Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ......

141
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

Transcript of Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ......

Page 1: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 2: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

0( ) nt R 0x x(IC)

( ) ( ),t f tx x q(Σ) {(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

Page 3: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

0( ) nt R 0x x(IC)

( ) ( ),t f tx x q(Σ) {(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

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nR( )2

x 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

Page 5: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

( ) ( )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

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( , ) 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

Page 7: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

( , ) 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

Page 8: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 9: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 10: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 11: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

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Epidemic Models

Susceptible Infected

Removed

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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

Page 14: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 15: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 16: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

SIR Model: Equilibrium

NONE ARE ISOLATED

2R

1x

2x

EQUILIBRIUM

1

2

2

: 0

xx x

xex

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SIR Model

x2(t)

x1(t)

x1(t) + x2(t) N = 1

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( ) ( )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

Page 19: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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.t

t

ex x

Fundamental Stability Theorem

Page 20: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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)

Page 21: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 22: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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 ?? e

x 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

Page 23: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 24: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

“LOOKS”

ASYMPTOTICALLY STABLE

Example 5.1

Page 25: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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 ?? e

x 0

Example 5.2

Page 26: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 27: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

ZOOM IN

Example 5.2

Page 28: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

“LOOKS”

ASYMPTOTICALLY STABLE

Example 5.2

Page 29: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

nR

x3

x1

e

x 0

ex

ˆe

x

( ) : nV H R R x

0 and , for 0n

e eH R x H x

0

an open set

nH R

Isolated Equilibrium

Page 30: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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

Page 31: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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 2

T n

nx x x R x

( ) : nV H R R x 1 2( ) ( , ,... )nV V x x xx

Lyapunov Functions

Page 32: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

( ) : nV H R R x is called a Lyapunov function

for the equilibrium of the system e

x 0

( ) ( )t f tx x(Σ)

( ) ( ) is positive definite in

( ) ( ) 0 for all

i V H

ii V x H

and

x

x

if

.

Lyapunov Functions

Page 33: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

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(Σ)

e

x 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(Σ)

e

x 0

( ) 0 and ( ) 0 for all , V V H 0 x x x 0,. .

Lyapunov Theorems

Page 34: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

31 1 1 2

32 2 1 2

( ) ( ) [ ( )] ( )

( ) ( ) ( ) [ ( )]

x t x t x t x tdf

x t x tdt x t x t

2 2

1 2 1 2( ) ( , ) [ ] [ ]V V x x x x x

2 is an open setH R 0

2 2

1 2 1 2and( ) 0 ( ) ( , ) [ ] [ ] 0 if V V V x x x x 0 x x 0

Hence, ( ) V is positive definitex

?? IS STABLE ?? e

x 0

Example 5.1 … AGAIN

Page 35: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

2 2

1 2 1 2( ) ( , ) [ ] [ ]V V x x x x x

1 21

1

( , )2

V x xx

x

1 22

2

( , )2

V x xx

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 12x 22x3

1 2( [ ] )x x 3

1 2( [ ] )x x

31 1 1 2 1 2

32 2 1 2 1 2

( , ) [ ]

( , ) [ ]

x f x x x xf

x f x x x x

Example 5.1 … AGAIN

Page 36: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 12x 22x3

1 2( [ ] )x x 3

1 2( [ ] )x x

4 4

1 2 1 1 2 1 2 2( , ) 2[ ] 2 2 2[ ]V x x x x x x x x .

4 4

1 2 1 2( , ) 2([ ] [ ] )V x x x x .

4 4(0,0) 2([0] [0] ) 0V .

4 4

1 2 1 2( , ) 2([ ] [ ] ) 0V x x x x .

1

2

0

0

x

x

and if , then Theorem L2 IS

ASYMPTOTICALLY

STABLE

e

x 0

Example 5.1 … AGAIN

Page 37: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

2 4

1 2 2 12

( ) ( , ) [ ] [ ]q

V V x x x x x

2 is an open setH R 0

2 4

1 2 2 12

and( ) 0 ( ) ( , ) [ ] [ ] 0 if q

V V V x x x x 0 x x 0

Hence, ( ) V is positive definitex

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

0q

Example 5.2 … AGAIN

Page 38: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 2 1 1 2

3

2 1 2 2 1 2

( , )

[ ] 3 ( , )

x x f x xf

x q x x f x x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

31 21

1

( , )2 [ ]

V x xq x

x

1 22

2

( , )2

V x xx

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 3

12 [ ]q x 22x2x 3

1 2( [ ] 3 )q x x

2 4

1 2 2 12

( ) ( , ) [ ] [ ]q

V V x x x x x

Example 5.2 … AGAIN

Page 39: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

3 3 2 2

1 2 1 2 1 2 2 2( , ) 2 [ ] 2 [ ] 6[ ] 6[ ]V x x q x x q x x x x .

