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MATH 4245 - FALL 2012
Intermediate Differential Equations
Stability and Bifurcation II
John A. Burns
Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0531
0( )nt R 0x x(IC)
( ) ( ),t f tx x q() {(IVP) ( , ) : n m nf D R R Rx q
1 1 1 2 1 2
2 2 1 2 1 2
1 2 1 2
( ) ( ( ), ( ),... ( ), , ,... )
( ) ( ( ), ( ),... ( ), , ,... )
( ) ( ( ), ( ),... ( ), , ,... )
n m
n m n
n n n m
x t f x t x t x t q q q
x t f x t x t x t q q qdR
dt
x t f x t x t x t q q q
Initial Value Problem
0( )nt R 0x x(IC)
( ) ( ),t f tx x q() {(IVP) ( , ) : n m nf D R R R x q
( , ) ( ( ), ) 0 f fe ex q x q q
Let xe= xe (q) be an equilibrium for some parameter q, i.e.
We will assume xe= xe (q) is an isolated equilibrium
Autonomous Systems
nR( )2x q
Isolated Equilibrium
( , ) ( ( ), ) 0, 1,2,3,... f f jj jx q x q q
( )1
x q
( )3
x q
( )4
x q
there exists a 0 such that ( ) if j jB i ji jx x ,
NON-ISOLATED EQUILIBRIUM CAN NOT BE
ASYMPTOTICALLY STABLE
( ) ( )x t qx t0( )
qtx t e x
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
( ) ( ) ( ( ), ) x t qx t f x t q
q = 0
q = 1.0
q = -.5
q = .5
q = -1.0
First Order Linear
( , ) 0 qx f x q 0ex
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0ex
q < 0
Equilibrium xe = 0, q < 0: Stable
( , ) 0 qx f x q 0ex q > 0
0 0.5 1 1.5 2 2.5-15
-10
-5
0
5
10
15
0ex
Equilibrium xe = 0, q > 0: Unstable
Equilibrium xe = 0, q = 0: Stable
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0ex
( , ) 0 qx f x q ex xq = 0
.25ex
.25ex
.1ex
.1ex
ANY
0 is NOT isolatedex
1 2( ) ( )x t x t3
2 1 2 1( ) ( ) ( ) [ ( )] x t qx t x t x t
1 2
3
2 1 2 1
0
[ ] 0
x xf
x qx x x
2 0x 2
1 1( [ ] ) 0 x q x
2
1 10 or ( [ ] ) 0 x q x
Example 4.2
2
2
1 1
0
( [ ] ) 0
x
x q x
1
2
0
0
x
x
q 0
1
2 0
x q
xq > 0
Example 4.2
1 ( , ) : ( , ) 0x y f x y y
3
2
3
( , ) : ( , ) 0
( , ) :
x y f x y ax y x
x y y ax x
1
3
2
( , ) 0 ( , ) :
( , ) 0
f x y yx y
f x y ax y x
0
0
Example 4.2
q = 1
Epidemic Models
Susceptible Infected
Removed
Epidemic Models
SIR Models (Kermak McKendrick, 1927)
Susceptible Infected Recovered/Removed
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) constant
dS t S t I t
dt
dI t S t I t I t
dt
dR t I t
dt
S t I t R t N
SIR Models
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
dS t S t I t
dt
dI t S t I t I t I t S t
dt
( ) ( ) and ( ) ( ) ( ) constant d
R t I t S t I t R t Ndt
1 1 1 2
2 1 1 2 2 2
( ) ( ) ( )
( ) ( ) ( ) ( )
x t q x t x td
x t q x t x t q x tdt
1 2( ) ( ), ( ) ( ) and T
x t S t x t I t q
SIR Model: Equilibrium
1 1 1 2
2 1 1 2 2 2
( ) ( ) ( )
( ) ( ) ( ) ( )
x t q x t x td
x t q x t x t q x tdt
1 1 2 1 1 2
1 1 2 2 2 1 1 2 2
0
( ) 0
q x x q x x
q x x q x q x q x
1 1 2 0 q x x 1 20 or 0 x x
2 1 20 hence 0 q x x1 0x
1 can be any valuex2 0x
SIR Model: Equilibrium
NONE ARE ISOLATED
2R
1x
2x
EQUILIBRIUM
1
2
2
: 0
xx x
xex
SIR Model
x2(t)
x1(t)
x1(t) + x2(t) N = 1
( ) ( )x t f x t 0ef x
x0
2
t
0
xe
(t)= xe
x(t)
( ) et x
? HOW DO WE KNOW IF xe IS ASYMPTOTICALLY STABLE ?
