Development of an Innovative Methodology of Research of Systems on an Example of a Shadow Economy as an Invariant of System of Economic Security of State

The object of research is the shadow economy as structural component (invariant) of economic security of state on example of Ukraine. The level of shadowing of Ukrainian economy requires improvement of both: the system of indicators and mechanism of influence on the level of shadowing on the basis of using of fundamental constants of mathematics of harmony and principle of structural harmony of systems.Under research, fundamental provisions and methods of the theory of safety studies, economic theory, theories of systems and structural harmony of systems, institutionalism, transdisciplinary nature were used. The theme of management of the system of economic security of the state is investigated from various scientific positions of classical science. At the same time, practically ignores the theory of synergetic, which is the synthesis of transdisciplinary approach, price approach, classical theory of systems, structural harmony of systems, theory of fractals and attractors and other components of theory of complexity.Obviously, any system has metric characteristics inherent. The most widespread math constant found in nature is the Phidias number (Golden section). Meаsure is the most important component of all and any knowledge, without it any description of processes and phenomena will be incorrect. The law of measures, or the law of proportional distribution with its derivatives can be defined as law of projecting and organizing sustainable systems at all levels of hierarchical management. The structure of the system is the most conservative characteristic of it as opposed to the state of the system. The properties of the system as a whole are determined not only the properties of its individual elements, but also the properties of the structure of the system as whole. They depend on the existence of the system. The existence of the system retains these parameters in the process of adapting the system to external conditions and thus maintains the existence of the system itself.Matching the metric quality system of economic security of the state and obtaining a conclusion on its stability and abilities to self-organization under the conditions of shadowing of the economy can be carried out by comparison. It is carried out using attractors representing a recurrent number of gold sections: 0.500 ...; 0.618 ...; 0.682 ...; 0.725, as well as – distractors: 0.5698 ...; 0.6540 ...; 0.7053. An integral indicator characterizing the state of any system is the relative information entropy by Shannon. Entropy testing of the system (calculation of relative information entropy) is a logical component of the innovative model of system research