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.
Highlights
Dynamic systems with an entropy operator are widely used for mathematical modeling of real processes
Wilson [1], who suggested a thermodynamic approach to the mathematical modeling of transport and regional systems based on the hypothesis about the random nature of exchange processes and the existence of a stationary state that maximizes entropy
The PDESO mathematical model contains the simplest Boltzmann entropy operator whose mathematical model taking into account the interval constraints on the variables transforms into the Fermi information entropy maximization problem
Summary
Dynamic systems with an entropy operator are widely used for mathematical modeling of real processes Research in this field was pioneered by A.J. Wilson [1] , who suggested a thermodynamic approach to the mathematical modeling of transport and regional systems based on the hypothesis about the random nature of exchange processes and the existence of a stationary state that maximizes entropy. The first idea introduced in [8] was to describe a “fast” process in terms of the local maximum of entropy that depends on the state of a corresponding “slow” process This approach turned out to be very fruitful as well, and it was successfully applied in many problems, such as population dynamics modeling [9], the spatiotemporal development of settlements [10,11,12] and the dynamic entropy-based procedures of image restoration [13]. The proposed methods are used to study stationary states and their stability in an economic system that exchanges investments
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