Semion 6 Applications of Qualitative Analysis

Plan Competing SpeciesPredator PreyThe Non Linear Pendulum

o

The Non Linear Pendulum

31T IT 0 IT 31T Newton's EgZ A Damping const m d e f

d doa S o e o

l const Oct

f fg t foye fois ds fg mg sincea so If o om fol d S ol e o

toi Ixmgsince fg TV mg i 8 9.81 mfs

0 021T 0 25 ml 0 my sirs d L O1 1

I s0NL Io

Remark Small oscillations

10 1 K L Sirs o e 0 Taylor expans

O e See 0 Im O

f mo t ol O t 8em_0 O

k mSpring Huerta Ifat.hu L

a

d4 1

I ymy tdy'tkY

D8

Sketch phase portrait of sols N P

I 0 sin to 0mL O i Oct L

EI i write L as a first order systemSod

X Ct Oct angular positionEct Ect angular remix

LXI o Sirs I Im o

Le Sind Inxs 1X's Xa

X's since Lm xSFO NL DE

DE

Example i Sketch a phase portrait of sols of

X Xs

Xs Sin Xs ol Xz

Re mati For simplicity we chose L m L

Sofvector Field

I n dXz

critical points 22 0E E sirs 1 0

Sina 01 2 0X HIT

M O Il ILXn CnIT

Ph use Spaces physicalSpacectX A O IT2

ZAl Ii fI lI ooo D Dl O l e Y

21T IT IT 21T Xi l 0 Do mI

Derivative Matrix and Linearizations

I.si c fI I Ii

I c't

X C not O DFG O L

1 arena no

ask.ote.is

DFW

foin

f IDFHM ffyn.toC1J d

h n n_2k i no dotn 2 tt

f ghH f g2Rt1 1htt f 12kt

HI

1 1Cc 2K C l y2kt2

4 c 2RC l n j2CktD

1k fi gRtl

L L1 1

DEC xI fo f DFA t

10 fo 1 IXz 1

a It o x

every odd even odd even

A Linearizations No Friction o

odd critical points X 2kt c IT O

DFCX2kH LO L Eigenpairs some work

z i I't F Ia so A i 42kt saddle Nodes

Xz 1

SITE EERIE isKK a

21T CT 0 IT 21T

every odd even odd even

Even critical points X k 2kt O

DFCX2k e 0 I Ty Eigenpairs i ome workl O

t Fifth212k 2k IT O centers No Hartman Grobman

Further StudyDirected

Xz 1

EF At Xp BE

soot 5HA t a as X

21T CT 0 IT 21T

every odd even odd even

iii L il gtfo I 19ft i i

Eff t'da

a b

O Xz 1

e

ifa DBEBE a D

s I oYe f ga

sietr

No Friction

Linearization s 01 0 0th Friction

odd critical points i 212kt 2kt 1 IT 0

DFC 2ktLO to

Eigenpairs at C ol IVoKt4 II 11,10127

some worke

2 dz 110174 o

at off 110147 o

2kt Saddle Nodes

Xz 1

5.9 EERIE isHH a

21T CT 0 IT 21T

every odd even odd even

Even critical Points X k 2k IT O

DFCx2k Iz fo

Ecgoupairs i a octild F IIs 7,110127some work

Two cases CA 62 4 Zo ol 32

B of 420 0 0122

Case A i 0132 large friction d2 430

at 0 I 4012 4 Red

A LO i Xt 0 t 24 L O1 L lol

XK

Simes No oscillations

Xz 1

an Ee T h2 f EERIE aMX na x

OC 32 21T CT 0 IT 21Tlarge friction

even odd even odd even

case B 0 Lol L2 Small friction 012 4 0

I E t4dn III 4401 iw Positive fReal Part complexnegatives

2kX Sink Spirals oscillatory

with decay

Xz 1

iA X

21T CT 0 IT 21T

even odd even odd even

0 Lol 22 Smallfriction

2kt saddle Nodes

212k oleo centers osogeocaccyations without

da sscjy.nk.is Coggeikayations with

d 2 Sink decay withoutNodes oscillations

case O L ol L 2

Hq

a