Chaos and ergodicity in the one and two dimensional ... · Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell,

Chaos andergodicity inthe one and

twodimensional

drippinghandrailmodels

Masaya Sato,Katherine

Shelley, RonSidell, Anh

Thai

The DrippingHandrailproblem

The twodimensionalextension

Chaos and ergodicity in the one and twodimensional dripping handrail models

Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai

San Jose State UniversityCAMCOS

May 16, 2007


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Acknowledgements

Thanks to Dr. Jeffrey Scargle for his help and patience, andbringing this problem to our attention.


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Outline

1 The Dripping Handrail problemAn astronomical modelErgodicityErgodicity of the eDHRChaos

2 The two dimensional extensionA similar setupErgodicity in the 2DDHRConclusions and further questions


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An astronomicalmodel

Ergodicity

Ergodicity of theeDHR

Chaos


Dynamical systems

A dynamical system is a model of relationships that changeover time.A dynamical system is usually given by a function

f : M → M,

where M is the set of all states of the system we areconsidering.Two special characteristics of a dynamical system are fixedpoints, x ∈ M such that

f (x) = x ,

and periodic points, x ∈ M such that for some positive integerm,

f m(x) = x .


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Ergodicity


Chaos


Dynamical systems

We will be using a discrete dynamical system, called theDripping Handrail Model, to represent an astronomicalphenomena, a binary star system.


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Ergodicity


Chaos


The basic setup

Figure: R.S. Ophiuchi System; David A. Hardy


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Ergodicity


Chaos


The basic setup

Accretion disc

Small star

Large star

Accreting matter


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Ergodicity


Chaos


The top view of the accretion disc

Accretion disc

Small star

Individual cells

Dripping matter


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Ergodicity


Chaos


Goal

1 Investigate the behavior of cell densities over time.

2 Investigate possible presence of chaos.

To do this we use the extended Dripping Handrail model(eDHR) from the Spring 2006 CAMCOS class.


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Ergodicity


Chaos


Diffusion along the DHR

31 2 N

Γ Γ

ω

Matter moves from each cell to the neighboring cells asgoverned by Γ, that is, cells that are more dense diffuse matterto the neighboring cells that are less dense by a factor of Γ.Matter accretes onto the star at a rate of ω, the combinationof a constant accretion, ω0 and density related accretionparameter, given by α.


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Ergodicity


Chaos


Equations defining eDHR

If ρin is the density of cell i at time n, then after one time step

we would have:

ρin+1 = ρi

n + Γ(ρi−1n − ρi

n) + Γ(ρi+1n − ρi

n) + αρin + ω0.

We assume that each cell has a maximum capacity for matterand, once that capacity is reached, the matter will immediatelydrip onto the central star.


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Ergodicity


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Equations defining eDHR

To represent the dripping along the rail, we use a (mod 1)operation on the cells. Whenever the density in a cell reaches 1we set it equal to 0.So we have, 0 ≤ ρi

n < 1:

ρin+1 = ρi

n(1− 2Γ + α) + Γρi−1n + Γρi+1

n + ω0 (mod 1).


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Ergodicity


Chaos


The model

If we consider the densities along the rail as a N × 1 vector Xn,the dynamical system becomes:

f (Xn) = AXn + b (mod 1),

where

A =

δ Γ 0 · · · ΓΓ δ Γ · · · 0

. . .. . .

. . .

Γ 0 · · · Γ δ

, b =

ω0

ω0...

ω0

and δ = 1− 2Γ + α.


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Ergodicity


Chaos


Eigenvalues

A characteristic of the matrix A is a set of numbers calledeigenvalues, denoted λ, such that for particular vectors X ∈ M,

AX = λX .

It can be shown that if A has an eigenvalue such that λ > 1,then part of the system expands, or is unstable. Similarly, forthose eigenvalues that are less than one, the system contracts,or is stable.


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Stability and instability

In particular, A has at least one eigenvalue greater than one,

λ = 1 + α,

and eigenvalues less than one. So the system exhibitsexpansion and contraction.The combination of expansion and contraction indicates thatwe may be dealing with a chaotic system.We will now be considering chaos and ergodicity.


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Ergodicity


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Ergodicity

Definition

A system is ergodic if it cannot be decomposed intosubsystems, that is, the system mixes things up as in thefollowing illustration of Arnold’s cat map:

Figure: Arnold’s cat map


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Ergodicity


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Why study ergodicity?

Ergodic systems make randomness out of order and that is acharacteristic of chaos.In a complicated system such as the eDHR we investigateergodic behavior because it is easier to consider chaos“statistically,” in other words, we want to consider averagevalues of the eDHR.


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Ergodicity


Chaos


The time average

Choosing a random initial X0, and a function ϕ : M → R,called an observable, we can find the time average of a discretedynamical system f :

An(X0) =ϕ(X0) + ϕ(X1) + ... + ϕ(Xn−1)

n,

where f (Xi ) = Xi+1.


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Ergodicity


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When is a system ergodic?

Theorem (Birkhoff’s Ergodic Theorem)

The following are equivalent:

1 The dynamical system f : M → M is ergodic.

2 The time average of f converges as n approaches ∞.


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When is the eDHR ergodic?

