Front dynamics in a delayed-feedback system with external forcing

Nizette, Michel

doi:10.1016/s0167-2789(03)00175-1

Cited by 24 publications

(32 citation statements)

References 37 publications

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“…The decomposition of the perturbation problem into a series of iterative maps is typical of delay differential equations with large delays [38][39][40][41]. Rewriting the above equation at time s − π and using the fact that x m (s − 2π ) = x m (s), we obtain Expressions (4.6) and (4.7) can be used to implement an automatic resolution of the perturbation analysis up to any order using a symbolic software such as MATHEMATICA.…”

Section: Singular Hopf Bifurcation (Kmentioning

confidence: 99%

Singular Hopf bifurcation in a differential equation with large state-dependent delay

Kozyreff

Erneux

2014

Proc. R. Soc. A.

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We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol's equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

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Section: Singular Hopf Bifurcation (Kmentioning

confidence: 99%

Singular Hopf bifurcation in a differential equation with large state-dependent delay

Kozyreff

Erneux

2014

Proc. R. Soc. A.

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“…For time-delay devices with passive nonlinearity and DC-coupled feedback, i.e. devices that can be model through scalar DDEs of Ikeda-type, many of these questions have been addressed both experimentally [Derstine, 1983;Liu, 1991;Goedgebuer, 1998] and theoretically [Ikeda, 1979;Nardone, 1986;Ikeda, 1987;Hale, 1996;Giannakopoulos, 1999;Nizette, 2004;Erneux, 2004], starting in 1979 with the pioneering work of Ikeda [Ikeda, 1979]. On the other hand, for DDEs of the form of Eq.…”

Section: Theorymentioning

confidence: 99%

Time-Delay Systems with Band-Limited Feedback

Illing

Blakely²,

Gauthier

2005

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Fast nonlinear devices with time-delayed feedback, developed for applications such as communications and ranging, typically include components that are ACcoupled, i.e. components that block zero frequencies.As an example of such a system, we describe a new opto-electronic device with band-limited feedback that uses a Mach-Zehnder interferometer as passive nonlinearity and a semiconductor laser as a current-to-opticalfrequency converter. Our implementation of the device produces oscillations in the frequency range of tens to hundreds of MHz. We observe periodic oscillations created through a Hopf bifurcation as well as quasiperiodic and high dimensional chaotic oscillations. Motivated by the experimental results, we investigate the steady-state solution and it's bifurcations in time-delay systems with band-limited feedback and arbitrary nonlinearity. We show that the steady state loses stability, generically, through a Hopf bifurcation, which can be either supercritical or subcritical. As a result of this investigation, we find that band-limited feedback introduces practical advantages, such as the ability to control the characteristic time-scale of the dynamics, and that it introduces differences to Ikeda-type systems already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental "period-2" mode found in Ikeda-type systems.

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“…A fundamental question is whether they may lead to stable oscillations past critical amplitudes. Nizette [27] showed that this is not likely to appear for the scalar first order DDEs treated by Ikeda and coworkers. More precisely, he examined the double limit of small amplitude solutions and large delays and showed that all primary Hopf branches of solutions except the first one are unstable.…”

Section: Introductionmentioning

confidence: 97%

Two distinct bifurcation routes for delayed optoelectronic oscillators

2017

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We investigate the coexistence of low-and high-frequency oscillations in a delayed optoelectronic oscillator. We identify two nearby Hopf bifurcation points exhibiting low and high frequencies and demonstrate analytically how they lead to stable solutions. We then show numerically that these two branches of solutions undergo higher order instabilities as the feedback rate is increased but remain separated in the bifurcation diagram. The two bifurcation routes can be followed independently by either progressively increasing or decreasing the bifurcation parameter.

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Front dynamics in a delayed-feedback system with external forcing

Cited by 24 publications

References 37 publications

Singular Hopf bifurcation in a differential equation with large state-dependent delay

Singular Hopf bifurcation in a differential equation with large state-dependent delay

Time-Delay Systems with Band-Limited Feedback

Two distinct bifurcation routes for delayed optoelectronic oscillators

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