### Abstract

The supercritical Hopf bifurcation is one of the simplest ways in which a stationary state of a nonlinear system can undergo a transition to stable self-sustained oscillations. At the bifurcation point, a small-amplitude limit cycle is born, which already at onset displays a finite frequency. If we consider a reaction-diffusion system that undergoes a supercritical Hopf bifurcation, its dynamics is described by the complex Ginzburg-Landau equation (CGLE). Here, we study such a system in the parameter regime where the CGLE shows spatio-temporal chaos. We review a type of time-delay feedback methods which is suitable to suppress chaos and replace it by other spatio-temporal solutions such as uniform oscillations, plane waves, standing waves, and the stationary state.

Original language | English |
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Title of host publication | SEMA SIMAI Springer Series |

Editors | J. Landeira, B. Escribano |

Publisher | Springer International Publishing AG |

Pages | 1-17 |

Number of pages | 17 |

ISBN (Electronic) | 978-3-030-16585-7 |

ISBN (Print) | 978-3-030-16584-0 |

DOIs | |

Publication status | E-pub ahead of print - 30 Apr 2019 |

### Publication series

Name | SEMA SIMAI Springer Series |
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Volume | 20 |

ISSN (Print) | 2199-3041 |

ISSN (Electronic) | 2199-305X |

### Fingerprint

### Cite this

*SEMA SIMAI Springer Series*(pp. 1-17). (SEMA SIMAI Springer Series; Vol. 20). Springer International Publishing AG. https://doi.org/10.1007/978-3-030-16585-7_1

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*SEMA SIMAI Springer Series.*SEMA SIMAI Springer Series, vol. 20, Springer International Publishing AG, pp. 1-17. https://doi.org/10.1007/978-3-030-16585-7_1

**Time-delay feedback control of an oscillatory medium.** / Stich, Michael; Beta, Carsten.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Time-delay feedback control of an oscillatory medium

AU - Stich, Michael

AU - Beta, Carsten

PY - 2019/4/30

Y1 - 2019/4/30

N2 - The supercritical Hopf bifurcation is one of the simplest ways in which a stationary state of a nonlinear system can undergo a transition to stable self-sustained oscillations. At the bifurcation point, a small-amplitude limit cycle is born, which already at onset displays a finite frequency. If we consider a reaction-diffusion system that undergoes a supercritical Hopf bifurcation, its dynamics is described by the complex Ginzburg-Landau equation (CGLE). Here, we study such a system in the parameter regime where the CGLE shows spatio-temporal chaos. We review a type of time-delay feedback methods which is suitable to suppress chaos and replace it by other spatio-temporal solutions such as uniform oscillations, plane waves, standing waves, and the stationary state.

AB - The supercritical Hopf bifurcation is one of the simplest ways in which a stationary state of a nonlinear system can undergo a transition to stable self-sustained oscillations. At the bifurcation point, a small-amplitude limit cycle is born, which already at onset displays a finite frequency. If we consider a reaction-diffusion system that undergoes a supercritical Hopf bifurcation, its dynamics is described by the complex Ginzburg-Landau equation (CGLE). Here, we study such a system in the parameter regime where the CGLE shows spatio-temporal chaos. We review a type of time-delay feedback methods which is suitable to suppress chaos and replace it by other spatio-temporal solutions such as uniform oscillations, plane waves, standing waves, and the stationary state.

UR - http://www.scopus.com/inward/record.url?scp=85065446320&partnerID=8YFLogxK

UR - https://link.springer.com/chapter/10.1007%2F978-3-030-16585-7_1

U2 - 10.1007/978-3-030-16585-7_1

DO - 10.1007/978-3-030-16585-7_1

M3 - Chapter

AN - SCOPUS:85065446320

SN - 978-3-030-16584-0

T3 - SEMA SIMAI Springer Series

SP - 1

EP - 17

BT - SEMA SIMAI Springer Series

A2 - Landeira, J.

A2 - Escribano, B.

PB - Springer International Publishing AG

ER -