Chaos and Ergodicity in the One and Two Dimensional Dripping Handrail Models M

Draft version June 15, 2007 A Preprint typeset using LTEX style emulateapj v. 12/14/05

CHAOS AND ERGODICITY IN THE ONE AND TWO DIMENSIONAL DRIPPING HANDRAIL MODELS M. Sato, K. Shelley, R. Sidell, A. Thai Draft version June 15, 2007

ABSTRACT The Dripping Handrail Model (DHR) is a discrete dynamical system used to model a binary star system in which matter accretes oﬀ of one star onto the other. With this model, we study the behavior of the matter as it collects in a ring around the star, called an accretion disc, builds up, and eventually drips onto the surface of the star. Considering the accretion disc as both a one dimensional and two dimensional “rail” divided into discrete cells, we investigate numerical evidence of chaotic behavior on a certain invariant subset, and ergodic behavior of the system as a whole. Subject headings: binary star system; accretion disc; chaotic; ergodic; invariant subset

1. INTRODUCTION havior. We will conclude in section five with some closing 1.1. Introduction remarks. A binary star system is an astronomical phenomenon 1.3. Dynamical systems in which two stars orbit each other. In a binary system one star, the larger of the two, gives off, or For the following standard material we follow accretes, matter which then collects in a ring around (HSD04). the second star, much like the rings of Saturn. The A dynamical system is a system that changes over matter along the ring, called an accretion disc, con- time, which can be discrete or continuous, depending denses as more and more matter joins. Eventually, the on whether the time parameter is considered discrete density of matter will have condensed so much that sec- or continuous. A discrete dynamical system (like the tions of matter fall, or “drip” onto the surface of the star. one we will be dealing with in this paper) is simply a map f : M → M, where M is the state (or phase) The Dripping Handrail Model (DHR) is a discrete space, that is, the space of all the possible states of dynamical system used to model various physical phe- the system. Usually M is a metric space or a smooth manifold. An orbit of a point x ∈ M is the set of iterates nomena such as a binary star system (cf.,(SY96)). With k this model, we study the behavior of the matter as it col- {f (x) : k = 0, 1, 2,...} of x. The orbit of x represents lects in a ring around the star, builds up, and eventually all future states of the system whose initial state is x. drips onto the surface. Considering the accretion disc as both a one dimensional and two dimensional “rail” Given a dynamical system f, the goal is to describe the divided into discrete cells, we will investigate chaos and global orbit structure of f. That is, what happens in the ergodicity, a characteristic of chaos, by considering an long run for most points x? In this paper, our goal is two- invariant subspace of the one dimensional system, and fold: first, starting with the DHR and based on existing time averages of various observables in both the one and astronomical observations, to find a more suitable dy- two dimensional cases. namical system model for the astrophysical phenomenon of interest, and second, to understand the asymptotic In this paper we will be defining the necessary terms behavior of orbits of that system. and results for understanding ergodicity and chaos, and To achieve that goal, it is useful to consider the behav- how it pertains to the DHR model. ior of the system on subsets of the state space which are left invariant under the system. Formally, a set A ⊂ M is called invariant under f if f(A)= A. This means that 1.2. Overview if the initial state of the system is in the set A, then ev- In the first section we will consider some background ery future state of the system will be in A. The simplest material about dynamical systems, the basic setup of the invariant sets are fixed and periodic points. DHR as well as the extended DHR (eDHR) proposed by (CAMCOS06), and the binary star system the eDHR is Definition 1.1. A point x ∈ M is called a fixed point of used to represent. the system if f(x) = x. The point x is called a periodic N n In the second section we will discuss ergodicity, its rela- point if there exists a n ∈ such that f (x) = x. The tion to the eDHR, and we will consider numerical analysis smallest such n is called the period. of possible ergodic behavior in the eDHR. 1.4. In the third section we will discuss chaos, invariant sub- The dripping handrail model spaces of the eDHR and the relation to chaos, and con- The dripping handrail model, denoted DHR, is a dis- sider numerical analysis of possible chaotic behavior in crete dynamical system in which a one dimensional the eDHR. handrail, usually considered circular, is divided into N In the fourth section we will consider the two dimensional discrete spatial steps, and time is taken to be discrete, extension of the eDHR (2DDHR), ergodicity within the n = 1, 2, .... The DHR models the change in density 2DDHR, and look at numerical evidence of ergodic be- across the cells through diffusion and accretion from an 2 outside source. It is assumed that each cell has a max- system. Although the eDHR model is very complicated imal density value and when that value is reached the to analyze with traditional methods, we can however, matter in the cell drains out or “drips.” still tell a great deal about the system it represents. By In the original DHR investigated by (SY96), the model considering the change in the whole system on average we can make some statistical, or measure-theoretic ω conjectures about the behavior of the system as a whole. In short, we would like to know if the eDHR system is ΓΓ ergodic. For the following material we follow (BS02).