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 3

12 [ ]q x 22x2x 3

1 2( [ ] 3 )q x x

e

x 0

Hence, 2 2

2( ) 6[ ] 0 for all V x H R x x.

and Theorem L1 implies that is stable

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

Example 5.2 … AGAIN

Page 40: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

2

1 2 2( , ) 6[ ] 0V x x x .

2(0,0) 6[0] 0V .

. 2(1,0) 6[0] 0V

BUT

SO

( ) 0 for all , V H x x x 0.

Theorem L2 does not apply

NEED A BETTER THEOREM

Example 5.2 … AGAIN

Page 41: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 1 2

3

2 2 1 2

( ) ( ) ( )

( ) ( ) ( ) [ ( )]

x t x t x tdf

x t x t x t x tdt

1 2

3

2 1 2

0

[ ] 0

x xf

x x x

1

2

0

0

x

x

e

x

1 2

2

2 0, 0

0 1 0 1( )

1 [ ] 1 0x

x x

fx

J 0

Try the linearization Theorem

?? IS STABLE ?? e

x 0

Example 5.3

Page 42: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

20 0 1 1

det ( ) det( ) det 10 1 0 1

I

J 0

2 1 0

1 i Re( ) 0 for 1,2i i

Linearization Theorems do not apply

BUT

Example 5.3

Page 43: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

ZOOM IN

Example 5.3

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“LOOKS”

LIKE A CENTER

Example 5.3

Page 45: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 1 2

3

2 2 1 2

( ) ( ) ( )

( ) ( ) ( ) [ ( )]

x t x t x tdf

x t x t x t x tdt

2 2

1 2 1 2( ) ( , ) [ ] [ ]V V x x x x x

1 2

3

2 1 2

0

[ ] 0

x xf

x x x

1

2

0

0

x

x

e

x

2 is an open setH R 0

2 2

1 2 1 2and( ) 0 ( ) ( , ) [ ] [ ] 0 if V V V x x x x 0 x x 0

Hence, ( ) V is positive definitex

Example 5.3

Page 46: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 2 1 1 2

3

2 1 2 2 1 2

( , )

[ ] ( , )

x x f x xf

x x x f x x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

2 2

1 2 1 2( ) ( , ) [ ] [ ]V V x x x x x

1 21

1

( , )2

V x xx

x

1 22

2

( , )2

V x xx

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 12x 22x2x 3

1 2( [ ] )x x

Example 5.3

Page 47: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 12x 22x2x 3

1 2( [ ] )x x

3

1 2 1 2 2 1 2( , ) 2 2 ( [ ] )V x x x x x x x .

4 4

1 2 1 2 2 1 2 2( , ) 2 2 2[ ] 2[ ]V x x x x x x x x .

4 2

2( ) 2[ ] 0 for all V x H R x x.

e

x 0

Hence,

and Theorem L1 implies that is stable

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

Example 5.3

Page 48: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

4

1 2 2( , ) 2[ ] 0V x x x .

4(0,0) 2[0] 0V .

. 4(1,0) 2[0] 0V

BUT

SO

( ) 0 for all , V H x x x 0.

Theorem L2 does not apply

NEED A BETTER THEOREM

Example 5.3

Page 49: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

nR

x3

x1

M

0x

0x

0( ) ( ; ) for all t t M t t 0

x x x

M0

x

0( ) nt R 0x x(IC)

( ) , ( )t f t tx x(Σ)

Positively Invariant Sets

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( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

dS t S t I t

dt

dI t S t I t I t I t S t

dt

If ( ) , then

( ) ( ) ( ) 0

S t

dI t I t S t

dt

NOT ISOLATED

NSI ee 0 ,0

Equilibrium

SIR Models

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I(t)

S(t)

S(t) + I(t) N = 1

M

M0

LOTS OF (POSITIVELY)

INVARIANT SETS

SIR Models

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0( ; , ) nt t R0x x

0( ; , ) nt t R0x x

nR0x

nR0x

Trajectories

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Given , the (positive) trajectory through

