Consider a polyhedral coil and a pivot which is
placed in the interior of . The pivot can move within the
plane according to any arbitrary continuous rule and the coil
remains fixed as long as the pivot moves only in its interior. If,
however, the pivot touches the coil, the latter starts moving. We
assume that only translations of the coil may occur, i.e.,
sides remain parallel to themselves.
The relationships between variable position of the pivot
, and the corresponding movement of a reference point of the
coil describes a hysteresis nonlinearity which is called two
dimensional play operator with
the characteristic . This nonlinearity, its straightforward multidimensional analogs, and further
modifications, are extremely important in
various subject areas including mechanics and network analysis.
For a rigorous definition of this nonlinearity and detailed studies of
of its properties see, for instance,
- M. Krasnosel'skii and A. Pokrovskii, ``Systems with
Hysteresis'', Springer, 1989,
- P. Krejci and A. Vladimirov. Lipschitz continuity of polyhedral Skorokhod maps. J. Analysis
Appl., 20:817844, 2001.
- P. Krejci and A. Vladimirov.
Polyhedral sweeping processes with oblique reflection in the space
of regulated functions. Set-Valued Anal., 11:91110, 2003.
- Nedaiborshch, K. Nikolaev, and A. Vladimirov. Lipschitz
continuity and unique solvability of fluid models of queueing
networks. Information Processes, Electronic Scientific
Journal, 3:138150, 2003.