2.1. 1 23 N Ergodicity Fig. 1.— The eDHR Intuitively, a dynamical system is ergodic if it is in a certain measure-theoretic sense indecomposable. Mix- ing, a property similar to but stronger than ergodicity, is was defined as follows: easier to explain: a mixing system “mixes” any two sets of initial conditions, therefore creating randomness out ρi i n 1. n = density in cell at time . of order. Formally speaking, ergodicity is defined as follows. 2. Xn = N × 1 density vector at time n. 3. Γ = diffusion parameter. Definition 2.1. A measure µ on M is said to be invariant with respect to f : M → M if for every measurable − 4. ω = constant accretion rate. set A ⊂ M, µ(f 1(A)) = µ(A). −1 Diffusion along the rail is represented by a differencing Here f (A) denotes the pre-image of A by f (not the by Γ. That is, after one time step: inverse of f, as f may not be bijective). One can think of a measure µ as a mass distributed in a certain way i i i−1 i i+1 i ρn+1 = ρn + Γ(ρn − ρn)+Γ(ρn − ρn)+ ω. (1) over the space M. If µ(M)= 1, µ is called a probability measure (the total mass is one). We are interested in (See Figure 1.) The extended DHR, or eDHR, proposed what the action of a dynamical system f does to the by (CAMCOS06) included the physical interpretation of distribution of this mass. If it preserves it (in the sense a density related accretion function: above), µ is called invariant and f is said to be a measure i i ωn = αρn + ω0, (2) preserving dynamical system. // Assume now that f preserves a probability measure µ. where α is the density related accretion parameter and Then: ω0 is a constant accretion rate. Hence, (1) becomes i i i−1 i+1 Definition 2.2. f is called ergodic if each invariant set ρn+1 = ρn(1 − 2Γ+ α)+Γρn + ρn + ω0. (3) has measure either zero or one. 1.5. An astronomical phenomenon That is, if f(A)= A, then µ(A) ∈{0, 1}. In particular, We use the eDHR to model the astronomical phenom- it is impossible to write the state space M as the disjoint ena of a binary star system. In a binary star system two union of two sets, A and B, such that both µ(A) and stars orbit each other while one star emits or accretes µ(B) are strictly between 0 and 1. If this were possible, matter towards the second star. This accreted matter we could decompose our dynamical system f in a non- collects into a ring around the star. Just as in the eDHR, trivial way into two decoupled systems: one on A and matter accumulates along the ring of matter, called the the other one on B. Arnold’s Cat map is an example of ergodic system; see accretion disc, builds up in each cell, then drips onto the 2 star. (See Figure 2.) Figure 3. This map takes the torus T to itself and is defined by the matrix Large star 2 1 A = . Accretion disc 1 1 Accreting matter Individual cells Since A preserves the integer lattice Z2, A can be thought 2 Small star of as a transformation of T . From Figure 3, we can see that after many iterations of an ergodic map, the figure Small star of a cat is unrecognizable. Accretion disc Dripping matter Another characterization of ergodicity is given by Fig. 2.— The binary system Birkhoff’s ergodic theorem. Suppose that φ : M → R is a continuous function. We can think of φ as a mea- surement of the state of the system, or, in the language 2. ERGODICITY AND THE EXTENDED DRIPPING of physics, an observable. For an initial state x, con- HANDRAIL MODEL sider the time average of the observable over the initial Ideally, we would like to know what happens to a segment of the orbit of x: single point in the system over time. This can be near impossible, however, due to the complicated nature φ(x)+ φ(f(x)) + ... + φ(f n−1(x)) A (x)= . of the setup and the possible chaotic behavior of the n n 3

