The theory of dynamical systems of conflict in the framework of functional analysis

Abstract

In this article, we give an introduction to the mathematical setting of problems related to the phenomenon of conflict in terms of constructions in Hilbert spaces. The struggle (conflict, game) between opponents (adversaries, players) will be represented by operator transformations of vectors in Hilbert spaces and probabilistic distributions on the territory of life resources. The phenomenon of conflict as a contradiction between opponents appears in mathematical terms as an intersection the domains of definition for operators and overlapping of corresponding measures. Conflict interaction between opponents in the physical sense is described by the specific transformation of states in a Hilbert space. In turn, this is a mapping that changes the spectral measurements. Thus, a complex dynamical system arises, which we call a dynamical system of conflict. Then the following main problems arise as fundamental questions. What reasonable law of engagement (game or war) should be adopted to resolve the initial intersections? What is a fair limiting distribution of the resource territory? In a more general formulation, solving conflict problems means the detailed describing of all possible outcomes on opponents states of the type: victories, defeats, states of equilibrium, compromises as fixed points together with their basins of attraction.