New hysteresis operators with applications to counterterrorism

We define two models of hysteresis that generalize the Preisach model. The first model is deterministic, the second model is stochastic and it utilizes discontinuous transition probabilities that satisfy impulsive differential equations. For the first model we prove, among other things, a local version of the "wiping out" property; for the stochastic model, we give methods for the construction of solutions of impulsive differential equations that determine the discontinuous transition probabilities. We also present a game-theoretic problem utilizing a generalized hysteresis operator. These hysteresis operators are motivated by questions of modelling the dynamics of decision making processes of networks of loosely knit terrorist groups.