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Dynamics of the Shapovalov mid-size firm model
Forecasting and analyses of the dynamics of financial and economic processes such as deviations of macroeconomic aggregates (GDP, unemployment, and inflation) from their long-term trends, asset markets volatility, etc., are challenging because of the complexity of these processes. Important related research questions include, first, how to determine the qualitative properties of the dynamics of these processes, namely, whether the process is stable, unstable, chaotic (deterministic), or stochastic; and second, how best to estimate its quantitative indicators including dimension, entropy, and correlation characteristics.
These questions can be studied both empirically and theoretically. In the empirical approach, researchers consider real data expressed in terms of time series, identify the patterns of their dynamics, and then forecast the short- and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamical models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models.
To implement these approaches, either numerical or analytical methods can be used. While numerical methods make it possible to study dynamical models, the possibility of obtaining reliable results using them is significantly limited due to the necessity of performing calculations only over finite time intervals, rounding-off errors in numerical methods, and the unbounded space of initial data sets. Analytical methods allow researchers to overcome these limitations and to identify the exact qualitative and quantitative characteristics of the dynamics of the process. However, effective analytical applications are often limited to low-dimensional models (in the literature, two-dimensional dynamical systems are most often studied).
In this paper, we develop analytical methods for the study of deterministic dynamical systems based on the Lyapunov stability theory and on chaos theory. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior (to localize self-excited and hidden attractors and identify multistability), but also to overcome difficulties related to implementing reliable numerical analysis of quantitative indicators such as Lyapunov exponents and the Lyapunov dimension. We demonstrate the effectiveness of the proposed methods using the mid-size firm model suggested by Shapovalov.