Is the set

nR0x 0x

0 0( ; , ) :nt t R t t 0x x

Given , the (negative) trajectory through

Is the set

nR0x 0x

0 0( ; , ) :nt t R t t 0x x

Given , the trajectory through

Is the set

nR0x 0x

0( ; , ) : ( , )nt t R t 0x x

Trajectories & Limit Sets

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Given , a point p belongs to the omega

limit set (-limit set) of

if for each and every there is a

such that

nR0x

0( ; , ) nt t R0x x

0 0T t t T

0( ; , )t t p 0x x <

( ) : is an -limit point of nR 0 0x p p x

-limit Sets

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( ) : is an -limit point of nR 0 0x p p x

0( ) : there is a sequence with lim ( ; , )k kk

t t t

0 0x p x x p

Theorem LIM1. If is bounded for

, then

is a non-empty, compact and connected positively

invariant set.

0( ; , ) nt t R0x x

0t t( ) 0x

0 0ˆ ˆˆ ˆ( ; , ) ( ) for all t t t t 0 0x x x

ˆ ( )0 0x x

-limit Sets

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nR

x3

x1

( ) ( ; )

as

t t M

t

0x x x

M

0

For any 0 there is a

( ) such that

if ( ), then there is

a point with

( ; )

T T t

t T

p M

t p

0x x <

Convergence to a Set

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nR

x3

x1

x2

M

0( ; , ) nt t R0x x

1 0( ; , )t t 0x xM1p

0( ( ); , )T t 0x x

2 0( ; , )t t 0x x

M2p

Convergence to a Set

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Example NS2

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M

Example NS2

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( ) ( )x t y t2 2( ) ( ) ([ ( )] [ ( )] 1) ( )y t x t x t y t y t

1 2

2 2

2 1 1 2 2

( ) ( )

( ) ( ) ([ ( )] [ ( )] 1) ( )

x t x td

x t x t x t x t x tdt

1 2

2 2

2 1 1 2 2

0

([ ] [ ] 1) 0

x xf

x x x x x

2 0x 1 0x

Example 4.1 .. again a =1 > 0

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Example 4.1 .. again a =1 > 0

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M

Example 4.1 .. again a =1 > 0

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M

Example 4.1 .. again a =1 > 0

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LIMIT CYCLE

M

Example 4.1 .. again a =1 > 0

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Theorem LIM2. If is bounded for

, then

i.e. approaches its -limit set.

0( ; , ) nt t R0x x

0t t0( ; , ) ( )t t 0 0x x x

0( ; , )t t 0x x

Theorem LIM3. If is bounded for

, and then

0( ; , ) nt t R0x x

0t t ( ) M 0x

0( ; , )t t M0x x

LaSalle Theorems

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0( ; , ) nt t R0x x

( ) 0x

0x

M

Theorem LIM2: Example NS2

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ˆLet be a bounded closed positively invariant setnH R

H

ˆ( ) : nV H R R x

minˆ ( ) ( ) > , for all

ˆ ( ) ( ) 0 for all

i V v H

ii V x H

and

x x

x.

ˆ : ( ) 0E H V x x.

E

ˆ is LARGEST invariant subset of E H E M

LaSalle’s Invariance Theorem

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0( ; , )t t 0x x M

Theorem LaSalle IP: If is a

function satisfying (i) and (ii) above and

is the largest

invariant subset of

, then

for each the trajectory

approaches M.

H0x

ˆ( ) : nV H R R x

ˆE H M

ˆ : ( ) 0E H V x x.

0( ; , )t t 0x x

H

0x

EM

Lets apply this to some previous examples

0( ; , )t t 0x x M

LaSalle’s Invariance Theorem

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2 4

1 2 2 12

( ) ( , ) [ ] [ ]q

V V x x x x x

2 is an open setH R 0

2 4

1 2 2 12

and( ) 0 ( ) ( , ) [ ] [ ] 0 if q

V V V x x x x 0 x x 0

Hence, ( ) V is positive definitex

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

0q

Example 5.2 … AGAIN

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1 2 1 1 2

3

2 1 2 2 1 2

( , )

[ ] 3 ( , )

x x f x xf

x q x x f x x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

31 21

1

( , )2 [ ]

V x xq x

x

1 22

2

( , )2

V x xx

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 3

12 [ ]q x 22x2x 3

1 2( [ ] 3 )q x x

2 4

1 2 2 12

( ) ( , ) [ ] [ ]q

V V x x x x x

Example 5.2 … AGAIN

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3 3 2 2

1 2 1 2 1 2 2 2( , ) 2 [ ] 2 [ ] 6[ ] 6[ ]V x x q x x q x x x x .

