Pattern formation in systems with multiple delayed feedbacks.
Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities. In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behavior in the case of two delays is shown to "encode" two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the attainment of the long-delay regime is introduced. We also demonstrate this phenomenon for a semiconductor laser with two delayed optical feedbacks.
Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany.
Serhiy
S
Yanchuk
CNR-Istituto dei Sistemi Complessi-via Madonna del Piano 10, I-50019 Sesto Fiorentino (FI), Italy.
Giovanni
G
Giacomelli