Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback

D. Puzyrev; S. Yanchuk; A. G. Vladimirov; S. V. Gurevich

doi:10.1137/130944643

Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback

D. Puzyrev
, S. Yanchuk
, A. G. Vladimirov
, S. V. Gurevich

Research output: Contribution to journal › Article › peer-review

Abstract

We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg-Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.

Original language	English
Pages (from-to)	986-1009
Number of pages	24
Journal	SIAM Journal on Applied Dynamical Systems
Volume	13
Issue number	2
DOIs	https://doi.org/10.1137/130944643
Publication status	Published - 2014
Externally published	Yes

Keywords

Delay differential equation
Large delay
Periodic solutions
Stability theory

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10.1137/130944643

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@article{92574bea4cdd4fbd831d0b4f7ff71266,

title = "Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback",

abstract = "We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg-Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.",

keywords = "Delay differential equation, Large delay, Periodic solutions, Stability theory",

author = "D. Puzyrev and S. Yanchuk and Vladimirov, \{A. G.\} and Gurevich, \{S. V.\}",

note = "Publisher Copyright: {\textcopyright} 2014 Society for Industrial and Applied Mathematics.",

year = "2014",

doi = "10.1137/130944643",

language = "English",

volume = "13",

pages = "986--1009",

journal = "SIAM Journal on Applied Dynamical Systems",

issn = "1536-0040",

publisher = "Society for Industrial and Applied Mathematics Publications",

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TY - JOUR

T1 - Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback

AU - Puzyrev, D.

AU - Yanchuk, S.

AU - Vladimirov, A. G.

AU - Gurevich, S. V.

PY - 2014

Y1 - 2014

N2 - We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg-Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.

AB - We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg-Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.

KW - Delay differential equation

KW - Large delay

KW - Periodic solutions

KW - Stability theory

UR - https://www.scopus.com/pages/publications/84907946090

U2 - 10.1137/130944643

DO - 10.1137/130944643

M3 - Article

AN - SCOPUS:84907946090

SN - 1536-0040

VL - 13

SP - 986

EP - 1009

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 2

ER -

Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback

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Keywords

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