The Framework of Mechanics for Dynamic Behaviors of Fractional-Order Stochastic Dynamic Systems

The main 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 starcoupled 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. Here the noise is modeled with the fractional power kernel function and analytical expressions for the first moment of the stability between system responses and parameters in the long-time (of asymptotic stability) are also given. The theoretic and simulation results show the non-monotonic dependence between the response output gain and the input signal frequency, noise parameters provided by fractional-order stochastic dynamics are significant different by comparing those exhibited under the traditional integer-order stochastic dynamics, which indicates that the bona fide resonance and the generalized SR phenomena would appear. Furthermore, the fluctuation noise, the number of the particles for the systems, and the fractional order work together, producing more complex dynamic phenomena compared with the traditional integral-order systems. The 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 principle. the ratio of SMEs growing up successfully is less than one third”, this is consistent with what the market has been observed commonly, but similar conclusion not available from the existing literature today. Finally, we like to point out that the framework established in this paper actually shows that under the basic model established in Section 2, through numerical simulation results given in sections 3 and 4, the fractional derivative in the interval (0; 1) as a basic tool, which can provide a new world with a more refined description of the market financial scene, such as in identifying risk factors or describing mechanics for enterprises’ growths more precisely with extra features compared with the traditional integer derivative one.