ndc slides 15
DESCRIPTION
Dynamical systemTRANSCRIPT
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20142015
EMAT33100
Nonlinear Dynamics & Chaos
Dr David A.W. Barton([email protected])
Department of Engineering Mathematics
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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?
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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)
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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 ()
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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 ()
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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)
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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)
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Nonlinear Dynamics & Chaos20142015
Bifurcations with two parameters
Examples
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Nonlinear Dynamics & Chaos20142015
Wheel shimmy
[Thota et al., Nonlinear Dynamics 57(3) 2009]
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Nonlinear Dynamics & Chaos20142015
Chaotic electronic oscillators
[Blakely and Corron, Chaos 14(4) 2004]
[Barton et al., Nonlinearity 20(4) 2007]
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Nonlinear Dynamics & Chaos20142015
Delay coupled lasers
[Erzgraber et al., Nonlinearity 22(3) 2009]
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Nonlinear Dynamics & Chaos20142015
A geography lesson simulation approach
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Nonlinear Dynamics & Chaos20142015
A geography lesson simulation approach
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Nonlinear Dynamics & Chaos20142015
A geography lesson Continuation approach
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Nonlinear Dynamics & Chaos20142015
A geography lesson Continuation approach
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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
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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
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Nonlinear Dynamics & Chaos20142015
Predict/correct
Predict next solution from previous solutionsCorrect solution (Newton iteration)Discretise periodic orbits (e.g., collocation)
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Nonlinear Dynamics & Chaos20142015
Predict/correct
Predict next solution from previous solutionsCorrect solution (Newton iteration)Discretise periodic orbits (e.g., collocation)
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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)
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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
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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
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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)