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Reaction-diff usion systems with hysteresis

PD Dr. Pavel Gurevich:

 

We give an overview of a number of models where hysteresis plays a role of a feedback control in a reaction-diff usion system and gives rise to spatio-temporal patterns. In particular, hysteresis on the boundary of a spatial domain may lead to time-periodic solutions, while hysteresis inside the domain may lead to spatial concentric-rings or sign-changing patterns. From the mathematical point of view, the challenge is that we have to deal with discontinuous and non-variational objects, where not only a long-term dynamics but even the well-posedness is a nontrivial issue.
We will also discuss spatially discretized reaction-di usion systems (dynamical systems on lattices or, more generally, on graphs) and see how hysteresis may lead to a certain self-organization. This will indicate a way to quantitatively describe the dynamics of a corresponding continuous model.
Questions to be addressed in the next funding period include delay control of periodic solutions, traveling-wave solutions in "hysteretic" slow-fast systems, and discrete-vs-continuous issues.

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