A Stability Analysing Method Of Stochastic Dynamic Al Systems

There are lots of systems in our world, being called dynamical systems,which have a common feature: all kind of statuses of one system drift with thetime flowing. The researches of dynamical systems have advanced a lot, sincethe knowledge with the system's stability has been increasing. The normalprocess is to construct ordinary differential equation and analyse theproperties, but at last the stochastic noise can never be voided under somesituations, which even could be the positive force to suggest a stable state.Potential function as a new stability analysing method is studied in thispaper, and the work related with fundamental definition and math tools isfocused specially, during which we have a purpose of making the potentialmethod be a standard method to solve the stability problem of SDE.Firstly, the background information of stochastic dynamical system isintroduced, and the construction process of potential function is discussed.An application by Ao is showed here, which is used to model the phageλ'slife circle. Adaptive fitness is a further application of the potential function,which strongly supports Wright's Shifting Balance Theory.Secondly, introduce several strong approximation solutions of stochasticdifferential equations based on Taylor approximation. Derive the potentialfunction's formula of 2-dimension and gradient development formula. Basedon the work of Liao X, bring the definition of Lyapunov stability from ODE toSDE, and show the limitations of Liao's definition. A new statement forstability is defined in this section.Potential function is used to analyse the stability feature of several typicalchaotic systems, i.e. a limit cycle—Vander Pol equation, a 2-dimensionbistable system—Duffing equation, and a typical chaotic system—Lorenz system. Compared to well-known Lyapunov fuctions, the potential functionmethod shows a strong ability to solve those tricky problems.In the end, from the point of MS-Stability, the relationship between timeinterval and convergence region of the stochastic numerical integral isanalysed. Another stochastic integral introduced by potential function methodis tested, order 0.5 strong scheme is showed here, and the time interval isreanalysed. At last the problems which need to be solved in the future ispointed out.