Stability of Equilibrium
det( ( )) 0k xI f J ex
Let 1 , 2 , 3 , n be the eigenvalues of Jxf(xe), i.e
(Re( ) , Im( ) )k k k k k k ki
Theorem S1: If Re(k) < 0 for all k=1,2, . n, then xe is an asymptotically stable equilibrium for the non-linear
system
In particular, there exist > 0 such that if
then
( ) ( ) .x t f x t
(0) , ex x lim ( ) 0.tt
ex x
Fundamental Stability Theorem
Theorem S2: If there is one eigenvalue p such that Re(p) > 0, then xe is an unstable equilibrium for the non-linear system
( ) ( ) .x t f x t
Non-Stability Theorem
The two theorems above may be found in:
Richard K. Miller and Anthony N. Michel, Ordinary Differential
Equations, Academic Press, 1982. (see pages 258 253)
and
Earl A. Coddington and Norman Levinson, Theory of Ordinary
Differential Equations, McGraw-Hill, 1955. (see pages 314 321)
Critical Case
If there is one eigenvalue p of such that Re(p) = 0, then xe the linearization theorems do not apply and other methods must be used to determine the
stability properties of the equilibrium for the nonlinear
system
( ) ( ),x t f x t q
[ ( , )]x f qJ ex
31 1 1 2
32 2 1 2
( ) ( ) [ ( )] ( )
( ) ( ) ( ) [ ( )]
x t x t x t x tdf
x t x tdt x t x t
31 1 2
32 1 2
0[ ]
0[ ]
x x xf
x x x
1
2
0
0
x
x
e
x
?? IS STABLE ?? ex 0
3
1 2
3
1 2
[ ]
and
[ ]
x x
x x
9 3
1 2 1
8
1
[ ] [ ]
or
[ ] 1
x x x
x
Example 5.1
1 2
2
1
2
2 0, 0
0 13[ ] 1( )
1 01 3[ ]
x
x x
xf
xJ 0
Try the linearization theorems
20 0 1 1
det ( ) det( ) det 10 1 3 1
I
J 0
2 1 0
1 i Re( ) 0 for 1,2i i
Theorem S1 and Theorem S2 do not apply
Example 5.1
LOOKS
ASYMPTOTICALLY STABLE
Example 5.1
1 1 2
3
2 2 1 2
( ) ( ) ( )
( ) ( ) [ ( )] 3 ( )
x t x t x tdf
x t x t q x t x tdt
1 2
3
2 1 2
0
[ ] 3 0
x xf
x q x x
1
2
0
0
x
x
e
x
1 2
2
1 0, 0
0 1 0 1( )
3 [ ] 3 0 3
x
x x
fq x
J 0
Try the linearization Theorem
?? IS ASYMPTOTICALLY STABLE ?? ex 0
Example 5.2
0 0 1 1
det ( ) det( ) det ( 3)0 0 3 0 3
I
J 0
( 3) 0
1 20 and 3 1Re( ) 0
Linearization Theorems do not apply
BUT
Example 5.2
ZOOM IN
Example 5.2
LOOKS
ASYMPTOTICALLY STABLE
Example 5.2
nR
x3
x1
e
x 0
ex
e
x
( ) : nV H R R x
0 and , for 0n e eH R x H x
0
an open set
nH R
Isolated Equilibrium
e
x 0
0
an open set
nH R ex
( ) : nV H R R x
x
1 2( ) ( , ,... )nV V x x xx
If and ( ) 0V 0 ( ) 0, V , when thenx x 0
( ) V is said to be positive definitex
Lyapunov Functions
1
2
( )
( )( )
( )
n
n
f
fdt R
dt
f
x
xx
x
We define the function by ( ) : nV H R R x.
1 2
1 2
( ) ( ) ( )( ) ( ) ( ) ... ( )
n
n
V V V
x x xV f f f
x x xx x x x
.
1 2T n
nx x x R x
( ) : nV H R R x 1 2( ) ( , ,... )nV V x x xx
Lyapunov Functions
( ) : nV H R R x is called a Lyapunov function
for the equilibrium of the system ex 0
( ) ( )t f tx x()
( ) ( ) is positive definite in
( ) ( ) 0 for all
i V H
ii V x H
and
x
x
if
.
Lyapunov Functions
Theorem L1. If there exists a Lyapunov function for
the equilibrium of the system
then the equilibrium is stable.
e
x 0
( ) ( ) ,t f tx x()
ex 0
Theorem L2. If there exists a Lyapunov function for
the equilibrium of the system
and
then the equilibrium is asymptotically stable.
e
x 0
( ) ( ) ,t f tx x()
e
x 0
( ) 0 and ( ) 0 for all , V V H 0 x x x 0,. .
Lyapunov Theorems
31 1 1 2
32 2 1 2
( ) ( ) [ ( )] ( )
( ) ( ) ( ) [ ( )]
x t x t x t x tdf
x t x tdt x t
