ε-随机规范形

Normal Form Theory for Random Differential Equations withParameters in Probability Engineering Mechanics

同确定性规范形理论相比，随机规范形理论涉及无穷维的讨论。本文提出随机普适形变、带参数随机规范形等概念。

There does not exist, to our best knowledge, any paper on the normal form theory of random differential equations with parameters. We succeeded in deriving theoretically such normal form. We obtained proof for eq.(8), which is different from what is generally employed in probability engineering mechanics. R(L m A) in eq.(8) means R(L m A) plus R(L m A) , but R(L m A) is generally disregarded in probability engineering mechanics. R(L m A) consists of the boundary points of R(L m A) . Now we present our mathematics. We discussed the normal forms of random matrices with parameters. We gave definitions of versal deformation and normal deformation respectively and proved eq.(6). We gave a definition of normal forms with parameters. For random differential equations, eq.(1), and for ε>0,2≤m≤r , we proved that there was a near identity random coordinate transformation x=h(θ tω,y,μ) with T mh∈L 2 m, 2≤m≤r, such that eq.(1) had normal form, eq.(9), by using L 2 m= R(L m A) N((L m A) ). Normal form can be very useful in analyzing nonlinear systems as it can greatly simplify them. Normal form theorem, eq.(11), is a sound theoretical base for application analysis of stochastic bifurcation.