In this study, a methodological approach based on the model of the differential equations system to determine the impact of the functions of public administration of the regional development in the context of digital transformation is proposed. As digital transformation involves changes in management models of systems at different levels in order to improve their efficiency, this also applies to public management of the regional development. Based on the system approach, social, reproductive and mobilizing functions of state management of the regional development were singled out. To determine the impact of public administration functions on the regional development, it is proposed to use a model of differential equations. Different weights of the influence of the functions of state management of the regional development in the conditions of digital transformation are characterized by dynamic changes in the amplitude of fluctuations of integral indices of the modules of the functionalities of social, reproductive and mobilization functions. Scientific novelty of this study is the development of a methodological approach based on the model of the differential equations system to determine social, reproductive and mobilizing functions of public administration of the regional development in the context of digital transformation.

Since modernization challenges of microeconomic systems are implemented through strategic management of the modernization potential development of enterprises, the rationale for choosing an exclusive innovation and investment strategy is proposed. An algorithm for making managerial decisions based on the model of "preferences in the implementation of investment and innovation strategies" ("PIIIS") is built. The model is based on the concept of strategic management, economic and mathematical hierarchy of potential levels of cognitive judgment, optimization of resources on demand and the structure of innovation and investment strategies. This allows to ensure the efficiency and resource conservation of individual innovation and investment strategies in modernization potential of the enterprise. The sequence of establishing preferences of innovation and investment strategies for activating modernization potential of microeconomic systems is proposed. To model the process of granting preferences to innovation and investment strategies, a system of coefficients based on the potential profile level using cognitive judgments is substantiated. Resource-providing, accumulating, implementation and resulting levels of the modernization potential development are singled out. According to the profile levers, target tasks and alternatives of preferences of innovation and investment strategies for the modernization potential development are identified. Decomposition of tasks of granting preferences to innovation and investment strategies to intensify modernization potential of microeconomic systems is substantiated. It is proposed to use mathematical tools for the analysis of hierarchies, based on mathematical and cognitive judgments based on the T. Saaty model. Application of the Pareto model for the distribution of the coefficients influence in the effectiveness of the components analysis of the potential profile is presented. The choice of innovation and investment strategies for the modernization potential development for enterprises in the infrastructure sector of Ukraine is analyzed and determined.

The study simulates the harmony of economic development of the regional economic systems in the context of sustainable development for the following successive stages: separation of indicators from the array of statistics in the regional context; standardization of indicators of economic development of the regional economic systems; use of correlation analysis to determine the integrated index of economic development of the regional economic systems; taking into account the coefficients of influence of indicators on the integrated index of economic development of the regional economic systems; determining the harmony of economic development of the regional economic systems using the hyperbolic Fibonacci cone; approbation of modeling the harmony of economic development of the regional economic systems. In accordance with the outlined stages, the proposed method was tested for the regions of Ukraine and an example was given in relation to the economic component of sustainable development. The highest integrated index of economic development is observed in Kyiv, and the lowest in Luhansk region, the difference between the values is 17.6 times. According to the results of calculations, the regions are grouped according to the harmonies of economic development, and cartographic analysis is given.

Households play one of the key roles in the development of the financial system of any country and the national economy in general. It is the understanding of the behavior of these economic entities in the market of financial services that makes it possible to predict the development of such a market, to understand the mechanisms of the emergence of dissipative processes in the interaction of households and financial institutions, which can form crisis phenomena in the development of such a system and restrain the innovative development of the national economy. This determines the importance and relevance of further research in this direction. Within the article, the impact of modern digital technologies on the development of financial services, in particular financial behavior of households, in the conditions of the formation and active development of the innovative economy is considered. Significant attention is paid to the specification of the methodology for determining the impact of the digitization index and the index of the model transformation of financial behavior of households. It is established that the outlined models are specific and different between various regions in any country. That is why the above method of calculation was used on the example of Ukraine. As a result, information was obtained on the digitalization index, and the transformation index of the financial behavior model of households in twenty-four regions of Ukraine. Based on the use of econometric analysis, algebraic equations of the dependence of the transformation index of the model of financial behavior of households on the digital technologies development in each of the outlined regions were determined.

Within the presented study, applied aspects of the resource provision of innovation and investment strategies for modernization of microeconomic systems in the context of digitalization are submitted. It is substantiated that the modernization of microeconomic systems is a certain activation of the potential modernization aimed at improving the efficiency of processes, technologies, management of products, services and more. Modernization challenges of microeconomic systems are implemented through strategic management of the enterprise development provided resource provision of the potential modernization. Initial conditions of the resource provision of the microeconomic systems modernization in the conditions of digitalization are singled out, which include: efficiency; availability and clarity of the algorithm analysis formed by information base and tools; clear calculations of the cost of attracting resources to the modernization process; forecasting and assessing the risks of incomplete implementation of modernization. It is proved that in the methodological support of resource management of microeconomic systems in the design, the method of subjectivity of distributions or cost centres is used. The methodical toolkit of the resource provision optimization of modernization designing of microeconomic systems in the conditions of digitalization with application of the competitive selection model of modernization projects for the carrying the chosen strategy of the resource supply is offered. The proposed approach to the optimization of resource provision of modernization design of microeconomic systems in the context of digitalization involves: formation of a set of applications based on the preferential principle and priority of innovation and investment strategies; assessment of actual indicators of resource efficiency in modernization projects by each participant; decision-making on providing resources to the modernization project; determining the completeness and redundancy of the resource allocation between projects; distribution of residual resources.