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 ( $p\le n$ , 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.

中文翻译：

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一类带熵算子的正动力系统的平衡与稳定性：在投资动力学建模中的应用

带熵算子的动力学系统（DSEO）构成了一类特殊的动力学系统，其非线性特性由带有熵目标函数的摄动数学规划问题描述。DSEO的一个子类是具有正状态坐标（PDSEO）的系统，该系统被用作人口统计学和经济过程的时空演变，计算机断层扫描和机器学习中的动态图像恢复程序的数学模型。构建了具有连接性参数的PDSEO数学模型，该连接性参数表征了熵算子对系统动态特性的影响。根据状态向量中正分量的数量，PDSEO可以具有各种类别的正稳态。p类 状态向量的正分量（ $p\le ñ$ ，其中n是系统的阶数）。建立了形式幂级数的框架和形成稳态存在条件的逐次逼近方法。存在条件以系统参数之间的关系形式获得。我们使用微分Bellman不等式的方法来研究相空间有限区域中稳态的类的稳定性。零稳态和p的不稳定性的参数条件根据连接参数获得正稳态。提出了形式幂级数的框架和存在条件形成及稳态分类的逐次逼近方法。基于微分Bellman不等式的方法，获得了“大”稳态下的稳定性条件。通过对具有随机投资交换的经济模型的动态特性进行分析，说明了存在，分类和稳定性的发展方法。正稳态代表了经济子系统的盈利能力。获得了系统中所有子系统及其各个组的获利条件及其稳定性。