Finally, we consider the geometric mean observable. That is, for each time step n

N φ(X )= N ρi . (8) n v n ui=1 uY t For our investigation of ergodicity we consider the N = 32 case, where Γ = 0.3333, and α = 0.001. In Figures 4, 5, 6, and 7 we plot the time average of the total density, quadratic mean, power mean, and geometric mean observables, respectively.

Time average; γ = 0.33333, and α = 0.1 18 Fig. 3.— Arnold’s cat map

17.5

∞ 17 It is natural to ask: do the time averages, {An(x)}n=1 converge to a limit? If so, how does that limit depend 16.5 on x and on φ? The answer is given by the following Density 16 Theorem 2.3 (Birkhoﬀ’s Ergodic Theorem). The dynamical system f : M → M is ergodic if and only if 15.5 R 15 for every continuous function φ : M → , there exists a 0 500 1000 1500 2000 2500 constant A(φ) such that for µ-almost every x ∈ X, Time

n−1 Fig. 4.— Time average of total density 1 lim φ(f i(x)) = A(φ). (4) n→∞ n i=0 X Time average; γ = 0.33333, and α = 0.1 It turns out that if f is ergodic, then A(φ) is just the 0.62 integral of the function φ over M relative to the measure 0.61 µ, which can be thought of as the space average of φ. In 0.6 other words, f is ergodic if and only if the time averages of φ converge to its space average. 0.59

Density 0.58 2.2. Ergodicity of the eDHR 0.57

Ergodicity creates randomness out of order, and that 0.56 is a characteristic of chaos. In order to prove that 0.55 0 500 1000 1500 2000 2500 the eDHR is chaotic we can ﬁrst investigate whether Time the system is ergodic, which is numerically simpler to do. Fig. 5.— Time average of quadratic mean We will choose four diﬀerent observables, φ, to investigate the time average of the system, in order to make

−3 γ α a strong claim of ergodic behavior. x 10 Time average; = 0.33333, and = 0.1 2.5 2.3. Numerical analysis and conclusions 2 The ﬁrst observable we consider is the total density along the rail, in other words, for each time step n 1.5 Density N 1 i φ(Xn)= ρn. (5) i=1 0.5 X

0 Next, we consider the quadratic mean observable. 0 500 1000 1500 2000 2500 That is, for each time step n Time Fig. 6.— Time average of power mean N 1 φ(X )= (ρi )2. (6) n vN n u i=1 We can see in each case the time average settles and u X t seems to converge to a limit. Hence we can conjecture Third, we consider an observable, let’s call it the power that, by Theorem 2.3, the eDHR shows ergodic behavior. mean, i.e., for each time step n N Next we will consider chaos directly by considering the 1 φ(X )= (ρi )i. (7) necessary conditions for a chaotic system as it pertains n N n i=1 to an invariant subset of the eDHR. X 4

Time average; γ = 0.33333, and α = 0.1 s 0.4 can associate a subset of M denoted by W (p), called the stable manifold of p. This set is characterized by the 0.38 property that for every q ∈ W s(p), the distance between n n 0.36 f (q) and f (p) tends to zero, as n → ∞. When f is invertible, we can similarly deﬁne the unstable manifold, 0.34 u Density W (p), by requiring that the above limit be zero as n s u 0.32 converges to −∞. If W (p) and W (p) intersect each other at a nonzero angle at some point q, we call q a 0.3 homoclinic point. It turns out that the existence of a

0 500 1000 1500 2000 2500 homoclinic point implies the presence of chaos in the Time system. This is, more or less, the content of the famous Fig. 7.— Time average of geometric mean Birkhoﬀ-Smale theorem.

