Frequency locking of modulated waves

Lutz Recke; Anatoly Samoilenko; Alexey Teplinsky; Viktor Tkachenko; Serhiy Yanchuk

doi:10.3934/dcds.2011.31.847

Article Contents

2011, Volume 31, Issue 3: 847-875. Doi: 10.3934/dcds.2011.31.847

This issue Previous Article An equivalent path functional formulation of branched transportation problems Next Article Stability for the modified and fourth-order Benjamin-Bona-Mahony equations

Frequency locking of modulated waves

1.
Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin
2.
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska St., 01601 Kiev
3.
Institute of Mathematics, Humboldt University of Berlin, Rudower Chaussee 25, 12489, Berlin

Received: May 31, 2010

Revised: August 31, 2010

Published: August 2011

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Abstract

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.

Keywords:

Mathematics Subject Classification: Primary: 34C30, 34C14, 34C15; Secondary: 34C29, 34C60, 34D35, 34D06.

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