Dynamics of the Shapovalov mid-size firm model

Abstract

One of the main tasks in the study of financial and economic processes is forecasting and analysis of the dynamics of these processes. Within this task lie important research questions including how to determine the qualitative properties of the dynamics and how best to estimate quantitative indicators. These questions can be studied both empirically and theoretically. In the empirical approach, one considers the real data represented by time series, identifies patterns of their dynamics, and then forecasts short- and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamic models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models. To implement these approaches, both numerical and analytical methods can be used. It should be noted that while numerical methods make it possible to study complex models, the possibility of obtaining reliable results using them is significantly limited due to calculations being performed only over finite-time intervals, numerical errors, and the unbounded space of initial data sets. In turn, analytical methods allow researchers to overcome these problems and to obtain exact qualitative and quantitative characteristics of the process dynamics. However, their effective applications are often limited to low-dimensional models. In this paper, we develop analytical methods for the study of deterministic dynamic systems. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior, but also to overcome the difficulties related to implementing reliable numerical analysis of quantitative indicators. We demonstrate the effectiveness of the proposed methods using the mid-size firm model suggested recently by V.I. Shapovalov.