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

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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