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 3

12 [ ]q x 22x2x 3

1 2( [ ] 3 )q x x

e

x 0

Hence, 2 2

2( ) 6[ ] 0 for all V x H R x x.

and Theorem L1 implies that is stable

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

Example 5.2 … AGAIN

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?? IS ASYMPTOTICALLY STABLE ?? e

x 0

2

1 2 2( , ) 6[ ] 0V x x x .

2(0,0) 6[0] 0V .

. 2(1,0) 6[0] 0V

BUT

SO

( ) 0 for all , V H x x x 0.

Theorem L2 does not apply

APPLY LaSALLE’s Theorem

Example 5.2 … AGAIN

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1 2( , ) : ( ) 2ˆ . 4 x x VH x

2 4

1 2 2 12

( ) ( , ) [ ] [ ]q

V V x x x x x

Example 5.2 … AGAIN q= -.5

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Example 5.2 … AGAIN q= -.5

ˆ : ( ) 0E H V x x.

1 2 2ˆ( , ) : 0 E x x H xx

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1 2

3

2 1 2

( ) ( )

( ) [ ( )] 3 ( )

x t x td

x t q x t x tdt

1x

2x

2

1 2 2( , ) : 6[ ] 0x x xE x { } e

M x 0

2IF ( ) 0, thenx t 3

2 1 20 ( ) [ ( )] 3 ( )x t q x t x t

1( ) 0x t

1

2

0

0

x

x

e

x

Invariant Sets in E

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1x

2x

2

1 2 2( , ) : 6[ ] 0x x xE x { } e

M x 0

ˆ is LARGEST invariant subset of E EH M

0( ; , )t t 0 ex x M = x = 0

LaSalle’s Invariance Theorem Implies

Hence IS ASYMPTOTICALLY STABLE e

x 0

Example 5.2 … AGAIN

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1 1 2

3

2 2 1 2

( ) ( ) ( )

( ) ( ) ( ) [ ( )]

x t x t x tdf

x t x t x t x tdt

1 2

3

2 1 2

0

[ ] 0

x xf

x x x

1

2

0

0

x

x

e

x

1 2

2

2 0, 0

0 1 0 1( )

1 [ ] 1 0x

x x

fx

J 0

Try the linearization Theorem

?? IS STABLE ?? e

x 0

Example 5.3 … AGAIN

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20 0 1 1

det ( ) det( ) det 10 1 0 1

I

J 0

2 1 0

1 i Re( ) 0 for 1,2i i

Linearization Theorems do not apply

BUT

Example 5.3 … AGAIN

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ZOOM IN

Example 5.3 … AGAIN

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“LOOKS”

LIKE A CENTER

Example 5.3 … AGAIN

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1 1 2

3

2 2 1 2

( ) ( ) ( )

( ) ( ) ( ) [ ( )]

x t x t x tdf

x t x t x t x tdt

2 2

1 2 1 2( ) ( , ) [ ] [ ]V V x x x x x

1 2

3

2 1 2

0

[ ] 0

x xf

x x x

1

2

0

0

x

x

e

x

2 is an open setH R 0

2 2

1 2 1 2and( ) 0 ( ) ( , ) [ ] [ ] 0 if V V V x x x x 0 x x 0

Hence, ( ) V is positive definitex

Example 5.3 … AGAIN

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1 2 1 1 2

3

2 1 2 2 1 2

( , )

[ ] ( , )

x x f x xf

x x x f x x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

2 2

1 2 1 2( ) ( , ) [ ] [ ]V V x x x x x

1 21

1

( , )2

V x xx

x

1 22

2

( , )2

V x xx

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 12x 22x2x 3

1 2( [ ] )x x

Example 5.3 … AGAIN

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1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 12x 22x2x 3

1 2( [ ] )x x

3

1 2 1 2 2 1 2( , ) 2 2 ( [ ] )V x x x x x x x .

4 4

1 2 1 2 2 1 2 2( , ) 2 2 2[ ] 2[ ]V x x x x x x x x .

e

x 0

Hence, 4 2

2( ) 2[ ] 0 for all V x H R x x.

and Theorem L1 implies that is stable

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

Example 5.3 … AGAIN

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?? IS ASYMPTOTICALLY STABLE ?? e

x 0

4

1 2 2( , ) 2[ ] 0V x x x .

