The mathematical models of dynamics in the innovation process management

Abstract

The article considers the main approaches to synergetic management of innovation processes, which determine the sustainable development of the modern economy as a knowledge economy. In this case, economic development is understood as a significantly nonlinear process that occurs in an open system and can have a jump-like nature when transitioning from one stationary state to another or even to a chaotic state. The feasibility of using the synergetic principles to create efficient mechanisms for managing innovation activities is substantiated. The main factors that determine the effectiveness of implementing new technologies and innovative products in the conditions of a knowledge economy are identified. A mathematical model of nonlinear dynamics is proposed that describes the process of new technologies diffusion in self-organizing systems. For this purpose, a logistic curve was used, the determination of which was carried out using an ordinary differential equation with respect to the first-order derivative of a technologically and economically significant indicator characterizing the new technology. Formally, synergistic control is implemented as additional negative feedback in the basic differential equation describing the state of the system. To characterize this control effect, the quadratic function of the significant indicator is used. The analysis of the influence of the parameters of this function on the presence of equilibrium states in the economic system and the stability of these states is carried out. The conditions under which a catastrophic failure of stability may be observed due to the appearance of bifurcations of various nature are determined. Therefore, the presence of nonlinearity in the structure of the controlling influence on the innovation process dynamics radically changes the behavioral properties of the studied system, making it structurally unstable. Based on the conclusions regarding synergistic management, according to the proposed mathematical model, it is possible to prevent negative trends in the evolution of the innovation process to ensure the implementation of the economic strategy of sustainable development.