分数阶随机动力系统动力学行为的力学框架
The Framework of Mechanics for Dynamic Behaviors of Fractional-Order Stochastic Dynamic Systems
摘要：
本文的目的是建立一个通用的框架，耦合分数阶随机动力学系统的动力学行为的颗粒，使用星耦合模型。特别地，通过考虑耦合、记忆效应、同步的发生与诱导随机共振的发生相联系，建立了粘性介质中星耦合谐振子在倍增涨落和周期力作用下的随机共振动力学行为的一般力学模型.将乘性噪声建模为二分噪声，用分数幂核函数表征粘性介质的记忆性。利用Shapiro-Loginov公式和拉普拉斯变换，建立了系统稳态响应的一阶矩、系统稳定性以及系统响应与系统参数在长时间极限内的渐近稳定关系的解析表达式。所有的理论分析都得到了相应的数值模拟的支持，从而为熊彼特创新与资本范式下的经济发展理论及相关学科的研究提供了一个可能的基础数学框架。特别是，本文所建立的框架使我们第一次从逻辑上得出结论：一般来说，中小企业成功成长的比率小于小于，这与市场上普遍观察到的情况相一致，也与现有文献中的结论相一致。
英文摘要：
The goal of this paper is to establish a general framework for dynamic behaviors of coupled fractional-order stochastic dynamic systems of particles by using star-coupled models. In particular, the general mechanics on the dynamic behaviors related to the stochastic resonance (SR) phenomenon of a star-coupled harmonic oscillator subject to multiplicative fluctuation and periodic force in viscous media are established by considering couplings, memory effects, the occurring of synchronization linked to the occurring of SR induced. The multiplicative noise is modeled as dichotomous noise and the memory of viscous media is characterized by fractional power kernel function. By using the Shapiro-Loginov formula and Laplace transform, the analytical expressions for the first moment of the steady state response, the stability and relationship between the system response and the system parameters in the long-time limit in terms of systems'asymptotic stability are also established. All theoretical analyses are supported by the corresponding numerical simulations, and thus it seems that the results established in this paper would provide a possible fundamental mathematical framework for the study of Schumpeter's theory on the economic development under the innovation and capital paradigm and related disciplines. In particular, the framework established by this paper allows us at the first time logically concluding that in general, the ratio for SMEs grow up successfully is less than one third, this is consistent with what the market has been observed commonly, and the similar conclusion available from the existing literature today.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
