Chaos and ergodicity in the one and two dimensional dripping Chaos and ergodicity in the one and two handrail models dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem San Jos´ e State University The two CAMCOS dimensional extension May 16, 2007
Acknowledgements Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, Katherine Shelley, Ron Thanks to Dr. Jeffrey Scargle for his help and patience, and Sidell, Anh Thai bringing this problem to our attention. The Dripping Handrail problem The two dimensional extension
Outline Chaos and ergodicity in the one and two dimensional 1 The Dripping Handrail problem dripping handrail An astronomical model models Ergodicity Masaya Sato, Katherine Ergodicity of the eDHR Shelley, Ron Sidell, Anh Chaos Thai The Dripping Handrail 2 The two dimensional extension problem A similar setup The two dimensional Ergodicity in the 2DDHR extension Conclusions and further questions
Dynamical systems Chaos and A dynamical system is a model of relationships that change ergodicity in the one and over time. two A dynamical system is usually given by a function dimensional dripping handrail models f : M → M , Masaya Sato, Katherine Shelley, Ron where M is the set of all states of the system we are Sidell, Anh Thai considering. Two special characteristics of a dynamical system are fixed The Dripping Handrail points , x ∈ M such that problem An astronomical model Ergodicity f ( x ) = x , Ergodicity of the eDHR Chaos and periodic points , x ∈ M such that for some positive integer The two dimensional m , extension f m ( x ) = x .
Dynamical systems We will be using a discrete dynamical system, called the Chaos and ergodicity in Dripping Handrail Model, to represent an astronomical the one and two phenomena, a binary star system. 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 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 Figure: R.S. Ophiuchi System; David A. Hardy
The basic setup Chaos and ergodicity in Large star the one and two dimensional dripping handrail models Masaya Sato, Katherine Accreting matter Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem Small star An astronomical model Ergodicity Ergodicity of the eDHR Chaos The two dimensional Accretion disc extension
The top view of the accretion disc Chaos and ergodicity in the one and two Accretion disc dimensional dripping handrail models Individual cells Masaya Sato, Katherine Shelley, Ron Sidell, Anh Thai The Dripping Handrail problem An astronomical model Ergodicity Small star Ergodicity of the eDHR Chaos Dripping matter The two dimensional extension
Goal Chaos and ergodicity in the one and two dimensional dripping handrail models Masaya Sato, 1 Investigate the behavior of cell densities over time. Katherine Shelley, Ron 2 Investigate possible presence of chaos. Sidell, Anh Thai To do this we use the extended Dripping Handrail model The Dripping Handrail (eDHR) from the Spring 2006 CAMCOS class. problem An astronomical model Ergodicity Ergodicity of the eDHR Chaos The two dimensional extension
Diffusion along the DHR Chaos and ergodicity in the one and two ω dimensional dripping handrail Γ Γ models Masaya Sato, Katherine Shelley, Ron Sidell, Anh 1 2 3 N Thai The Dripping Handrail Matter moves from each cell to the neighboring cells as problem governed by Γ, that is, cells that are more dense diffuse matter An astronomical model Ergodicity to the neighboring cells that are less dense by a factor of Γ. Ergodicity of the eDHR Matter accretes onto the star at a rate of ω , the combination Chaos The two of a constant accretion, ω 0 and density related accretion dimensional parameter, given by α . extension
Equations defining eDHR Chaos and ergodicity in the one and two dimensional dripping handrail If ρ i n is the density of cell i at time n , then after one time step models we would have: Masaya Sato, Katherine Shelley, Ron n + Γ( ρ i − 1 n ) + Γ( ρ i +1 Sidell, Anh ρ i n +1 = ρ i − ρ i − ρ i n ) + αρ i n + ω 0 . Thai n n The Dripping We assume that each cell has a maximum capacity for matter Handrail problem and, once that capacity is reached, the matter will immediately An astronomical model drip onto the central star. Ergodicity Ergodicity of the eDHR Chaos The two dimensional extension
