86 research outputs found
Spatially discrete reaction-diffusion equations with discontinuous hysteresis
We address the question: Why may reaction-diffusion equations with hysteretic
nonlinearities become ill-posed and how to amend this? To do so, we discretize
the spatial variable and obtain a lattice dynamical system with a hysteretic
nonlinearity. We analyze a new mechanism that leads to appearance of a
spatio-temporal pattern called {\it rattling}: the solution exhibits a
propagation phenomenon different from the classical traveling wave, while the
hysteretic nonlinearity, loosely speaking, takes a different value at every
second spatial point, independently of the grid size. Such a dynamics indicates
how one should redefine hysteresis to make the continuous problem well-posed
and how the solution will then behave. In the present paper, we develop main
tools for the analysis of the spatially discrete model and apply them to a
prototype case. In particular, we prove that the propagation velocity is of
order as and explicitly find the rate .Comment: 46 pages, 8 figures, Update bibliographi reference
Shadowing for actions of some finitely generated groups
We introduce a notion of shadowing property for actions of finitely generated
groups and study its basic properties. We formulate and prove a shadowing lemma
for actions of nilpotent groups. We construct an example of a faithful linear
action of a solvable Baumslag-Solitar group and show that the shadowing
property depends on quantitative characteristics of hyperbolicity. Finally we
show that any linear action of a non-abelian free group does not have the
shadowing property.Comment: 18 page
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