Comments on multiple oscillatory solutions in systems with time-delay feedback

Research output: Contribution to journalArticle

Abstract

A complex Ginzburg-Landau equation subjected to local and global
time-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding the
disappearance of limit cycle solutions, derive analytical criteria for frequency
degeneration, amplitude degeneration, frequency extrema. Furthermore, we
discuss the influence of the phase shift parameter and show analytically that
the stabilization of the steady state and the decay of all oscillations (amplitude
death) cannot happen for global feedback only. Finally, we explain the onset
of traveling wave patterns close to the regime of amplitude death.
Original languageEnglish
Pages (from-to)99-109
Number of pages11
JournalElectronic Journal of Differential Equations
Publication statusPublished - 20 Nov 2015
Event2014 Madrid Conference on Applied Mathematics - Universidad Politécnica de Madrid, Madrid, Spain
Duration: 14 Jun 201415 Jun 2014

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Oscillatory Solution
Time Delay
Feedback Delay
Complex Ginzburg-Landau Equation
Degeneration
Extremum
Phase Shift
Traveling Wave
Limit Cycle
Stabilization
Decay
Oscillation
Term
Influence

Bibliographical note

Proceedings of the 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal. Universidad Politécnica de Madrid, Madrid, Spain, June 14-15, 2014.

This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles in this journal without asking prior permission from the publisher or the author.

Keywords

  • pattern formation
  • reaction-diffusion system

Cite this

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title = "Comments on multiple oscillatory solutions in systems with time-delay feedback",
abstract = "A complex Ginzburg-Landau equation subjected to local and globaltime-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding thedisappearance of limit cycle solutions, derive analytical criteria for frequencydegeneration, amplitude degeneration, frequency extrema. Furthermore, wediscuss the influence of the phase shift parameter and show analytically thatthe stabilization of the steady state and the decay of all oscillations (amplitudedeath) cannot happen for global feedback only. Finally, we explain the onsetof traveling wave patterns close to the regime of amplitude death.",
keywords = "pattern formation, reaction-diffusion system",
author = "Michael Stich",
note = "Proceedings of the 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal. Universidad Polit{\'e}cnica de Madrid, Madrid, Spain, June 14-15, 2014. This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles in this journal without asking prior permission from the publisher or the author.",
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N2 - A complex Ginzburg-Landau equation subjected to local and globaltime-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding thedisappearance of limit cycle solutions, derive analytical criteria for frequencydegeneration, amplitude degeneration, frequency extrema. Furthermore, wediscuss the influence of the phase shift parameter and show analytically thatthe stabilization of the steady state and the decay of all oscillations (amplitudedeath) cannot happen for global feedback only. Finally, we explain the onsetof traveling wave patterns close to the regime of amplitude death.

AB - A complex Ginzburg-Landau equation subjected to local and globaltime-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding thedisappearance of limit cycle solutions, derive analytical criteria for frequencydegeneration, amplitude degeneration, frequency extrema. Furthermore, wediscuss the influence of the phase shift parameter and show analytically thatthe stabilization of the steady state and the decay of all oscillations (amplitudedeath) cannot happen for global feedback only. Finally, we explain the onsetof traveling wave patterns close to the regime of amplitude death.

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JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

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