Equations defining eDHR Chaos and ergodicity in the one and two dimensional dripping handrail models To represent the dripping along the rail, we use a (mod 1) Masaya Sato, operation on the cells. Whenever the density in a cell reaches 1 Katherine Shelley, Ron we set it equal to 0. Sidell, Anh Thai So we have, 0 ≤ ρ i n < 1: The Dripping n (1 − 2Γ + α ) + Γ ρ i − 1 + Γ ρ i +1 Handrail ρ i n +1 = ρ i + ω 0 (mod 1) . problem n n An astronomical model Ergodicity Ergodicity of the eDHR Chaos The two dimensional extension
The model Chaos and ergodicity in the one and If we consider the densities along the rail as a N × 1 vector X n , two dimensional dripping the dynamical system becomes: handrail models f ( X n ) = AX n + b (mod 1) , Masaya Sato, Katherine Shelley, Ron Sidell, Anh where Thai The Dripping δ Γ 0 Γ ω 0 · · · Handrail problem Γ Γ 0 δ · · · ω 0 An astronomical A = , b = . ... ... ... model . . Ergodicity Ergodicity of the eDHR Γ 0 Γ δ ω 0 · · · Chaos The two and δ = 1 − 2Γ + α . dimensional extension
Eigenvalues Chaos and ergodicity in the one and two dimensional dripping A characteristic of the matrix A is a set of numbers called handrail models eigenvalues , denoted λ , such that for particular vectors X ∈ M , Masaya Sato, Katherine Shelley, Ron AX = λ X . Sidell, Anh Thai It can be shown that if A has an eigenvalue such that λ > 1, The Dripping Handrail then part of the system expands, or is unstable . Similarly, for problem An astronomical those eigenvalues that are less than one, the system contracts, model Ergodicity or is stable . Ergodicity of the eDHR Chaos The two dimensional extension
Stability and instability Chaos and ergodicity in the one and two dimensional dripping In particular, A has at least one eigenvalue greater than one, handrail models Masaya Sato, λ = 1 + α, Katherine Shelley, Ron Sidell, Anh Thai and eigenvalues less than one. So the system exhibits expansion and contraction. The Dripping Handrail The combination of expansion and contraction indicates that problem An astronomical we may be dealing with a chaotic system. model Ergodicity We will now be considering chaos and ergodicity. Ergodicity of the eDHR Chaos The two dimensional extension
Ergodicity Chaos and Definition ergodicity in the one and A system is ergodic if it cannot be decomposed into two dimensional subsystems, that is, the system mixes things up as in the dripping handrail following illustration of Arnold’s cat map: 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? Chaos and ergodicity in the one and two dimensional dripping handrail models Ergodic systems make randomness out of order and that is a Masaya Sato, Katherine characteristic of chaos. Shelley, Ron Sidell, Anh In a complicated system such as the eDHR we investigate Thai ergodic behavior because it is easier to consider chaos The Dripping “statistically,” in other words, we want to consider average Handrail problem values of the eDHR. An astronomical model Ergodicity Ergodicity of the eDHR Chaos The two dimensional extension
The time average Chaos and ergodicity in the one and two dimensional dripping handrail models Choosing a random initial X 0 , and a function ϕ : M → R , Masaya Sato, called an observable , we can find the time average of a discrete Katherine Shelley, Ron dynamical system f : Sidell, Anh Thai A n ( X 0 ) = ϕ ( X 0 ) + ϕ ( X 1 ) + ... + ϕ ( X n − 1 ) The Dripping , Handrail n problem where f ( X i ) = X i +1 . An astronomical model Ergodicity Ergodicity of the eDHR Chaos The two dimensional extension
When is a system ergodic? Chaos and ergodicity in the one and two dimensional dripping handrail models Theorem (Birkhoff’s Ergodic Theorem) Masaya Sato, Katherine Shelley, Ron The following are equivalent: Sidell, Anh Thai 1 The dynamical system f : M → M is ergodic. The Dripping 2 The time average of f converges as n approaches ∞ . Handrail problem An astronomical model Ergodicity Ergodicity of the eDHR Chaos The two dimensional extension
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