4(0,0) 2[0] 0V .

. 4(1,0) 2[0] 0V

BUT

SO

( ) 0 for all , V H x x x 0.

Theorem L2 does not apply

APPLY LaSALLE’s Theorem

Example 5.3 … AGAIN

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1 2( , ) : ( ) 1ˆ . x x VH x

2 2

1 2 2 1( ) ( , ) [ ] [ ] V V x x x xx

Example 5.2 … AGAIN q= -.5

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H

4

1 2 2( , ) : 2[ ] 0x x xE x

ˆ : ( ) 0H VE x x.

Example 5.3 … AGAIN

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1 2

3

2 1 2

( ) ( )

( ) ( ) 3[ ( )]

x t x td

x t x t x tdt

1x

2x

4

1 2 2( , ) : 2[ ] 0x x xE x { } e

M x 0

2IF ( ) 0, thenx t 3

2 1 20 ( ) ( ) [ ( )]x t x t x t

1( ) 0x t

1

2

0

0

x

x

e

x

Example 5.3 … AGAIN

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1x

2x

4

1 2 2( , ) : 2[ ] 0x x xE x { } e

M x 0

ˆ is LARGEST invariant subset of E EH M

0( ; , )t t 0 ex x M = x = 0

LaSalle’s Invariance Theorem Implies

Hence IS ASYMPTOTICALLY STABLE e

x 0

Example 5.3 … AGAIN

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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 … AGAIN

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1 2

3

2 1 2 1[ ]

x xf

x qx x x

1

2

2 1

0 1( )

3[ ] 1

x x

xf f

x q xJ Jx

1

2

0

0e

x

x

x0q

Example 4.2 … AGAIN

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1

2

2 1

0 1( )

3[ ] 1

x x

xf f

x q xJ Jx

1

2

0

0e

x

x

x

0 0 1( )

0 1J Jx e xf f

q

x

Example 4.2: q 0 … AGAIN

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1 0 0 1 1

0 1 1 1q q

2 0 q

Example 4.2: q < 0 … AGAIN

1 0

( ( )0 1

J f

x 0

Theorem S1 IMPLIES 0

0

ex is asymptotically stable

1 2( ) 0 and ( ) 0real real

IN ALL CASES WHEN 0q

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Theorem S1 IMPLIES

0

0

ex is asymptotically stable

IN ALL CASES WHEN 0q

Example 4.2: q > 0

Also, we found that

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1 2 2

3 3

2 1 2 1 2 1[ ] [ ]

x x xf

x qx x x x x

1

2

0

0

x

x

ex

0 0 1 0 1( )

0 1 0 1J Jx xf f

q

ex

Example 4.2: q = 0

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0 0 1( )

0 0 1J Jx xf f

ex

1 0 0 1 1

0 1 0 1 0 1

1det ( 1) 0

0 1

2 0 1 0 1 1

Example 4.2: q = 0

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2 4

1 2 2 1

1

2( ) ( , ) [ ] [ ]V V x x x x x

2 is an open setH R 0

2 4

1 2 2 1

1

2and( ) 0 ( ) ( , ) [ ] [ ] 0 if V V V x x x x 0 x x 0

Hence, ( ) V is positive definitex

1 2 2

3 3

2 1 2 1 2 1[ ] [ ]

x x xf

x qx x x x x

Example 4.2: q = 0

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1 2 1 1 2

3

2 1 2 2 1 2

( , )

[ ] ( , )

x x f x xf

x x x f x x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

31 21

1

( , )2[ ]

V x xx

x

1 22

2

( , )2

V x xx

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 3

12[ ]x 22x2x 3

1 2( [ ] )x x

2 4

1 2 2 1

1

2( ) ( , ) [ ] [ ]V V x x x x x

Example 4.2: q = 0

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1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

. 3

12[ ]x 22x2x 3

1 2( [ ] )x x

3 3 2 2

1 2 1 2 1 2 2 2( , ) 2[ ] 2[ ] 2[ ] [ ]V x x x x x x x x .

e

x 0

Hence, 2 2

2( ) 2[ ] 0 for all V x H R x x.

and Theorem L1 implies that is stable

?? IS ASYMPTOTICALLY STABLE ?? e

x 0

Example 4.2: q = 0

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1 2

3

2 1 2

( ) ( )

