The mechanics of dynamic behaviors for SMEs’ growth by using fractional-order dynamic system approach

Ruibin Ren and George X. Yuan
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Ruibin Ren: College of Mathematics, Southwest Jiaotong University, Chengdu 610031, P. R. China
George X. Yuan: ��School of Economics, Guangxi University, Nanning 530004, P. R. China‡Sino-British Blockchain Industry Research Center, Guangxi University, Nanning 530004, P. R. China§College of Science, Chongqing University of Technology, Chongqing 400054, P. R. China¶Business School, Sun Yat-Sen University, Guangzhou 510275, P. R. China∥Business School, Chengdu University, Chengdu 610106, P. R. China

International Journal of Financial Engineering (IJFE), 2022, vol. 09, issue 01, 1-28

Abstract: The goal of this paper is to establish a general framework for the development of small- or medium-sized enterprises’ (SMEs) growth path by using star-coupled stochastic dynamic system approach. The dynamic behaviors are related to the stochastic resonance (SR) phenomenon subject to multiplicative fluctuation and periodic force which could be interpreted as uncertain environments faced by SMEs (enterprises) during their growth process. The multiplicative noise is modeled as dichotomous noise (uncertainty under the innovation and capital paradigm) and the memory of risk environments is characterized by using fractional kernel function. By using the mathematical derivation, the analytical expressions for the first moment of the steady state response, the stability in the long-time limit in terms of (enterprises) systems’ asymptotic stability is obtained. Theoretic and simulation results show the nonmonotonic dependence between the output gain and the input signal frequency, noise parameters for SMEs dynamic behaviors. Furthermore, the fluctuation noise, the number of related parties (partners) for SMEs, and the fractional-order work together, producing more complex dynamic phenomena compared with the traditional integral-order systems (for enterprises). Finally, theoretical analyses with corresponding numerical simulations results established in this paper would provide a possible fundamental mathematical framework for the study of Schumpeter’s theory on the development for SMEs’ growth under the “innovation and capital paradigm†and related disciplines. In particular, the framework allows us, for the first time, to logically conclude that “in general, the ratio for SMEs’ growth successfully is around less than one-third†, this is actually consistent with what the market has been observing commonly in general.

Keywords: Dynamic behavior; SMEs’ growth; star-coupled structure; fractional-order dynamic system; stochastic resonance (SR); U-Shaping Phenomenon; Schumpeter’s theory of economic development (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S2424786322500049

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