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Transcript of Theory of Ordinary Differential 4245 - FALL 2012 Theory of Ordinary Differential Equations Stability

• MATH 4245 - FALL 2012

Theory of Ordinary 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 td