DCDS-B
Periodic solutions of isotone hybrid systems
Thomas I. Seidman Olaf Klein
Discrete & Continuous Dynamical Systems - B 2013, 18(2): 483-493 doi: 10.3934/dcdsb.2013.18.483
Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work [7] of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.
keywords: fixed point isotone Periodic discontinuous hybrid system calcium waves. hysteresis
DCDS
On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions
Olaf Klein
Discrete & Continuous Dynamical Systems - A 2015, 35(6): 2591-2614 doi: 10.3934/dcds.2015.35.2591
In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e. being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function.
    In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.
keywords: Hysteresis operators vectorial hysteresis string representation discontinuous inputs.

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