# ndc slides 15

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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

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()

x = x(s)

= (s)

• Nonlinear Dynamics & Chaos20142015

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

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)

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