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Rattling in reaction-diffusion systems with hysteresis


It is known that bi-stable slow-fast systems can be often approximated by limiting systems evolving on the slow time scale, while the fast time scale is replaced by hysteresis. We consider a class of systems in which the hysteretic (fast) component forms a spatially oscillating pattern that originates at a point and propagates outwards. The profile of a slow component, at the same time, exhibits a nontrivial dynamics involving oscillations in time. This phenomenon, which we call the "rattling'', is not related to the Turing instability or traveling waves and persists in both discrete and continuous media.

In the talk, we explain the dynamics of a prototype model for a scalar discrete reaction-diffusion equation in one spatial dimension. In the simplest realization, hysteresis takes two values: $1$ and $-1$. In this case, the "rattling'' implies that the propagating spatial pattern of the hysteretic component is represented by the function that takes value $1$ at each even point and $-1$ at each odd point. The phenomenon is quite stable under various perturbations of the system and has analogs for more general implementations of hysteresis.

The results are obtained jointly with Pavel Gurevich.

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