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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Spectrum and amplitude equations for scalar delay-differential equations with large delay

Pages: 537 - 553, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.537

 
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Serhiy Yanchuk - Humboldt-University of Berlin, Institute of Mathematics, Unter den Linden 6, 10099, Berlin, Germany (email)
Leonhard L├╝cken - Humboldt-University of Berlin, Institute of Mathematics, Unter den Linden 6, 10099, Berlin, Germany (email)
Matthias Wolfrum - Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany (email)
Alexander Mielke - Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany (email)

Abstract: The subject of the paper is scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.

Keywords:  Amplitude equations, scalar delay differential equations, Ginzburg-Landau equation, large delay, pseudo-continuous spectrum.
Mathematics Subject Classification:  Primary: 34K18, 34K08, 34K20, 34K25; Secondary: 34K05, 34K06.

Received: August 2013;      Revised: June 2014;      Available Online: August 2014.

 References