2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal,
Electron. J. Diff. Eqns., Conference 22 (2015), pp. 99-109.

Comments on multiple oscillatory solutions in systems with time-delay feedback

Michael Stich

A complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding the disappearance of limit cycle solutions, derive analytical criteria for frequency degeneration, amplitude degeneration, and frequency extrema. Furthermore, we discuss the influence of the phase shift parameter and show analytically that the stabilization of the steady state and the decay of all oscillations (amplitude death) cannot happen for global feedback only. Finally, we explain the onset of traveling wave patterns close to the regime of amplitude death.

Published November 20, 2015.
Math Subject Classifications: 35K57, 35B10, 35Q92.
Key Words: Pattern formation; reaction-diffusion system; control.

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Michael Stich
Non-linearity and Complexity Research Group
School of Engineering and Applied Science
Aston University
Aston Triangle, Birmingham B4 7ET, UK
email: m.stich@aston.ac.uk

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