Computational Complexities of Modeling of Dynamical Systems with Anticipation

Abstract

The concept of anticipation foresees the dependence of future states not only on the past but also on their future states. One of the main reasons causing urgency of the study of anticipatory systems is the ultimate resource capacity of the problem of modeling of the systems with multiple potential scenarios since anticipatory systems often provide for multivalued solutions. Not great number of publications in this field of computer science is also often caused by ill-posed problem statements due to the existence of several potential solutions. In that way, the systems with anticipation represent a new direction in cybernetics and the models based on anticipation can formally describe a great number of existing systems and processes with higher accuracy in comparison with classical models with delay. We consider the following nonlinear discrete dynamical systems with strong anticipation, where future states can be represented by an explicit dependence on the past ones by means of the Hutchinson operator. The evolution of such dynamical systems is carried out in the Hausdorff metric space. We consider the fundamental problem of such systems modeling, i.e., the volume of use of computation resources. A number of definitions were introduced to study the dynamics of anticipation systems. The necessary notions of the theory of computational complexity were presented. An important tool of studying the dynamics of systems is the map of dynamic modes, which construction requires adaptation of procedures for finding periodic trajectories of such type systems. Procedures of determination of periodic trajectories of dynamical systems with anticipation were proposed and described in detail. Time and spatial complexities of construction of states, trajectories, and these procedures, in general, were obtained successively. Representation of states of the corresponding dynamical systems is substantiated by multisets with the purpose of minimization of time complexity on simulation of systems with anticipation. For further optimization of computational costs, it is necessary to take into account the structure of the phase space of a dynamical system with anticipation combining the proposed procedures.