3. CHAOS AND THE EXTENDED DRIPPING HANDRAIL From (CAMCOS06), there is at least one eigenvalue MODEL greater than one: Due to the appearance of possible ergodic behavior in λ =1+ α, (10) the eDHR, we would like to continue to investigate the corresponding to the eigenvector presence of chaos in the eDHR. 1 Considering the densities along the rail at time n 1 . as a N × 1 vector, Xn, the discrete dynamical sys- 1 =  .  tem f : HN → HN (HN is the unit hypercube, i.e. . N  1  H = {(x1, x2, ..., xN ) | 0 ≤ xi < 1}.) becomes     Xn+1 = f(Xn)= AXn + b (mod 1), (9) There are also eigenvalues, λ< 1. So the system exhibits both expansion and contraction. where, δ Γ 0 · · · 0 Γ Although we were not able to detect the presence of a Γ δ Γ · · · 0 0 homoclinic orbit in our system of interest, we note that A =  . . .  , ...... the presence of both expansion and contraction, and the  Γ 0 · · · 0 Γ δ  nonlinear nature of the system, is a good indication of the   possible chaotic behavior. In fact, we will give numerical   δ = 1 − 2Γ + α, and b is the N × 1 constant accretion evidence that the eDHR system is indeed chaotic on a vector 1-dimensional invariant subset of the state space. ω0 ω0 3.2. Chaos in the eDHR  .  . . The eigenvector 1 defines an invariant subspace ∆ of  ω  the system f. Since the system as a whole is complicated,  0    we focus on this invariant subset. We can think of ∆ as 3.1. Chaos the diagonal of HN . If we parameterize ∆ by t, then f We will now continue our investigation of chaos in the acts on ∆ in the following way: eDHR by considering the definition of a chaotic system directly. We follow (Devaney86). f(t1)= A(t1)+b (mod 1) = {(1+α)t+ω}1 (mod 1). Definition 3.1. A dynamical system is called chaotic if: Define

(a) The set of its periodic points is dense in M. g(t)=(1+ α)t + ω (mod 1). (b) It is transitive. That is, there exists a dense orbit. Then Recall that a set A is dense in M if for every open set f(t1)= g(t)1. U in M, A ∩ U is nonempty. Therefore, we can think of the restriction of f on ∆ as Intuitively, chaotic systems depend very sensitively on the map g : [0, 1) → [0, 1). More precisely, f and g are initial conditions, as in the famous “butterfly effect”: smoothly conjugate, so they have the same dynamical even though two initial states of the system may be properties. It is well known that for any m ∈ N, m> 1, extremely close, after a while the states will have nothing the map Em : [0, 1) → [0, 1) defined by to do with each other.

Em(x)= mx (mod 1) Suppose that p is a ﬁxed point of f, f(p) = p, and let A = Df(p) be the linearization (i.e., the total is chaotic. It is very likely that for similar reasons, for derivative) of f at p. If the eigenvalues of the matrix A most values of α and ω, g is chaotic, but we do not yet lie oﬀ the unit circle in the complex plane, p is called have a proof for this. We therefore consider the map g, hyperbolic. To the eigenvalues λ of A with |λ| < 1 we which is easier to analyze numerically. 5

Graph of g after 5 iterations; α = 0.1763, ω = 0.4057 Graph of g after 10 iterations; α = 0.1763, ω = 0.4057 3.3. Numerical analysis and conclusions 1 1 0.9 0.9

0.8 0.8 In Figures 8, and 9 we plot the graph of g, the ﬁfth 0.7 0.7 iteration, g5, and the tenth iteration, g10, respectively, 0.6 0.6 0.5 0.5 g(t) g(t) with α =0.41027 and ω0 =0.89365. We can see that as 0.4 0.4 0.3 0.3 the number of iterations increases the number of periodic 0.2 0.2 points and orbits of g become more dense. 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t t Graph of g after 1 iteration; α = 0.41027, ω = 0.89365 5 10 1 Fig. 11.— g and g 0.9