( ) [ ( )] ( )

x t x td

x t x t x tdt

1x

2x

2

1 2 2( , ) : 2[ ] 0x x xE x { } e

M x 0

2IF ( ) 0, thenx t 3

2 1 20 ( ) [ ( )] ( )x t x t x t

1( ) 0x t

1

2

0

0

x

x

e

x

Example 4.2: q = 0

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1x

2x

4

1 2 2( , ) : 2[ ] 0x x xE x { } e

M x 0

ˆ is LARGEST invariant subset of E EH M

0( ; , )t t 0 ex x M = x = 0

LaSalle’s Invariance Theorem Implies

Hence IS ASYMPTOTICALLY STABLE e

x 0

Example 4.2: q = 0

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q 0

R2

0

0

0x

0

q

1x

0

q

2x

STABLE UNSTABLE

STABLE

STABLE

0

0

0x

LaSalle’s Invariance

Theorem Implies ? EXPONENTIALLY?

?STABLE?

Bifurcation Diagram: Example 4.2

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( ) ( )x t y t2 2( ) ( ) ([ ( )] [ ( )] 1) ( )y t qx t x t y t y t

1 2

2 2

2 1 1 2 2

( ) ( )

( ) ( ) ([ ( )] [ ( )] 1) ( )

x t x td

x t qx t x t x t x tdt

1 2

2 2

2 1 1 2 2

0

([ ] [ ] 1) 0

x xf

x qx x x x

2 0x 1 0x

Example 4.1 q < 0

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Example 4.1 q = -1

0

0

1 ( , ) : ( , ) 0x y f x y y

2 2

2 ( , ) : ( , ) ( 1) 0x y f x y qx x y y

1

2 2

2

( , ) 0 ( , ) :

( , ) ( 1) 0

f x y yx y

f x y qx x y y

1q

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Example 4.1 q = -1

1

2

( ) sin( )

( ) cos( )

x t t

x t t

PERIODIC SOLUTION

1q

2 2

1 2([ ( )] [ ( )] 1) 0x t x t

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Example 4.1 q = -1

1

2

( ) sin( )

( ) cos( )

x t t

x t t

1q

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LIMIT CYCLE

Example 4.1 q = -1

1q

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Example 4.1 q = -1

1 2

2 2

2 1 1 2 2

( ) ( )

( ) ( ) ([ ( )] [ ( )] 1) ( )

x t x td

x t x t x t x t x tdt

1 2

2 2

2 1 1 2 2

0

([ ] [ ] 1) 0

x xf

x x x x x

2 0x 1 0x

2 2 2

1 2 1 2

1

4( ) ( , ) ([ ] [ ] 1)V V x x x x x

Page 109: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

Example 4.1 q = -1

2 21 21 1 2

1

( , )([ ] [ ] 1)

V x xx x x

x

2 21 22 1 2

2

( , )([ ] [ ] 1)

V x xx x x

x

1 2 1 2

1 2

1 2 1 1 2 2 1 2

( , ) ( , )( , ) ( , ) ( , )

V x x V x x

x xV x x f x x f x x

.

1 2

2 2

2 1 1 2 2

0

([ ] [ ] 1) 0

x xf

x x x x x

2 2 2

1 2 1 2

1

4( ) ( , ) ([ ] [ ] 1)V V x x x x x

2 2 2 2 2 2 2 2

1 2 1 2 1 2 1 2 1 2 1 2 2( , ) ([ ] [ ] 1) ([ ] [ ] 1) ([ ] [ ] 1)V x x x x x x x x x x x x x .

2 2

1 1 2 2( ([ ] [ ] 1) )x x x x 2x2 2

1 1 2([ ] [ ] 1)x x x 2 2

2 1 2([ ] [ ] 1)x x x

Page 110: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

Example 4.1 q = -1

2 2 2 2 2 2 2 2

1 2 1 2 1 2 1 2 1 2 1 2 2( , ) ([ ] [ ] 1) ([ ] [ ] 1) ([ ] [ ] 1)V x x x x x x x x x x x x x .

2 2 2 2

1 2 1 2 2( , ) ([ ] [ ] 1) 0V x x x x x .

2 2 2 2

1 2 1 2 2ˆ( , ) : ([ ] [ ] 1) 0x x x xE H x x

ˆ : ( ) 0H VE x x.

