DEM IR if Q Es only eqmlib.pt

April^{19,2021 Lentine 27}

Suppose

DEM

IR and ^{I D} Rn be a G vatu

field

and not ^an

equilibrium pt of

^us

Suppose we can

find

^a Lyapunov

funton

Q

^{V s R at mis This means 970 Qin 0}

if

I

ni

^{mis CV E D}

Q

^{Is and ee on v}

Ent

and Get ^{E O}

for

^{EE V ios where}

Ee

LIQ

^{CB Pcsos}

fer

^{se EV}

Ent

and

^CUT7

8

^we showed last time that

in care NT ^{is a stable}

equilibrium point of B

^Ee

given ^{e 0} there exists 870 smh that

0 ^{t E BCmd E At O whenua n E} Boos J

Canreplace

by

^{bcEOD Cros}

we also claimed that

if

^{Q is a} struct Lyapunov

futon

then

NT

^is asymptotically ^{stable Here} shirt hyapuvor

means that in addition ^to Q being Lyapunov

Q

^{Es O}

fer

^{R E} V ENT ^We haveto prove ^this Remark

If

^a stunt Lyapunov

futon

^{exists at nT}

then not ^{is the}

only eqmlib.pt

^{in domain Q}

Indeed if

^{n EV wig then nd co ie}

Q ^Es ECE ^O

where ICED 48

PT O S

Let

Q

^{V IR be a strict} _Lyapunov

fruition for

^{it at mt so in}

particular

^{V E}

D

Let ^{B be} a bull in

D

^{centered at mt s t}

B ^C

D

and ^let 8 ⁰ be smh that

Fas

^CA _hit^{E B} _E

for

^{all b 30}

B mise Can ^{assure BCndo} EV

The _{proof is easier} with the notation

of

^flows

So

when

convenient ^we write

gta effect

Sime is _negative on

V't

^V _{Smt and surie}

G gta LIQ gta gta

LIQ Facts F

^effects

IQ Fact Facts

dq of

^{Ct Chair rule}

therefore

^Q

gta

^{is demeaning}

for

^{TECO so}

In particular if

D _e

wit

^QCgtas

ECOD

then BE ^{EO D We have a} sequence

of tune

_points

0 to ^{t c} tz ^{c D}

smh that

9 gta to B

^The

sequins

gtk

^{of unret have a} erumpent subsequence

mince

BT

^is

compart By

^replacing

gtfo by this convergent subsequence Cif neumary

we

may

^assume

gta

^{is convergent}

hit ⁿ him

k

gtkn

Since god is contruous in

fait

^{le thenfne}

gta't lyg zgtgtkn lgjggtmn.es

Wehave

QCg1n

^{EQ Est}

by

^{our earlier}

reasoning

^{Or in} greater detail

QC

^ghg.tkas

gamine

^Q

gta

ⁱ

On

taking

^{hints we}

get

giant

^e

Q CB't

^z

Guinn ^{E 0} there enists K O smh that

QCgdI

^E

Q

g

gta EQ gtx He

t ka k There exists 170 smh that

te tie

¹

Herne

gta't QCn

^e

gtfo

Cg

Gtk

^Is

g1gtkn

E Q

ghost

^te

byes

So 9 great

^E

QCn

^E

Qcg1n

^eE

fer

every ^{E 0 Heure}

Q gta't Acre't

This

can

only

^happen

if

ⁿ

27

In

_{particular p} ^O

We want _{to prove} that _fringe

gto

^me

hit _E 0 be

given

^{Assure E small enough that}

B

nose

^{EV Let}

r _Pescage

life 9 ye

where

Scout E

FER Uys

^{mish E}

Assume _E small _enough

that

^{B not E EV} Pick

n

^{O smh that 11}

Qty

^{Il e I}₂

fer

all

JE Blut y

^{We are}

using

^the_Fox

continuity of Q

^{and the}

feet

^that

9

^{Cmt 0 Let ice Boat}

7

brine

lining

gtkn

^{nd then exits k I}

s t

11

gtk

^{nd ni}

Il M fer

^{k K}

Suire gta

^Q

gta fer

^t

tie

therefore

g gta

^{e I}_Z ^{t ta}

Ek

Tt follows

that gtr

^{cannot lie on}

Soos

^e

fr

^{all b}

Btk

^Thus

gtnc

^{B.cm E V}

t7tk

This

means

bing.tn nT

e so

Theorems we hope to _prove next time hit NT ^{be an}

equilib pt

Wh 0h assume

not 8 hit

⁸ be ee

hit ^A DF Es DT ^arts Want to compare

in

^FCE

with the homies linin

egen

je AF

If

^{not is a}

hyperbole eqnilib.pt qP nie

^all

the eigenvalues

of

^A

hare

^non

zero

^real

parts

then there is ^a clone ^convention Thurs we'll do

I

If

^{A has even one} eigenvalue with ^the real pent ^then

ni

is ^an unstable

eqrulib.pt

^{of F}

2

If KT

^is hyperbolic and ^all the eigenvalues

of

^A home negative real parts then

mi

^is

asymptotically

^stable

forts