0.8 Graph of g after 20 iterations; α = 0.1763, ω = 0.4057 0.7 1

0.6 0.9 0.5 g(t) 0.8

0.4 0.7

0.3 0.6

0.2 0.5 g(t) 0.1 0.4 0 0 0.2 0.4 0.6 0.8 1 0.3 t 0.2

Fig. 8.— g(t)=(1+ α)t + ω0 (mod 1) 0.1 0 0 0.2 0.4 0.6 0.8 1 t

Fig. 12.— g20 Graph of g after 5 iterations; α = 0.41027, ω = 0.89365 Graph of g after 10 iterations; α = 0.41027, ω = 0.89365 1 1 0.9 0.9 We would like to reﬁne the extended dripping handrail 0.8 0.8

0.7 0.7 model to two spatial dimensions, (2DDHR). We consider 0.6 0.6 a two-dimensional disc with the star in the center. What 0.5 0.5 g(t) g(t) 0.4 0.4 was referred to in the original DHR model as the accre- 0.3 0.3 0.2 0.2 tion disc is the inner ring of the disc. We can divide the 0.1 0.1 two-dimensional disc into M rows and N columns. (See 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t t Figure 13.) Fig. 9.— g5 and g10 1 2 3 N 1 Similarly, in Figures 10, 11, and 12, we plot the graph 2 of g, the ﬁfth iteration, g5, the tenth iteration, g10, and the twentieth iteration, g20, with α = 0.1763 and ω0 = 0.4057. We can see that, although not all points M in [0, 1) × [0, 1) are hit by g, as the number of iterations increases the number of periodic points and orbits grow Fig. 13.— M × N accretion model more dense.

Graph of g after 1 iteration; α = 0.1763, ω = 0.4057 We use the following notation, similar to the DHR: 1 i,j 0.9 ρn = density in cell i, j at time n. 0.8 n =1, 2, 3, ... = discrete time steps. 0.7

0.6 i =1, 2, ..., M = vertical spatial dimension.

0.5 g(t) j =1, 2, ..., N = horizontal spatial dimension. 0.4 i,j 0.3 Xn = [ρn ] 0.2 = MN × 1 density vector at time n. 0.1 1,j 0 ω = initial accretion into cells ρ . 0 0.2 0.4 0.6 0.8 1 0 n t We have a similar setup for the 2DDHR. Horizontal Fig. 10.— g(t)=(1+ α)t + ω0 (mod 1) diﬀusion is given by β in the same manner that Γ acted on the eDHR.

So the invariant subspace of the eDHR given by 1, the Vertically we consider two cases: eigenvector corresponding to λ = 1+ α, shows numerical evidence of chaotic behavior, given by Definition 3.1. 1. Vertical diffusion given by γ. We can thus conjecture that the eDHR exhibits chaotic behavior. 2. Vertical effusion given by . (See Figures 14 and 15.) 4. THE TWO DIMENSIONAL DRIPPING HANDRAIL MODEL In the 2DDHR we allow dripping from the entire rail, 4.1. A similar setup taking every cell density to be between 0 and 1, but matter only accretes onto the outer edge of the rail, i = 1. 6

ω j+1 i,j i,j i,j−1 i,j+1 i−1,k i+1,k γ γ γ ρn+1 = ρn δ2+β(ρn +ρn )+ γ(ρn +ρn ), k j− =X 1 β β (16) where δ2 = δ1 − 3γ − α. γγ γ The vertical differencing by γ in cell M, j at time n is given by Fig. 14.— γ diffusion j+1 − γ(ρM 1,k − ρM,j). (17) ω n n k j− =X1 ε Combining (11) and (17), we get the change in density at cell M, j after one time step, β β j+1 M,j M,j M,j−1 M,j+1 M−1,k ε ρn+1 = ρn δ3 + β(ρn + ρn )+ γρn , k j− =X1 (18) Fig. 15.— effusion where δ3 = δ1 − α.

