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The goal of this paper is to establish a general framework for dynamicbehaviors of coupled fractional-order stochastic dynamic systems of particlesby using star-coupled models. In particular, the general mechanics on thedynamic behaviors related to the stochastic resonance (SR) phenomenon of astar-coupled harmonic oscillator subject to multiplicative fluctuation andperiodic force in viscous media are established by considering couplings,memory effects, the occurring of synchronization linked to the occurring of SRinduced. The multiplicative noise is modeled as dichotomous noise and thememory of viscous media is characterized by fractional power kernel function.By using the Shapiro-Loginov formula and Laplace transform, the analyticalexpressions for the first moment of the steady state response, the stabilityand relationship between the system response and the system parameters in thelong-time limit in terms of systems'asymptotic stability are also established.  All theoretical analyses are supported by the corresponding numericalsimulations, and thus it seems that the results established in this paper wouldprovide a possible fundamental mathematical framework for the study ofSchumpeter's theory on the economic development under the innovation andcapital paradigm and related disciplines. In particular, the frameworkestablished by this paper allows us at the first time logically concluding thatin 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 thesimilar conclusion available from the existing literature today.

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