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Page 1: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Chaotic Stellar Dynamo Models

Math 638 Final Project

Rodrigo Negreiros

Ron Caplan

Page 2: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Overview• Background: What are stellar dynamos?• Formulation of the model

– Desirable Dynamics– Step by Step Formulation and Analysis

• 1-D, 2-D– Formulation– Lorentz Force adds Hopf Bifurcations– Breaking degeneracy of 2nd Hopf

• 3-D– Symmetry breaking– Chaos!– New breaking term for Reversibility

• Numerical Results• Summary

Page 3: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Stellar Dynamos

Page 4: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Formulation of the Model

Page 5: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Begin with 1D

• Simplify all hydrodynamical behavior of a star into a single variable, z

• We want to describe two steady convecting velocity fields, so we model z by:

2zz This gives rise to a saddle-node bifurcation, with fixed points:

UnstablezP

StablezP

_:

__:

Page 6: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Magnetic Field

• Toroidal field Bt = x• Poloidal field Bp = y• Set q = x + iy = reiФ

• r = (x2 + y2)0.5 Strength of magnetic field• Now we have,

)()()(2qqOzqibaqiq zqibaqiq )()(

Page 7: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Magnetic Field cont.• Using the definition of q, and reordering, we

obtain the following system:

xbzyazy

ybzxazx

zz

)()(

)()(

2

xyazy

yxazx

zz

)(

)(

2

And in cylindrical coordinates:

bz

azrrr

zz

2

azrrr

zz 2

Page 8: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

What do we have now?

azrrr

zz 2

azrrr

zz

2

azar

zJ

02

0

*

2/1

r

z

)(),(22/1 ae

))(2( aT

a2)(2

:P

:P

a a

Sink Saddle

) a + 4 + a (4 + ) ( a) 2 + (44 222 T

a a

Saddle Source

Page 9: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Lorentz Force

• Need to add back-reaction of magnetic field on the flow

• This force is proportional to B, so we add a term to z-dot (carefully):

xyazy

yxazx

zz

)(

)(

2

azrrr

zz 2

xyazy

yxazx

yxzz

)(

)(

)( 222

azrrr

rzz 22

Page 10: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

What do we have now?

azrrr

rzz

22

azar

rzJ

22

0

*

2/1

r

z

2

2

*

3

a

ar

z

0

22

),(

2

2

2

2

*3

aa

aarzJ

02

a

T )(2

2

2

aa

)8a(4

42

2

2

22

aaT

02 a

Saddlea 02

000

UnstableCenterStable

a

a

a

a

21

2

21

2 33

Spirals

Page 11: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Quick Reality Check

• We are analyzing r vs. z

• Fixed point in r (r ≠ 0) means? Periodic orbit in x and y!

• Periodic orbit in r means?

• Toroidal orbit in x and y (and z)!

Page 12: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Bifurcation Diagram

Page 13: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Breaking Degeneracy

• We want a torris that will break into chaos, so first we need a viable torris that is maintained in parameter space!

• To do this, a cubic term is added to z-dot, breaking the symmetry that caused the degeneracy.

• Now our system: (c<0)

xyazy

yxazx

yxzz

)(

)(

)( 222

xyazy

yxazx

czyxzz

)(

)(

)( 3222

azrrr

czrzz 322

Page 14: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

What do we have now?(1)

• New fixed point, and total remap of three old ones.

• No degeneracy

• Heteroclinic Connection

• Stable, unique toroidal orbits, shown as limit cycles in r-z plane

Page 15: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

What do we have now? (2)

azrrr

czrzz

322

0

023321

*

3,2,1

μzcz:z

r

z ,,

3

3

2

2

*

4

ac

a

ar

z

azar

rczzJ

232 2

0

232),(

3

3

2

2

3

3

2

2

2

2

4

ac

aa

ac

aac

arzJ

2

2

32ac

aT

)(2 323

2 caa

a

8a-)84

()128

(94 22

3324

422

aacacaacT

0

c

a

3

2

Hopf

Hopf

a2

227

82

c

aa

0

227

4

c

Page 16: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Z1 = [1/9*(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)+100/9/(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)+10/9]

Z2 =

[ -1/18*(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)-50/9/(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)+10/9+1/2*i*3^(1/2)*(1/9*(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)-100/9/(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3))]

Z3 =

[ -1/18*(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)-50/9/(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)+10/9-1/2*i*3^(1/2)*(1/9*(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3)-100/9/(1000-1215*u+45*(-1200*u+729*u^2)^(1/2))^(1/3))] Lambda1 = [1/2/a^2*(2*lam*a+3*c*lam^2+(4*lam^2*a^2+12*lam^3*a*c+9*c^2*lam^4+8*a^3*lam^2+8*a^2*c*lam^3-8*a^5*u)^(1/2))]

Lambda2 = [ 1/2*(2*lam*a+3*c*lam^2-(4*lam^2*a^2+12*lam^3*a*c+9*c^2*lam^4+8*a^3*lam^2+8*a^2*c*lam^3-8*a^5*u)^(1/2))/a^2]

Page 17: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Bifurcations Revisited

Page 18: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

What about Chaos?

• Since system is essentially 2-D, no chaos possible.

• To break axisymmetric property of system, we add cubic term to toroidal field (x).

• Finally(?), our system:

xyazy

yxazx

czyxzz

)(

)(

)( 3222

sin

cos2

322

drz

zdrazrrr

czrzz

xyazy

yxdzyxazx

czyxzz

)(

)()(

)(22

3222

Page 19: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

One is Better than Two• In order to simplify our numerical

experiments, we want to only have one variable parameter.

• This is done by creating a parametric curve in the lambda-mu plane, which crosses into all the interesting different qualitative regions.

Page 20: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

As if this Wasn’t Enough!• After 10 years, an improvement has been made on the model. • We last left our model after adding a symmetry-breaking cubic term to x• This unfortunately breaks the y->-y, x->-x reversibility of the system• Another possibility for achieving the same result, without losing

reversibility is:

)3()(

)3()(

)(

32

23

3222

yyxdxyazy

yxxdyxazx

czyxzz

sin

cos2

322

drz

zdrazrrr

czrzz

2sin

2cos2

3

322

dr

drazrrr

czrzz

xyazy

yxdzyxazx

czyxzz

)(

)()(

)(22

3222

Page 21: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Numerical Results

• 1 - All trajectories collapsing to the fixed point P+.

1

Page 22: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.
Page 23: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Numerical Results

• 2 - First Hopf bifurcation.36.38 10

Page 24: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.
Page 25: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Numerical Results

• 3 - Second Hopf bifurcation.36.38 10

Page 26: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Numerical Results

• 3 - Second Hopf bifurcation. • Poincaré Plane

36.38 10

Page 27: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.
Page 28: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Numerical Results

• 4 - Torus folding onto a chaotic attractor.

0.49

Page 29: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.
Page 30: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Numerical Results

• 4 - Torus folding onto a chaotic attractor.

0.85

Page 31: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.
Page 32: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Bifucartions numerically

Page 33: Chaotic Stellar Dynamo Models Math 638 Final Project Rodrigo Negreiros Ron Caplan.

Summary

• We re-derived the model found in the paper, and in addition we did a detailed analysis of the bifurcations occurring in the system.

• We could see that the model is fairly successful reproducing the different qualitative regimes of magnetic activity in the star.

• Even being an artificial model, it might be very helpful to understand the processes occurring in such complex system.

• Furthermore is a very rich non-linear model in which a great number of features.