4.2. The 2DDHR with γ diffusion So, just as in the eDHR, we have a discrete dynamical In the first case we have vertical diffusion by γ. We system given by assume this diffusion is to be given by a differencing Xn+1 = F (Xn)= AXn + b (mod 1), (19) similar to the differencing by Γ in the eDHR. We must consider a few cases separately, in particular, the change such that in density along the “top” row, i = 1; the “middle” rows, 2 ≤ i ≤ M − 1; and the “bottom” row, i = M. D G 0 · · · 0 G D − (3γ + α)I G · · · 0 The horizontal differencing by β in cell i, j at time n   A = ...... , is given by . . .  0 · · · G D − (3γ + α)I G  i,j−1 i,j i,j+1 i,j   β(ρn − ρn )+ β(ρn − ρn ). (11)  0 · · · 0 G D − αI    The vertical differencing by γ in cell 1, j at time n is is anMN × MN matrix, where,  given by j+1 γ γ 0 0 · · · γ ,k ,j γ γ γ 0 · · · 0 γ(ρ2 − ρ1 ). (12) n n  . . .  k=j−1 G = ...... , X  γ γ γ  We must also consider the dynamic accretion into the  0 · · · 0   γ · · · 0 0 γ γ  top row, cells 1, j given by   and   αρ1,j ω . n + 0 (13) δ1 β 0 0 · · · β So, combining (11), (12), and (13), we have the change β δ1 β 0 · · · 0  . . .  in density in cell 1, j after one time step, D = ...... , j+1  0 · · · 0 β δ1 β  ,j −   1 1,j i,j 1 i,j+1 2,k  β · · · 0 0 β δ1  ρn+1 = ρn δ1+β(ρn +ρn )+ γρn +ω0, (14)   k j−   =X1 where δ1 =1 − 2β − 3γ + α. with dimension N × N, such that b is the MN × 1 constant accretion vector with ω0 in the first N entries, The vertical differencing by γ in cell i, j, for 2 ≤ i ≤ and zero elsewhere. M − 1, at time n is given by 4.3. The 2DDHR with effusion

j+1 j+1 In the second case we have vertical effusion, or a − γ(ρi 1,k − ρi,j )+ γ(ρi+1,k − ρi,j ). (15) gravity related diffusion, given by . We still allow n n n n β k=j−1 k=j−1 differencing by horizontally along the rail, but now X X gravity “pulls” the matter downward towards the So, combining (11) and (15), we have the change in accretion disc at a rate of . Again we must consider the density in cell i, j, for 2 ≤ i ≤ M − 1, after one time top, middle, and bottom rows separately. step, 7

Time average; γ = 0.33333, β = 0.1, and α = 0.1 In the top row of the rail, i = 1, the vertical eﬀusion 34 is given by 33 ρ1,j. − n (20) 32

Since horizontal diﬀusion is given by (11), and dynamic 31 accretion is given by (13), with (20), we have the change Density 30 , j in density in cell 1 after one time step, 29

1,j 1,j 1,j−1 1,j+1 28 ρn+1 = ρn (1−2β−+α)+βρn +βρn +ω0. (21) 27 0 1000 2000 3000 4000 5000 In the middle rows of the rail, 2 ≤ i ≤ M − 1, the Time vertical eﬀusion is given by Fig. 16.— Time average of total density with γ diﬀusion. i−1,j i,j ρn − ρn . (22)

Time average; γ = 0.33333, β = 0.1, and α = 0.1 Combining (11) and (22), we have the change in den- 18.6 sity in cell i, j, 2 ≤ i ≤ M − 1, after one time step, 18.4

i,j i,j i,j−1 i,j+1 i−1,j 18.2 ρn+1 = ρn (1−2β −)+βρn +βρn +ρn . (23) 18

In the bottom row of the rail, i = M, the vertical Density 17.8 eﬀusion is given by M−1,j 17.6 ρn . (24) 17.4

17.2 Combining (11) and (24), we have the change in den- 0 1000 2000 3000 4000 5000 sity in cell M, j after one time step, Time

M,j M,j M,j−1 M,j+1 M−1,j Fig. 17.— Time average of quadratic mean with γ diﬀusion. ρn+1 = ρn (1 − 2β)+ βρn + βρn + ρn . (25) Time average; γ = 0.33333, β = 0.1, and α = 0.1 So, just as before, we have a discrete dynamical system 1.8 given by, 1.6

1.4 Xn = F (Xn)= AXn + b (mod 1), (26) +1 1.2 where, 1

Density 0.8

D 0 0 · · · 0 0.6 E D − αI 0 · · · 0 0.4 A =  ......  , 0.2 . . . 0 0 2000 4000 6000 8000 10000   Time  0 · · · E D − αI 0   0 · · · 0 E D − (α − )I    Fig. 18.— Time average of power mean with γ diﬀusion.   is an MN × MN matrix with,

