⑮
a)
Write
"Besset
the
ij
a
as
b) Write
=
2-D
the
-
equation" (THER
Ey
-
(1
-
,
+
so)
7 )y
system
of
first order
generat
N-th
order
ODEs
ODE
= F(u,
as
a
N-D
system
of first order
ODE's
=)
a) ODE
·
Equilibria
:
F(xY)
equilibria
=
nification
·
(PF(c0
↳
↳
=
1
4
a
A+
=
b) o E(x)
=>
·
=
x2
Eld
Letwit
=>
=
=>
67
-
30
-
,
x
=
-
0
1
-
( 1)(4-1)
=
-
= 0
eigenrates
are
source
-
12
=0
[4 3
IPF 16-71) =( 1)
Es
=2
1-5 . -16
15** -1)
-1)
is
50 = 0
and
,
=
&2
E)
DF =
&
(0, 0)
=
0
F(xy
(0 01
:
-
1= 2
=>>
=
are
0
,
=:
5= 0
& 1. 0
Es
=
=
( +(4-4)
-
te
(8 -16) is
,
-
to
16
3 Ve
a
saddle
#
.
,
=
&
0
be
EE(u)
a
=
E +0
Exto
solution
of
2xi
=
-
i
=
=>
(i) of Def 7 5
.
.
-u ?
zu40fu +
0
(ii) of Def 7 5,
E is a strict
ThE2UX
.
.
.
function
asymptotically stable
Lyapanor
=
0
is
⑱
y
Note,
·
has
+
=
be
.
,
.
Clearly Ely)
and
=
=
,
xx)
2x4)
#22
x
-
-
+
choosing
* E(x y)
,
Exy)
=
=
X
=
1
2y(3x
+
+
we
+
g
Since
Ey)
is
VW ge
R2
we
,
attraction
I
-0
,
Oxy(
+
-
y)
-
- +
1)
-
problematic
,
(2x
-
as
·
20
Thus
long
as
2yi
+
(2x(-yy
=
5.0
E)
0
20xk
=
FC, + (0,0
0
>
El0, %
E(x, y)
·
#
equitibrium
a
·
=>
10, 0) as an
E(y) ox + y2 M
for constants
0 :0
candidate for a
determined, as
Lyapnor function
Define
to
((
=
is
a
-
strict Lyapanov function
a
strict
is
20
2yz0
conclude
of 10,01
-
that
have
224)
2yz
-
alt
.
Lyapanor function
that
the
of
IR2
basin
of
.
↳
(i)
S
sofre
itas
(ii)
-
-
My
=
y + xy
0
E)
X =0
or
y= 7
0
E)
x= 1
or
y
=
equifitrium
the
Show
N
that
solutions
E(1)
·
=
0
10, 0) and (1 1)
are
,
E(7) > 0
&
0
=
F(xy + (7 11
in
/
a
neighborhood of (1, 1)
E(H y())10
·
on
,
We
non
E(x, y) = X
·
=
El 1)
-
+
X
=
has
a
-
y
(n(y)
-
el
+
e
-enc-
2
=
-
E
wa
=O
E= 2
=)
(2m(x) 1)
+
-
+
y
(m(y) 1) >
+
-
0
close enough to (1, 11
since f(x) X-Enx
,
locat minimum in 5 1 (Use I derivative test
E(xH y(1)
,
((t), y(t))
properties"
E(t, 1) = 0
F(X, y)
·
1
=
,
E(x, y)
en(
-
Thus
·
these
prove
sola's
=
=
=
i
=
(1 E)k
-
-
Ex
+
+
y
(1
-
-
.
ti
E)i
)
.
=
O(-E) Fry)
El,yH
=>
In
(iii)
of
Since
(x 1(( y)
=
0
this
summary ,
function
=>
the
there
that
,
-
+
(1 y)( y xy)
+
(y 1)(x 1)
-
+
-
-
=
is
a
a
Lyapanor
.
hyapunor function
a
0
0
,
=
exist
=
(*(H y(t)
proves that Elig) is
ODE (1)
when
E 2
,
,
(7 1)
-
-
solutions
on
GEAH H
with
-
-
y(x 1)
X(1 y)
,
Thm
stable
7 2
.
implies
equitibrium
.
↳
·
El
=
=>
E
=>
II
E(20
&
0
has
Let
,
SE
,
<
SElul
be
=
Er ha
o
solution
a
E'(ni
=
with
#
F(N)
=
F(A) co f
<
Elke
0
=in
A
similar
This
proves
=
=
Ux
that UX
U0EIL
·
)
, by (
*
with
(x)
fel
do > U*.
fu(H + 4x
>
UX
Ux
U
for any solution with
initial data
shows
argument
im PH no
,
>
,
Eful F(u)
> 0
·
U*
I
,
interval , open with U
E0 F * U*, wel
UH = H no
Now
at
minimum
local
a
such that
·
**
No U*
that
for aft HU
is
asymptotically stable
.
↳
59) Write
the
"Besset
i
Ey
=
(1 -
-
system
2-D
a
as
equation" (CHER
y
= :
,
+
>0)
f(t y y)
,
first order
of
(x)
,
ODEs
Solution :
Define
the
variables
new
[i] =
(NE)
42)
+, m ,
=
This
is
5b)
[i]
[ ] [2]
=
(2x2)-system
generat
the
Write
Eyz
=
N-th
f(t )
-
,
order
,
.
(1 Y2)u
-
it-order
of
(n [i])
=
=
=:
-
linear
a
3
Zu
=
Un : y
&
U:= y
ODE's.
ODE
= F(u,
as
N-D
a
system
of first order
(** )
ODE's
solution :
Define
Them :
-
()
Un :
=
y
,
12 :
Eine
=
i ,
=
-r
,
Un
:=
Um
ymu
=
f(t, un
,
Un- ,
-
..,
(n)
-
=:
f(t, i)
=:
<=
(H)
=
for i :
E(t u(H)
,
-[i]
&
*
,
Hal I
E
#