The Local Dynamics Model Of Complex System

Complex networks research focused on how to construct a reasonable mathematical model to describe its dynamic characteristics, and through a variety of iterative methods to simulate the evolution process to predict trends. Thepurpose of this paper is to predict the trend of development by ananlyzing the random factor in the complex system. The framework introducted in this work bases on the stochastic differential equation and dynamic system theory, in which the network is described to be stochastic differential equations, and a potential function is build out. Then the steady-state distribution of complex system is the Boltzmann-Gibbs distribution with energy fucntion that is just the potential function.With the novel framework of construction of potential function, and some sophisicated numer-ical integration algorithm such as Euler-Maruyama, Milstein and one order Runge-Kutta method, the work interprates that the oscillation in a prototypical two-component activator-repressor generic circuit with both positive and negative feedback is a kind of bistability phenomena, then confirms the prediction of existence of extrinisc noise in the system, and discovers the extrinisc noise is the real source of oscillatin, at last, support a theoretical explanation for the mechanism of noise.The most important contribution to the research of complex network is finding a framwork of construction of potential function in non-conserved system by introducing the noise factor into the dynamic system, and improving the Prigogine's dissiptive theory. In the engineering domain, the potential function and the steady-state distributionis helpful to estimate the final optimal so-lution, the effeciency of iterated algorithm also could be improved by adjusting some important parameters in potential function.