I 7 Linear plan or system

荆⻔ 三 AM ________inx t y Lt Unknown functions

A hi 品 constant matrix

If I n.li is an eisenpair then

㑎⻔ ⼆ ⽐ El is asoli to ⼼

类1⽐ M ⼼ tl

teitlilze.it Alǔk inMeRecall i phase portrait for scalar autonomousODEguy o guy 70 g n o 类 9 y

33

fCCCI3ystationayptslg.es0

a Phase portrait for planar system

Dijon fieldfor ⻘们 I 您别

7 In ⼼ plane at every

fiíǐiiii

rares g x yX

Case I i real distinct eigenvalues

⼋ ⼋ z.COalong Ū directionf f ly e Mi

y⼋ ŪdirectionL along I direction

iÜ

iiiinr.rs x ⽚ C iˋ

Citir7 direction converges toOhodd sink l similar to D stablecaseI faster as tsx

For linear system I is a stationary pt

Case 2 i real distinct eigenvaluesOL ⼊z ⼊

n

1

n n7

rr 7 nodalsource

iii.it 品L 0ˇ 1 0 i

Case 3 real distinct eigenvalues⼋ o 2

1 V2 generalsolid7 7

iss.hessflsc.eMitczeidffii.ioTi

L rL r

LLCt.frL L Ysaddle unstable unless you start in i direction

When eigenvalues are complex

⼊ 2 r ⼟ ⼉ in 0 real

t.iiimassoli.iet i elm tint real t i Ǔimaset osu t t is in Ut Feat i i g

ttlosvt.vreaisinutiimeslzr r.ci

ieMtlosut.iimgtsinvtir.atentire ǓzǓ Ǔz is a fundamental set of Solis

Case 4 u 0

sj iiii.inf V G so counter clockwise

toLCenter

Case 5 i m co Case 6ˊ_ o

L

其

spiral sinkspiral source

Draw phase portrait for

⻘⽵ A15 A 1名

and determine type of stationary pt to O

de t l E I A det E 品 2 1 2 4⼋ ⼆ I ⼊z 4 wrsnodg

i 测 测 ⼼ 仙 派 13 v 2ˇ2 Y

了n 2 n 4 V

zvz 2ˇˋ 2Vz V1n_n

i Nil ⻔

n s.zffiresestse.inLL tf