Picking random initial cell densities, we can find the timeaverage of our observable, φ.

An(X0) =φ(X0) + φ(X1) + ... + φ(Xn−1)

n.

We would like to know if An(X0) converges.


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Ergodicity


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Numerical analysis of ergodicity in the eDHR

Figure: φ = total density along the rail


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Ergodicity


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Numerically, we can see convergence, thus we can conjecturethat the eDHR shows possible ergodic behavior.


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Ergodicity


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Chaos

Definition

1 Mathematically, a system is chaotic if the set of periodicpoints of the system is dense, and there is an orbit that isdense. Compare with: the rational numbers are dense inthe reals.

2 Intuitively, a chaotic system depends sensitively on initialconditions, that is, although we may start from two almostidentical states, after a while the corresponding states areentirely different. For example, the “Butterfly effect.”


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Ergodicity


Chaos


Chaos in the eDHR

Since the system as a whole is complicated, we focus oninvariant subsets of the system.The eigenvalue λ = 1 + α defines an invariant subset.The corresponding vector, or eigenvector, of 1 + α is

1 =

11...1

.


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An easier system to consider

We can think of 1 as the diagonal, ∆. If we parameterize ∆ byt, then the eDHR acts on ∆ as the function

g(t) = (1 + α)t + ω0 (mod 1).

So we consider the map g , which is easier to analyzenumerically.


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Ergodicity


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Numerical analysis of chaos in the eDHR

By taking α = 0.5, and ω = 0.25 we graph g :


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Ergodicity


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We graph several iterates of g :

Figure: g2 and g3


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Ergodicity


Chaos



The periodic points of g and the orbits of g are dense.

Figure: g10 and g25


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Ergodicity


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From the appearance of ergodicity and dense periodic orbits onan invariant subset of the system we can conjecture that theeDHR might exhibit chaotic behavior.Next, we will discuss the two dimensional extension of theeDHR, and investigate possible ergodic behavior.


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A similar setup

Ergodicity in the2DDHR

Conclusions andfurther questions

The two dimensional extension

We have a similar setup for the two dimensional DHR, or2DDHR. Horizontal diffusion is given by β in the same mannerthat Γ acted on the eDHR.Vertically we consider two cases:

1 Vertical diffusion given by γ.

2 Vertical effusion given by ε.


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A similar setup



Two dimensional diffusion

β β

ω

γγ γ

γγγ

Figure: γ diffusion


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A similar setup



Two dimensional diffusion

β β

ω

ε

ε

Figure: ε effusion


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A similar setup



A similar setup

As with the eDHR, this gives a discrete dynamical system ofthe form

F (Xn) = AXn + b (mod 1).

In the 2DDHR we allow dripping from the entire rail, takingevery cell density to be between 0 and 1.


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A similar setup



Goal

We investigate possible ergodic behavior in the 2DDHR byconsidering the time average of the total density, just as before.We consider the 2× 32 case.


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A similar setup



Numerical analysis of ergodicity in the 2DDHR

We map the time average of the total density of the rail in the2× 32 case with γ diffusion:


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A similar setup




We map the time average of the total density of the rail in the2× 32 case with ε effusion:


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A similar setup




Because the time average of our observable settles and seemsto converge to a limit, by Birkhoff’s Ergodic Theorem discussedearlier, we can conclude that the 2× 32 model shows possibleergodic behavior.


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A similar setup



Conclusions

What we have investigated:

1 The one dimensional eDHR.

1 Time averages and ergodicity.2 Chaos in invariant subspaces.

2 The two dimensional 2DDHR.

1 Ergodicity in 2DDHR with diffusion everywhere.2 Ergodicity in 2DDHR with effusion.


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A similar setup



Conclusions

What we have concluded:

1 The eDHR exhibits evidence of chaos;

1 From convergence of the time average.2 From dense periodic points and orbits in an invariant

subspace.

2 The 2DDHR exhibits evidence of chaos;

1 From convergence of the time average with full diffusion.2 From convergence of the time average with effusion.


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A similar setup



Further research

1 Rigorously prove eDHR is chaotic by proving f acting onthe invariant subspace of ∆ is chaotic.

2 Consider other cases for the 2DDHR model such as:

1 Effusion allowed in “diagonal” directions.2 Higher order cases, M × N.

3 Rigorously prove ergodicity in both the eDHR and 2DDHRmodels.


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A similar setup



References

Scargle, J.D., and Young, K., The Dripping HandrailModel: Transient Chaos in Accretion Systems, TheAstrophysical Journal, 468, 1996, 617–632.

Dey, A., Low, M., Rensi, E., Tan, E., Thorsen, J.,Vartanian, M., and Wu, W., The Dripping Handrail: AnAtrophysical Accretion Model, Presented to San Jose StateUniversity and the NASA Ames research center, June 14,2006.

Brin, M., and Stuck, G., Introduction to DynamicalSystems, Cambridge University Press, 2002.

Devaney, R., An Introduction to Chaotic DynamicalSystems, Benajmin Cummings,1986.


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A similar setup



The End

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Documents

Chaos and ergodicity in the one and two dimensional ... · Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell,