1.1 Simulation as a method for studying complex systems

Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics.

The fundamental tenet of Model Theory is that mathematical truth, like all truth, is relative. A statement may be true or false, depending on how and where it is interpreted. This isn’t necessarily due to mathematics itself, but is a consequence of the language that we use to express mathematical ideas The very notion of undergoing the same model as the concept of the system, a certain evolution. Evolution models reflects the evolution of the process of cognition.

So, in the early stages of a model to understand some of the physical device (object), which in certain circumstances replaces another object. Examples of such devices can serve as model airplanes, ships, cars, different layouts, templates, prostheses, etc.

In the next step below the model object understood to substitute that reflects only interested researchers properties and characteristics of the original object. In this model before the object has benefits such as clarity, simplicity, visibility, accessibility for the experiment , the ability to identify , etc. The very notion of a model already significantly expanded and includes drawings, tables, specifications, charts , drawings, cartographic representations , various forms of device descriptions , etc.

At the third stage, the same concept models include only on real (physical, material), but also abstract (ideal) construction. Example of the latter ideas, hypotheses, theories, mathematical, logical, and simulations. In the form of a mathematical model can describe the typical activities of a human operator in the organizational and technical systems. The very process of thinking can be interpreted as a process of successive transition from one to the other abstract models. In this model acts as a form of existence and representation of knowledge about the object, phenomenon, process, system. Thus, knowledge of the material world goes through the model, and purposeful human activity is impossible without simulation.

We indicate to other properties of the models. First, the model is very informative, and this information is presented in a highly compressed form. Second, the model is hierarchical, i.e. model has a higher level (for example, a model management system) and lower (for example, the model elements of control systems).

Third, the model is refined and corrected in the simulation, i.e. disadvantages of the model cannot predict in advance. Fourthly, the model can serve as a reference, idealized different forms of activity: management, planning, decision making, forecasting, monitoring, etc. For example, in adaptive control systems of technical objects implementing the principle of control by the reference model. From this perspective, the objective of the management often acts as a model for the future (desired) state of the system. And the algorithm can be seen as a model of formation control actions aimed at the transfer of control object from one state to another. In this model, the drive is considered as a performance model of control actions by moving the regulatory authorities, and model information elements (sensors) - as a model for the processing and transformation of primary information. Thus, the research management system is through the construction of models of its elements and to study the properties of the system by simulating its behavior under various conditions.

Also consider the main drawback of the method for modeling, which is that the simulation results can be obtained, not related to the properties of the system under study or incorrectly reflecting properties of the real system. This is an objective reason: the model is not always accurate and adequately reflects the real object. But advantages of a mathematical modeling method more than disadvantages.

There are the following advantages:

1. Models practical, they are always built so that there are easier and more convenient for research than the original objects. Models can be put such experiments conducted on real objects which are either too expensive or dangerous to personnel and the environment. For example, models of the aircraft engine control can study its properties at various speeds and altitudes LA instead of studying these properties on expensive high-rise stands.

2. Certain phenomena can only be studied in their models. For example, nuclear explosions, electrical discharges of lightning, are flying LA in the development of an emergency on board as a result of failures of individual functional subsystems, fire, etc.

3. Models reproduce only the basic, most important for this study the properties of the system under study. This allows the simulation to reveal the mechanism of formation of these properties of the system to learn to predict these properties and purposefully change them in the desired direction.

4. When modeling systems may have side effects. For example, the model can reproduce such system properties which are adequate to real, but this model was not designed for this. This effect should be seen as an exception rather than a rule. These advantages make it the method of modeling the most effective method, both research and practical activity.