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