Equilibria and Stability of One Class of Positive Dynamic Systems with Entropy Operator: Application to Investment Dynamics Modeling

Abstract

Dynamical systems with entropy operator (DSEO) form a special class of dynamical systems whose nonlinear properties are described by the perturbed mathematical programming problem with entropy objective functions. A subclass of DSEO is the system with positive state coordinates (PDSEO), which are used as mathematical models of the spatiotemporal evolution of demographic and economic processes, dynamic image restoration procedures in computer tomography and machine learning. A mathematical model of the PDSEO with a connectivity parameter characterizing the influence of the entropy operator on the dynamic properties of the system is constructed. PDSEO can have positive stationary states of various classes depending on the number of positive components in the state vector. Classes with p positive components of the state vector ( p ≤ n , where n is the order of the system) are considered. The framework of formal power series and the method of successive approximations for the formation of existence conditions of stationary states are developed. The conditions of existence are obtained in the form of relations between the parameters of the system. We used the method of differential Bellman inequalities to study the stability of classes of stationary states in a limited region of phase space. The parametric conditions of instability of the zero stationary state and p positive stationary states depending on the connectivity parameter are obtained. The framework of formal power series and the method of successive approximations for the formation of existence conditions and classification of stationary states are developed. The stability conditions “in large” stationary states are obtained, based on the method of differential Bellman inequalities. The developed methods of existence, classification and stability are illustrated by the analysis of the dynamic properties of the economic model with stochastic investment exchange. Positive stationary states characterize the profitability of economic subsystems. The conditions of profitability and their stability for all subsystems in the system and their various groups are obtained.