2 2

1 2 1 2 1 2 2ˆ ˆ( , ) : ([ ] [ ] 1) 0 ( , ) : 0x x H x x x x xE H x x

2 2

1 2 11 2ˆ( , ) : ([ ] [ ] 1) 0x x H x xE x

1 2E E E 1 22 2

ˆ( , ) : 0x x H xE x

WHAT IS H

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Example 4.1 q = -1

1 2( , ) : ( )ˆ 2x x VH x

2 2 2

1 2 1 2

1

4( ) ( , ) ([ ] [ ] 1)V V x x x x x

1E

2E

2 { } e

x 0M

11

2 2

2 1 2{ [ , ] ([ ] [ ] 1) }Tx x x x x =M : 0

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LIMIT CYCLE

Example 4.1 q = -1

1q

1 2( , ) : ( )ˆ 2x x VH x

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Return to Bifurcation Theory

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3( ) ( ) [ ( )]x t qx t x t 3( , , ) [ ] 0f t x q qx x

Supercritical Pitchfork Bifurcation

Bifurcation Theory: 1D

q < 0 q > 0 q = 0

xe = 0 xe = 0

xe = -q(1/2)

xe = 0

xe = q(1/2)

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q 0

R1

00x

1/ 2[ ]q1x

1/ 2[ ]q 2x

STABLE UNSTABLE

STABLE

STABLE

00x

Bifurcation Theory: 1D

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3 5( ) ( ) [ ( )] [ ( )]x t qx t x t x t

3 5( , ) [ ] [ ] 0f x q qx x x

Bifurcation: 1D

1x

1x4

x2

x

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3 5( ) ( ) [ ( )] [ ( )]x t qx t x t x t

3 5( , ) [ ] [ ] 0f x q qx x x

Bifurcation: 1D

1x

4x 2

x 3x

5x

1x

4x 2

x 3x

5x

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3 5( ) ( ) [ ( )] [ ( )]x t qx t x t x t

3 5( , ) [ ] [ ] 0f x q qx x x

Bifurcation: 1D

3x

5x 3

x5

x1x

1x

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q 0

R1

00x

3x

5x

STABLE UNSTABLE

STABLE

STABLE

00x

q = -.25

2x

4x

UNSTABLE

UNSTABLE

Subcritical Pitchfork Bifurcation

Bifurcation: 1D

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

q = -0.1

Bifurcation: 1D

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

q = 0

Bifurcation: 1D

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q 0

R1

00x

3x

5x

STABLE UNSTABLE

STABLE

STABLE

00x

q = -.25

2x

4xUNSTABLE

UNSTABLE

Subcritical Pitchfork Bifurcation = BIG JUMP!!!

Bifurcation: 1D

Page 123: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

( ([ ( )] [ ( )] )) ( )

x t q x t y t x t y t x t

b x t y t y t

2 2 2 2 2 2 2

2 2 2 2 2 2 2

0( ( ) ) ( ( ))

0( ( ) ) +( ( ))

x q x y x y x b x y yf

y q x y x y y b x y x

2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

+( ([ ( )] [ ( )] )) ( )

y t q x t y t x t y t y t

b x t y t x t

0, 0 and b q

OR

Typical Hopf Bifurcation

Page 124: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

2 2 2 2 2 2 2

2 2 2 2 2 2 2

0( ( ) ) ( ( ))

0( ( )) ( ( ) )

xq x y x y b x y

yb x y q x y x y

2 2 2 2 2 2 2

2 2 2 2 2 2 2

( ( ) ) ( ( ))det

( ( )) ( ( ) )

q x y x y b x y

b x y q x y x y

2 2 2 2 2 2 2 2 2( ( ) ) ( ( ))q x y x y b x y

0

0

x

y

ex

0, 0 and b q

2 2 2 2 2 2 2 2 2( ( ) ) ( ( )) 0q x y x y b x y

Typical Hopf Bifurcation

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0( )

0J Jx x

qf f

q

ex

2 2 2 2 2 2 2

2 2 2 2 2 2 2

( ( ) ) ( ( ))

( ( ) ) +( ( ))

x q x y x y x b x y yf

y q x y x y y b x y x

1 0

0 1

q q

q q

2 2det ( ) 0q

qq

Typical Hopf Bifurcation

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2 2det ( ) 0q

qq

2( )q

1 q i 2 q i

0q 0

0

ex 0 IS STABLE

0q 0

0

ex 0 IS UNSTABLE

( )iRe q

Typical Hopf Bifurcation

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2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

