ 1 -, and τ 2 -feedback rates, respectively. The instantaneous part of the system (without feedback) is known as the normal form for the Hopf bifurcation.The following basic questions arise: what kind of new phenomena can be observed in systems with several delayed feedbacks? Can one relate the dynamics of such systems to spatially extended systems with several spatial dimensions? In the case of positive answer, under which condi...

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