Abstract:

The content of the article develops the research methodology of complex systems in mathematical modeling of management processes based on efficiency and innovation development indices. Resource and energy conservation subsystems have been selected as the control object. The management system of efficiency of modernization subsystems and their high-tech innovation improvement has been determined as the subject of study. The research has revealed an essentially non-linear character of the cyclic changes of the studied property under imbalance of goals of efficiency and innovation development subsystems. To simulate the process the mathematical model of the Gibbs phenomenon is suggested to be modernized using parameters of efficiency for resource-saving technologies and products, which vary in levels of innovativeness. This allowed us to substantiate a hypothesis and concept of enhancement of managing efficiency of subsystems development by the criteria of stability of boundaries of the compromise zone of the stated purposes. The research has proven capabilities to simulate evolutionary processes of gradual modernization and stepwise changes of a high-tech innovation type by using mathematical step functions. This allowed us to ascertain advanced capabilities and objectivity of the proposed approach in comparison with the tools of O. Heaviside’s discontinuous functions. The identified features of the studied processes of resource-saving management enabled us to neutralize the negative manifestations of the Gibbs phenomenon. To eliminate them we propose a sequence of recursive periodic functions, which is grounded by numerical checks of studying the dynamic properties of the management mechanism of innovative resource-saving development. To do this, a substantially non-linear differential equation of the second order was selected. It is specified by the indexes of management quality, the speed of technologies and products implementation and others, enabling to differentiate processes. In the end, а numerical solution of the equation has been calculated and phase trajectories of the process have been constructed in comparison with the solutions of O. Heaviside’s model. Interpretation of the contours of the phase      portrait of the resource-saving system was realized. This enabled to justify the types of modernization strategies and high-tech innovation of resource and energy saving in compromise of purposes.