−274 γ β α x 10 Time average; = 0.33333, = 0.1, and = 0.1 δ β 0 · · · β 1 0.9

β δ β · · · 0 0.8

 . . .  0.7 D = ...... , 0.6  0 · · · β δ β    0.5 Density  β · · · 0 β δ  0.4     0.3 0.2 of dimension N × N, with δ =1 − 2β − , and E = I. 0.1 0 0 2 4 6 8 10 Also, b is deﬁned as before. 4 Time x 10

4.4. Erdogicity in the 2DDHR Fig. 19.— Time average of geometric mean with γ diffusion. In order to determine possible convergence in the 4.5. Numerical analysis and conclusions 2DDHR model, we will consider, just as with the eDHR, the time average of a given observable, φ. We will The first observable we consider is the total density consider both cases of diffusion; differencing given by γ, along the rail. That is, for each time step n and effusion given by . M N i,j φ(Xn)= ρ . (27) We consider four different observables for our investi- n i=1 j=1 gation of ergodicity in the 2DDHR in order to make a X X strong claim of ergodic behavior. Next, we consider the quadratic mean observable. 8

Time average; ε = 0.33333, β = 0.1, and α = 0.01 That is, for each time step n 21

20.8 M N 20.6 1 i,j 2 φ(Xn)= (ρn ) . (28) vMN 20.4 u i=1 j=1 20.2 u X X Third, we considert the observable we call the power Density 20 mean, i.e., for each time step n 19.8 19.6 MN 1 i,j k 19.4 φ(Xn)= (ρn ) , (29) MN 0 1000 2000 3000 4000 5000 k Time X=1 where the order of the densities runs along the rows Fig. 21.— Time average of quadratic mean with eﬀusion. ﬁrst, i.e., i = 1 and j =1,...,N, etc.

−3 ε β α x 10 Time average; = 0.33333, = 0.1, and = 0.01 Finally, we consider the geometric mean observable. 7 That is, for each time step n 6 i,j φ(Xn)= MN ρn (30) 5 i,j 4 sY Density For the first case of vertical diffusion, differencing 3 given by γ, we consider a specific case of the 2DDHR, 2 the 2 × 32 case where γ =0.3333, β =0.1, and α =0.01. 1 In Figures 16, 17, 18, and 19 we plot the time average of 100 200 300 400 500 600 700 800 the the total density, quadratic mean, power mean, and Time geometric mean, respectively. Fig. 22.— Time average of power mean with effusion. We can see that in each case, the time average of

−256 ε β α φ appears to settle and converge to a limit. So, by x 10 Time average; = 0.33333, = 0.1, and = 0.01 2

Theorem 2.3, we can make a strong conjecture that the 1.8

2DDHR with vertical diﬀerencing given by γ exhibits 1.6 ergodic behavior. 1.4

1.2 For the second case of vertical diffusion, effusion 1 Density given by , we will also consider the 2 × 32 case, with 0.8 γ = 0.3333, β = 0.1, and α = 0.01. In Figures 20, 21, 0.6 22, and 23, we plot the time average of the total density 0.4 0.2 of the rail, quadratic mean, power mean, and geometric 0 0 2 4 6 8 10 4 mean of the rail, respectively. Time x 10 Fig. 23.— Time average of geometric mean with effusion.

Time average; ε = 0.33333, β = 0.1, and α = 0.01 35.6

35.4 used to represent a binary star system over time, as

35.2 investigated by (SY96). In a binary system, matter 35 accretes oﬀ of one star onto the other, collecting into 34.8 a ring of matter, called an accretion disc. The matter 34.6

Density accumulates, condenses, and eventually drips from the 34.4 accretion disc onto the star. 34.2

