40 线性化和局部稳定性,临界点稳定定义和线性化准备——第十章

Chapter 10 Plane Autonomous

Systems and Stability

10.3 Linearization and Local Stability

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We will denote the distance between two points recall that if XandYbyX?Y.X?(x1,x2,?,xn)andY?(y1,y2,?,yn), then

X?Y?(x1?y1)?(x2?y2)???(xn?yn).222首页 上页 返回 下页 结束 铃 DEFINITION 10.1

Stable Critical Points

X1Let be a critical point of an autonomous

X?X(t)system, and let denote the

solution that satisfies the initial condition X(0)?X0,X0?X1.X1 where We say that is a stable critical point when, given any

??0,radius there is a corresponding ??0radius such that if the initial position

X0?X1??, satisfies X0首页 上页 返回 下页 结束 铃 Then the corresponding solution satisfies X(t)X(t)?X1??t?0. for all If, in addition, limX0?X1??, whenever t??X(t)?X1X1We call an asymptotically stable critical point(渐进稳定临界点).

?X0?(a) Figure 10.20

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Unstable Critical Points

X1Let be a critical point of an autonomous system, and let denote the X?X(t)solution that satisfies the initial condition X(0)?X0,X0?X1.X1 where We say that

is an unstable critical point if there is a disk of radius with the property that, for ??0,any , there is at least one initial ??0position that satisfies X0首页 上页 返回 下页 结束 铃