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Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

Chaos and ergodicity in the one and two dimensional dripping handrail models

Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai

San Jos´ e State University CAMCOS

May 16, 2007

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

Acknowledgements

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

Outline

1 The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

2 The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Dynamical systems

A dynamical system is a model of relationships that change

• ver 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 are considering. Two special characteristics of a dynamical system are fixed points, x ∈ M such that f (x) = x, and periodic points, x ∈ M such that for some positive integer m, f m(x) = x.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Dynamical systems

We will be using a discrete dynamical system, called the Dripping Handrail Model, to represent an astronomical phenomena, a binary star system.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

The basic setup

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

The basic setup

Accretion disc Small star Large star Accreting matter

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

The top view of the accretion disc

Accretion disc Small star Individual cells Dripping matter

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

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.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Diffusion along the DHR

3 1 2 N Γ Γ ω Matter moves from each cell to the neighboring cells as governed by Γ, that is, cells that are more dense diffuse matter to the neighboring cells that are less dense by a factor of Γ. Matter accretes onto the star at a rate of ω, the combination

• f a constant accretion, ω0 and density related accretion

parameter, given by α.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Equations defining eDHR

If ρi

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

we would have: ρi

n+1 = ρi n + Γ(ρi−1 n

− ρi

n) + Γ(ρi+1 n

− ρi

n) + αρi n + ω0.

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Equations defining eDHR

To represent the dripping along the rail, we use a (mod 1)

• peration on the cells. Whenever the density in a cell reaches 1

we set it equal to 0. So we have, 0 ≤ ρi

n < 1:

ρi

n+1 = ρi n(1 − 2Γ + α) + Γρi−1 n

+ Γρi+1

n

+ ω0 (mod 1).

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

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 · · · Γ Γ δ Γ · · · ... ... ... Γ · · · Γ δ      , b =      ω0 ω0 . . . ω0      and δ = 1 − 2Γ + α.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Eigenvalues

A characteristic of the matrix A is a set of numbers called eigenvalues, 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, for those eigenvalues that are less than one, the system contracts,

• r is stable.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

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 exhibits expansion and contraction. The combination of expansion and contraction indicates that we may be dealing with a chaotic system. We will now be considering chaos and ergodicity.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Ergodicity

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Why study ergodicity?

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

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 discrete dynamical system f : An(X0) = ϕ(X0) + ϕ(X1) + ... + ϕ(Xn−1) n , where f (Xi) = Xi+1.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

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 ∞.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

When is the eDHR ergodic?

Picking random initial cell densities, we can find the time average of our observable, φ. An(X0) = φ(X0) + φ(X1) + ... + φ(Xn−1) n . We would like to know if An(X0) converges.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Numerical analysis of ergodicity in the eDHR

Figure: φ = total density along the rail

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Chaos

Definition

1 Mathematically, a system is chaotic if the set of periodic

points of the system is dense, and there is an orbit that is

• dense. Compare with: the rational numbers are dense in

the reals.

2 Intuitively, a chaotic system depends sensitively on initial

conditions, that is, although we may start from two almost identical states, after a while the corresponding states are entirely different. For example, the “Butterfly effect.”

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Chaos in the eDHR

Since the system as a whole is complicated, we focus on invariant subsets of the system. The eigenvalue λ = 1 + α defines an invariant subset. The corresponding vector, or eigenvector, of 1 + α is 1 =      1 1 . . . 1      .

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

An easier system to consider

We can think of 1 as the diagonal, ∆. If we parameterize ∆ by t, 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 analyze numerically.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Numerical analysis of chaos in the eDHR

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Numerical analysis of chaos in the eDHR

We graph several iterates of g:

Figure: g 2 and g 3

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Numerical analysis of chaos in the eDHR

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

Figure: g 10 and g 25

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem

An astronomical model Ergodicity Ergodicity of the eDHR Chaos

The two dimensional extension

Numerical analysis of chaos in the eDHR

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

The two dimensional extension

We have a similar setup for the two dimensional DHR, or

• 2DDHR. Horizontal diffusion is given by β in the same manner

that Γ acted on the eDHR. Vertically we consider two cases:

1 Vertical diffusion given by γ. 2 Vertical effusion given by ǫ.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Two dimensional diffusion

β β

ω

γ γ γ γ γ γ

Figure: γ diffusion

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Two dimensional diffusion

β β

ω

ε ε

Figure: ǫ effusion

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

A similar setup

As with the eDHR, this gives a discrete dynamical system of the form F(Xn) = AXn + b (mod 1). In the 2DDHR we allow dripping from the entire rail, taking every cell density to be between 0 and 1.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Goal

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

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Numerical analysis of ergodicity in the 2DDHR

We map the time average of the total density of the rail in the 2 × 32 case with γ diffusion:

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Numerical analysis of ergodicity in the 2DDHR

We map the time average of the total density of the rail in the 2 × 32 case with ǫ effusion:

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Numerical analysis of ergodicity in the 2DDHR

Because the time average of our observable settles and seems to converge to a limit, by Birkhoff’s Ergodic Theorem discussed earlier, we can conclude that the 2 × 32 model shows possible ergodic behavior.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

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.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

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.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

Further research

1 Rigorously prove eDHR is chaotic by proving f acting on

the 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 2DDHR

models.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

References

Scargle, J.D., and Young, K., The Dripping Handrail Model: Transient Chaos in Accretion Systems, The Astrophysical Journal, 468, 1996, 617–632. Dey, A., Low, M., Rensi, E., Tan, E., Thorsen, J., Vartanian, M., and Wu, W., The Dripping Handrail: An Atrophysical Accretion Model, Presented to San Jose State University and the NASA Ames research center, June 14, 2006. Brin, M., and Stuck, G., Introduction to Dynamical Systems, Cambridge University Press, 2002. Devaney, R., An Introduction to Chaotic Dynamical Systems, Benajmin Cummings,1986.

Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem The two dimensional extension

A similar setup Ergodicity in the 2DDHR Conclusions and further questions

The End

Any questions?