( ([ ( )] [ ( )] )) ( )

x t q x t y t x t y t x t

b x t y t y t

2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

+( ([ ( )] [ ( )] )) ( )

y t q x t y t x t y t y t

b x t y t x t

( ) ( )cos( ( )) ( )sin( ( )) ( )

( ) ( )sin( ( )) ( )cos( ( )) ( )

x t r t t r t t t

y t r t t r t t t

( ) ( )cos( ( )) ( ) ( )sin( ( ))x t r t t y t r t t

Polar Coordinates

Page 128: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

( ([ ( )] [ ( )] )) ( )

x t q x t y t x t y t x t

b x t y t y t

2( )r t 4[ ( )]r t ( )cos( ( ))r t t

2( )r t ( )sin( ( ))r t t

2 4( ) ( [ ( )] ([ ( )] ) ( )r t q r t r t r t

2( ) ( [ ( )] )t b r t

2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

( ([ ( )] [ ( )] )) ( )

y t q x t y t x t y t y t

b x t y t x t

2( )r t 4[ ( )]r t

( )cos( ( ))r t t2( )r t

( )sin( ( ))r t t

Polar Coordinates

? HOW ?

WORK BACKWARDS

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2 4( ) ( [ ( )] ([ ( )] ) ( )r t q r t r t r t

2( ) ( [ ( )] )t b r t

2 4( ) ( [ ( )] ([ ( )] ) ( )cos( ( ))

( )sin( ( )) ( )

x t q r t r t r t t

r t t t

( ) ( )cos( ( )) ( )sin( ( )) ( )x t r t t r t t t

2 4

2

( ) ( [ ( )] ([ ( )] ) ( )cos( ( ))

( )sin( ( ))( [ ( )] )

x t q r t r t r t t

r t t b r t

Polar Coordinates

Page 130: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

2 2 2( ) cos( ) sin( )r x y x r y r

2 4

2

( ) ( [ ( )] ([ ( )] ) ( )cos( ( ))

( )sin( ( ))( [ ( )] )

x t q r t r t r t t

r t t b r t

2 2 2 2 2

2 2

( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( )cos( ( ))

( )sin( ( ))( ([ ( )] [ ( )] ))

x t q x t y t x t y t r t t

r t t b x t y t

2 2 2 2 2

2 2

( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( )

( )( ([ ( )] [ ( )] ))

x t q x t y t x t y t x t

y t b x t y t

Polar Coordinates

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2 2 2 2 2

2 2

( ) ( ([ ( )] [ ( )] ) (([ ( )] [ ( )] ) ) ( )

( )( ([ ( )] [ ( )] ))

x t q x t y t x t y t x t

y t b x t y t

2 2 2 2 2

2 2

( ) ( [ ( )] [ ( )] ([ ( )] [ ( )] ) ) ( )

+( ([ ( )] [ ( )] )) ( )

y t q x t y t x t y t y t

b x t y t x t

2 4( ) ( [ ( )] ([ ( )] ) ( )r t q r t r t r t

2( ) ( [ ( )] )t b r t

Polar Coordinates

SIMILARLY …

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2 4( ) ( [ ( )] ([ ( )] ) ( )r t q r t r t r t 2( ) ( [ ( )] )t b r t

3 5( ) ( ) [ ( )] [ ( )]r t qr t r t r t 2( ) ( [ ( )] )t b r t

3 5 qr r r 3 5 qr r r

Polar Coordinates

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3 5( ) ( ) [ ( )] [ ( )]x t qx t x t x t

3 5( , ) [ ] [ ] 0f x q qx x x

Recall 1D example

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3 5( ) ( ) [ ( )] [ ( )]r t qr t r t r t

2( ) ( [ ( )] )t b r t

3 5 qr r r 3 5 qr r r

Polar Coordinates

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q = -.5

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q = -.25

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q = -.2

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q = -.05

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q = 0

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q = 0.2

Page 141: Intermediate Differential Equations - math.vt.edu€¦ · Intermediate Differential Equations ... Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations,

q 0

R2

0

0

0x

( )ts_lcx

STABLE

UNSTABLE

STABLE LIMIT CYCLE

0

0

0x

q = -.25

( )tus_lcx

4x

UNSTABLE LIMIT CYCLE

Subcritical Hopf Bifurcation = BIG JUMP!!!

Hopf Bifurication