具有快速与慢速分量微分方程的稳定性定理

A Stability Theorem For Differential Equations With Fast And Slow Components

在正规线性空间上讨论微分方程系统X′(t)=F(t,x,y)X′(t)=ε.G(t,x,y),这里参数ε很小。证明了如果F和G满足Lipschitz条件,F(t,x,y)对y的小的值是指数稳定的,系统在x和y对1/ε阶时间周期的持久扰动是稳定的。考虑扰动系统X′(t)=F(t,x,y)+J(t),X′(t)=ε.G(t,x,y)+K(t),这里J(t)和K(t)从S到S+1的积分值很小。从而得到存在仅依赖于F和G的常数A,B,C和λ,使对σ≤λ,如果初始值和持久扰动比σ小,且ε≤σ,则解X(t)和Y(t)对一切时间t有界σAeBtε,使得σeBtε≤C。

This paper concerns system differential equations of the formX＇ （t）=F （t,x,y）Y＇ （t）=ε.G （t,x,y）In a normed linear space, where the parameter εis small. We prove a theorem which shows that ifF and G are Lipschitz and F（t,x,y） is exponentially stable for small values of Y, then the system is stable for persistent disturbances in X and Y for a period of time of order 1/8. We consider the perturbed systemX＇（t）=F（t,x,y）＋J（t）Y＇（t）=ε.[G（t,z,y）＋K（t）] Where all the integrals of J（t） and K（t） fi＇om s to s＋1 are small. The theorem states that there are constant A,B,C andλwhich depend only on F and G such that for any δ≤A, if the initial values and persistent disturbance, are smaller then δ and ε≤ δ then the solutions X（t） and Y（t） are bounded by δAe^Btε for all times t such that δAe^Btε ≤C