Bifurcation Analysis of Lasers with Delay

Krauskopf, Bernd

doi:10.1002/0470856211.ch5

Cited by 32 publications

(14 citation statements)

References 65 publications

(147 reference statements)

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“…It arises due to the finite travel time of light between components of the system and may lead to different types of dynamic behaviour including chaos; see, e.g., [47]. The numerical tools introduced in this chapter are very well suited for the study of nonlinear dynamics in lasers with delayed optical feedback; see also [49].…”

Section: A Laser With Filtered Optical Feedbackmentioning

confidence: 99%

“…The coupling of the laser with the filter is controlled by the parameter κ, while τ is the time that the light spends in the external feedback loop. The dynamics of the filter is modelled by (49). The feedback phase C p is the phase difference between the laser and the filter fields, and ∆ is the detuning between the filter center frequency Ω F and the solitary frequency Ω 0 of the laser.…”

Section: A Laser With Filtered Optical Feedbackmentioning

confidence: 99%

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Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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Mathematical modeling with delay differential equations (DDEs) is widely used in various application areas of science and engineering (e.g., in semiconductor lasers with delayed feedback, high-speed machining, communication networks, and control systems) and in the life sciences (e.g., in population dynamics, epidemiology, immunology, and physiology). Delay equations have an infinite-dimensional state space because their solution is unique only when an initial function is specified on a time interval of length equal to the largest delay. Consequently, analytical calculations are more difficult than for ordinary differential equations andnumerical methods are generally the only way to achieve a complete analysis, prediction and control of systems with time delays.Delay differential equations are a special type of functional differential equation (FDE). In FDEs the time evolution of the state variable can depend on the past in an arbitrary way as long as the dependence is a bounded function of the past. However, DDEs impose a constraint on this dependence, namely that the evolution depends only on certain past values of the state at discrete times. (We do not consider here the case of distributed delay.) The delays can be constant or state dependent. The equations can also involve delayed values of the derivative of the state, which leads to equations of neutral type.In this chapter we mainly discuss the simplest case, namely a finite number of constant delays. Specifically, we consider a nonlinear system of DDEs with constant delays τ j 0, j = 1, . . . , m, of the formwhere x(t) ∈ R n , and f : R (m+1)n+p → R n is a nonlinear smooth function depending on a number of (time-independent) parameters η ∈ R p . We assume that the delays are in increasing order and denote the maximal delay by

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Section: A Laser With Filtered Optical Feedbackmentioning

confidence: 99%

Section: A Laser With Filtered Optical Feedbackmentioning

confidence: 99%

Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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“…Feedback mechanisms that involve delays may be present in many forms, not just in climate systems. For example, DDEs have been used to describe the dynamics of epidemics [40], networks of neurons [51], coupled chemical oscillators [11], and lasers [36,41]. Delayed feedback is also successfully and widely used as a form of control [47].…”

Section: Introduction Elmentioning

confidence: 99%

Delayed Feedback Versus Seasonal Forcing: Resonance Phenomena in an El Nin͂o Southern Oscillation Model

Keane¹,

Krauskopf²,

Postlethwaite³

2015

SIAM J. Appl. Dyn. Syst.

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Climate models can take many different forms, from very detailed highly computational models with hundreds of thousands of variables, to more phenomenological models of only a few variables that are designed to investigate fundamental relationships in the climate system. Important ingredients in these models are the periodic forcing by the seasons, as well as global transport phenomena of quantities such as air or ocean temperature and salinity. We consider a phenomenological model for the El Niño Southern Oscillation system, where the delayed effects of oceanic waves are incorporated explicitly into the model. This gives a description by a delay differential equation, which models underlying fundamental processes of the interaction between internal delay-induced oscillations and the external forcing. The combination of delay and forcing in differential equations has also found application in other fields, such as ecology and gene networks. Specifically, we present exemplary stable solutions of the model and illustrate bistability in the form of one-parameter bifurcation diagrams for the seasonal forcing strength parameter. So-called maximum maps are calculated to show regions of bistability in a two-parameter plane for the seasonal forcing strength and oceanic wave delay time. To explain the observed solutions and their multistabilities, we conduct a bifurcation analysis of the model by means of dedicated continuation software. Knowing for which parameter values certain bifurcations take place allows us to explain and expand on some features of the model found in previous publications concerning the existence of unstable solutions, multistability, and chaos. We uncover surprisingly complicated behavior involving the interplay between seasonal forcing and delay-induced dynamics. Resonance tongues are found to be a prominent feature in the bifurcation diagrams and they are responsible for a high degree of multistability in the model. We find bistability within certain resonance tongues as a result of a symmetry property of the governing delay differential equation. We investigate the coexistence of stable tori, how they relate to each other, and bifurcate, which involves bifurcations of invariant tori.

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“…As entry points into nonlinear dynamics of semiconductor lasers with optical injection or feedback see, for example, Refs. [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of semiconductor lasers with filtered optical feedback

2006

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In the filtered optical feedback (FOF) scheme a part of the emission of the laser is spectrally filtered, for example by a Fabry-Pérot filter, and than fed back into the laser. If a semiconductor laser is subject to such delayed FOF qualitative different types of oscillations are possible: the well known relaxation oscillations and, more remarkably, frequency oscillations. We explain how the continuous wave operation of the FOF laser -the external filtered modes -lose their stability and the different types of oscillations arise due to the presence of the filter. This study is restricted to the case of a narrow filter. This means that there are only a few external filtered modes within the width of the filter, so that the influence of the feedback phase can be studied explicitly.

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Bifurcation Analysis of Lasers with Delay

Cited by 32 publications

References 65 publications

Continuation and Bifurcation Analysis of Delay Differential Equations

Continuation and Bifurcation Analysis of Delay Differential Equations

Delayed Feedback Versus Seasonal Forcing: Resonance Phenomena in an El Nin͂o Southern Oscillation Model

Dynamics of semiconductor lasers with filtered optical feedback

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