Complex Dynamics of Innovation Diffusion
Malуarets L. M., Voronin A. V., Lebedeva I. L., Lebedev S. S.

Malуarets, Lyudmyla M. et al. (2025) “Complex Dynamics of Innovation Diffusion.” The Problems of Economy 4:417–427.
https://doi.org/10.32983/2222-0712-2025-4-417-427

Section: Mathematical methods and models in economy

UDC 330.46

Abstract:
To search for innovative ways of sustainable growth within the trend of global economic expansion, it is necessary to have formalized approaches within the framework of the synergetic paradigm – the theory of self-organization in open non-equilibrium systems. One of the most important directions in this regard is the conception of innovation diffusion. In this study, the classical logistic model of the spread of an innovative product was considered. The development of the mathematical model was implemented based on the dynamic balance of «supply – demand» in the innovation market, both in discrete and continuous time. At the same time, the linear dependence of demand on the total volume of innovative products was taken into account, while on the supply side, the possibility of technological production constraints was considered, which is reflected in the form of a quadratic dependence of the supply function on the quantity of innovative products. In developing the discrete dynamic model, the basic balance equation was transformed into the form of the classical logistic equation with known properties, with a further detailed analysis provided in the study. The theoretical results were confirmed through corresponding numerical computations and simulation modeling, which illustrated important dynamic regimes such as limit cycles with period doubling, irregular chaotic behavior, and others. In continuous time, a mathematical model of innovation diffusion was constructed taking into account delays (distributed time lag), considered as a second-order dynamic process. The model was reduced to a system of two differential equations, in which limit cycles with varying stability characteristics can exist. Both mathematical models – discrete and continuous – have the same equilibrium states (fixed points), and the dynamics near these points significantly depend on the initial conditions.

Keywords: dynamic equilibrium of demand and supply for innovations, logistic curve, dynamic memory, stability of equilibrium positions (fixed points), limit cycle, attractor and repulsor, bifurcation, chaos.

Fig.: 6. Formulae: 32. Bibl.: 51.

Malуarets Lyudmyla M. – Doctor of Sciences (Economics), Professor, Head of the Department, Department of Economic and Mathematical Modeling, Simon Kuznets Kharkiv National University of Economics (9a Nauky Ave., Kharkіv, 61166, Ukraine)
Email: malyarets@ukr.net
Voronin Anatolii V. – Candidate of Sciences (Engineering), Associate Professor, Associate Professor, Department of Economic and Mathematical Modeling, Simon Kuznets Kharkiv National University of Economics (9a Nauky Ave., Kharkіv, 61166, Ukraine)
Email: voronin61@ ukr.net
Lebedeva Irina L. – Candidate of Sciences (Physics and Mathematics), Associate Professor, Associate Professor, Department of Economic and Mathematical Modeling, Simon Kuznets Kharkiv National University of Economics (9a Nauky Ave., Kharkіv, 61166, Ukraine)
Email: irina.lebedeva@hneu.net
Lebedev Stepan S. – Senior Lecturer, Department of Economic and Mathematical Modeling, Simon Kuznets Kharkiv National University of Economics (9a Nauky Ave., Kharkіv, 61166, Ukraine)
Email: stepan.lebedev@hneu.net

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