Elements of Geometric Theory of Complex Systems Behavior

Abstract

The paper proposes the elements of a geometric theory of behavior. The basis of this theory is Model Synthesis - an end-to-end technology for describing, synthesizing and implementing agent models. Its main achievement – is the formalization of an agent by a mathematical object - a species of structure in the sense of N. Bourbaki. Algorithms that implement actions of agents are included in the base sets of this species of structure. Let us consider morphisms of the base sets with the special attention to the actions mappings. It occurs that unrestricted mappings of base sets can radically change system’s behavior. We can use invariants as restrictions of mappings, if demand their preservation under admissible morphisms. As a result, we can obtain a classification of the systems behavior depending on the invariants that persist during the mappings.

Structural theory as a research language for open complex systems, including social ones

“…an auxiliary base set of natural numbers, from which the typical characterization selects a subset , which turns out to be just a natural number, with the help of the auxiliary axiom (meaningfully it is the number of processes in the model). The typical characterizations (10) and axioms (indirectly, their definitions also include typical characterizations (8) with axioms , determining the rules for elements switching in processes and typical characterizations (9), determining the initial elements of the processes) formally determine the processes.…”

“…Typical characterizations (3), (4) shows, from where the methods-elements and methods-events are taken, ( 5) - (7) shows what characteristics of the model are transmitted to methods-elements and accepted from them, typical characterization (8) sets the rules for switching elements under the influence of the events. Axioms specify the typical characterizations ( 1) - (8).…”

“…Secondly, a complex system carries out regular activities to maintain the structure and invariants (laws of existence). We called these activities a cult in [8]. For example, to keep a car alive, we need periodically to undergo maintenance, inspection, refuel, etc.…”

“…As a result, we can say that in complex open systems, conservation of laws instead of conservation laws prevails [8], in contrast to "simple" closed physical systems. The essence of conservation of laws and conservation laws is the same: conservation laws are invariants of certain symmetries: energythe uniformity of time, momentumthe uniformity of space, etc.…”

“…In [10] it is shown that the class of programs defined by the typical characterizations ( 8) -(10) and the axioms , is algorithmically complete. Therefore, for example, in a high-level language, we can talk about the "preservation of laws" in a complex open system ( [8] or section 4 of this work), as a rather complicated algorithm for the system controlling, nevertheless described by ( 8) -(10) and , p . As examples of humanitarian phrases that allow formal interpretation in the language of model species of structures, we can cite here well-known aphorisms not only from humanitarian discourse, but also from the fields of philosophy, religion, esotericism:…”