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Dynamical system

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  • 20142015

    EMAT33100

    Nonlinear Dynamics & Chaos

    Dr David A.W. Barton(david.barton@bristol.ac.uk)

    Department of Engineering Mathematics

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parameters

    So far we have looked at one parameter bifurcations:

    saddle-node bifurcation,I two equilibria collide and destroy each other,

    transcritical bifurcation,I requires an invariant equilibrium,I two equilibria collide and exchange stability,

    pitchfork bifurcation,I requires symmetry,I one equilibrium splits into three,

    Hopf bifurcation,I periodic orbits emerge from an equilibrium,

    What happens when we have two parameters that can be varied?

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersConsider a nonlinear mass-spring-damper

    which can be modelled by

    d2 x

    d t2+

    dx

    d t+ x+ x3 = cos(t)

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersIn general, the occurrence of bifurcations depends on multiple parameters

    The location of the saddle-node bifurcations below depend on the forcingamplitude as well as

    0

    2

    4

    6

    Size

    ofo

    rbit(r)

    0 0.5 1 1.5 2Frequency ()

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersIn general, the occurrence of bifurcations depends on multiple parameters

    The location of the saddle-node bifurcations below depend on the forcingamplitude as well as

    0

    2

    4

    6

    Size

    ofo

    rbit(r)

    0 0.5 1 1.5 2Frequency ()

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersIn general, the occurrence of bifurcations depends on multiple parameters

    The location of the saddle-node bifurcations below depend on the forcingamplitude as well as

    0

    2

    4

    6

    Size

    ofo

    rbit(r)

    0 0.5 1 1.5 2Frequency ()

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersIn general, the occurrence of bifurcations depends on multiple parameters

    The location of the saddle-node bifurcations below depend on the forcingamplitude as well as

    0

    2

    4

    6

    Size

    ofo

    rbit(r)

    0 0.5 1 1.5 2Frequency ()

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersIn general, the occurrence of bifurcations depends on multiple parameters

    The location of the saddle-node bifurcations below depend on the forcingamplitude as well as

    0

    2

    4

    6

    Size

    ofo

    rbit(r)

    0 0.5 1 1.5 2Frequency ()

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parametersIn general, think of a solution surface or hyper-surface when there are moreparameters

    forcing frequency (Hz) forcing amplitude (N)

    resp

    onse

    am

    plitu

    de (m

    m)

    (a)

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parameters cusp bifurcation

    Curves of saddle-node bifurcations can meet at a point and disappear a cuspbifurcation

    0

    2.5

    5

    7.5

    10

    12.5

    forcingam

    plitude(N

    )

    19 20 21 22 23forcing frequency (Hz)

  • Nonlinear Dynamics & Chaos20142015

    Bifurcations with two parameters

    Examples

  • Nonlinear Dynamics & Chaos20142015

    Wheel shimmy

    [Thota et al., Nonlinear Dynamics 57(3) 2009]

  • Nonlinear Dynamics & Chaos20142015

    Chaotic electronic oscillators

    [Blakely and Corron, Chaos 14(4) 2004]

    [Barton et al., Nonlinearity 20(4) 2007]

  • Nonlinear Dynamics & Chaos20142015

    Delay coupled lasers

    [Erzgraber et al., Nonlinearity 22(3) 2009]

  • Nonlinear Dynamics & Chaos20142015

    A geography lesson simulation approach

  • Nonlinear Dynamics & Chaos20142015

    A geography lesson simulation approach

  • Nonlinear Dynamics & Chaos20142015

    A geography lesson Continuation approach

  • Nonlinear Dynamics & Chaos20142015

    A geography lesson Continuation approach

  • Nonlinear Dynamics & Chaos20142015

    Mathematical details of numerical continuation

    All numerical continuation problems are set in the form

    f(x, ) = 0

    where f is an algebraic function, x is the state and is the system parameters.

    Think equilibrium problems:

    dx

    d t= f(x, ) = 0

  • Nonlinear Dynamics & Chaos20142015

    Implicit function theorem

    Numerical continuation relies on the implicit function theorem:

    Theorem (Implicit function theorem)

    Let f : Rn+m Rp be a continuously differentiable function of x and y. If theJacobian of partial derivatives of f is invertible then it is possible to find a functiong such that (at least locally):

    f(x, ) = 0 x = g().

    I.e., its possible to write down solution branches as functions of the parametersexcept in the vicinity of bifurcations

  • Nonlinear Dynamics & Chaos20142015

    Predict/correct

    Predict next solution from previous solutionsCorrect solution (Newton iteration)Discretise periodic orbits (e.g., collocation)

  • Nonlinear Dynamics & Chaos20142015

    Predict/correct

    Predict next solution from previous solutionsCorrect solution (Newton iteration)Discretise periodic orbits (e.g., collocation)

  • Nonlinear Dynamics & Chaos20142015

    Saddle-node bifurcations/folds

    Implicit function theorem fails at saddle-node bifurcations

    Need to re-parameterise the system: arc-length is a good parameter

    ||x||

    (x',')0 0

    i.e., rather than

    x = x()

    use instead

    x = x(s)

    = (s)

  • Nonlinear Dynamics & Chaos20142015

    Saddle-node bifurcations/folds

    Add the equation:(dx)2 + (d)2 = (d s)2

    This is nonlinear, so instead add the pseudo-arclength condition

    x0T(x x0) + 0T( 0) = s

    where (x0, 0) is the initial point and (x0, 0) is its tangent; this is a good(linear) approximation to the arclength condition

  • Nonlinear Dynamics & Chaos20142015

    Continuation of bifurcationsBifurcations can be continued in two (or more) parameters by adding testfunctionsAt saddle-node bifurcations of an ODE x(t) = f(x(t), ) we have

    g(x, ) = det(J(x, )) = 0

    where J is the Jacobian of partial derivatives

    J(x, ) =

    f1x1

    f1x2

    f2x1

    . . ....

    So the full system becomes

    f(x, ) = 0

    x0T(x x0) + 0T( 0) = s

    g(x, ) = 0

  • Nonlinear Dynamics & Chaos20142015

    Software

    (Almost) all examples were done with off-the-shelf softwareI AUTO-07p industry standard

    (Doedel, Oldeman, many more)I MatCont Matlab based, slow

    (Govaerts, Dhooge, plus more)I COCO Matlab based, much faster, multi-point problems

    (Dankowicz, Schilder)I LOCA massively parallelised, PDE discretisations

    (Sandia Labs: Salinger)

    I DDE-BIFTOOL for delay equations(Engelborghs, Samaey, Luzyanina)

    I KNUT for periodic delay equations(Szalai)