33.8 In the original DHR the accretion disc is represented

33.6 by a one dimensional rail divided in to N discrete cells, 0 1000 2000 3000 4000 5000 Time and n time steps, where matter diffuses across the cells, Fig. 20.— Time average of total density with effusion. represented by a differencing at a rate of Γ. In the extended DHR, eDHR, proposed by (CAMCOS06), which we investigated and extended, a density related accre- We can see that in each case, the time average of φ tion variable was added. The eDHR model is a matrix appears to settle and converge to a limit. So we can equation such that for all density vectors Xn, again make a strong conjecture that the 2DDHR with effusion exhibits ergodic behavior. Xn+1 = f(Xn)= AXn + b (mod 1). (31)

5. FINAL REMARKS We used this model to investigate ergodic behavior through time averages of different observables. Next, 5.1. What we discussed as another method of checking for chaotic behavior, we In this paper we discussed the Dripping Handrail investigated the action of f on an invariant subspace Model, or DHR. The DHR is a mathematical model of the system, namely ∆, the diagonal of the unit 9 hypercube. average of the four observables, φ, the total density, the quadratic mean, the power mean, and the geometric Finally, we extended the eDHR to two spatial dimen- mean of the rail in both cases of diffusion. Specifically, sions, the rail now represented by a two dimensional grid we considered the 2 × 32 case with γ = = 0.3333, with M rows and N columns. Horizontal diffusion was β =0.1, and α =0.01. represented as differencing by β, similar to the eDHR, and we considered two cases of vertical diffusion. The Through convergence of the time averages of all four first case was vertical differencing by γ, and the second observables we found numerical evidence of ergodic case was vertical effusion, or a gravity related diffusion, behavior. Just as with the eDHR, we made a claim that by . Again, this gave a matrix equation such that for the 2DDHR exhibits possible ergodic behavior. all density vectors Xn, Since all analysis on the eDHR and 2DDHR was Xn+1 = F (Xn)= AXn + b (mod 1). (32) done numerically, we cannot make a rigorous state- Similar to the eDHR, we used this model to investi- ment of ergodicity or chaos, only conjectures. This, gate ergodic behavior through time averages of different however, leads to many open problems left by this paper. observables.

5.2. What we found 5.3. Further questions and research In our investigation of ergodic behavior in the eDHR, In each investigation of chaos we could not make a rig- we plotted the time average of four diﬀerent observables, orous and concise claim for any of our ﬁndings because φ, the total density of the rail, the quadratic mean, the our evidence was strictly numerical. This leads to a nat- power mean, and the geometric mean of the rail. Specif- ural open problem. Below is a list of open problems and ically, we considered the N = 32 case with Γ = 0.3333, conjectures that can be investigated based on the results and α =0.001. of this paper.

Through convergence of the time averages in each case 1. Rigorously prove eDHR is chaotic by proving f act- of φ we found numerical evidence of ergodic behavior. ing on the invariant subspace of ∆ is chaotic. Because we found convergence in all four observables we made a conjecture that the eDHR shows possible 2. Consider other cases for the 2DDHR model such ergodic behavior. as:

Second, we considered the action of f on the (a) Effusion allowed in “diagonal” directions. eigenvector 1, considered the diagonal ∆, which we (b) Higher order cases, M × N. parameterized by t, as the function g(t)=(1+ α)t + ω0 mod (1). We plotted many iterates of g for different 3. Rigorously prove ergodicity in both the eDHR and values of α and ω0. We determined numerically in each 2DDHR models. case that the periodic points and orbits of g are dense. So we can claim that the eDHR exhibits chaotic behavior. 4. Investigate chaos rigorously in the 2DDHR by find- Finally, we checked for ergodic behavior in the two ing and investigating the invariant subspaces of F . dimensional 2DDHR model. We considered the time

REFERENCES Scargle, J.D., and Young, K., The Dripping Handrail Model: Devaney, R., An Introduction to Chaotic Dynamical Systems, Transient Chaos in Accretion Systems, The Astrophysical Benajmin Cummings,1986. Journal, 468, 1996, 617–632. Hirsch, M.W., and Smale, S., Devaney, R.L., Diﬀerential Dey, A., Low, M., Rensi, E., Tan, E., Thorsen, J., Vartanian, M., Equations, Dynamical Systems, and an Introduction to Chaos, and Wu, W., The Dripping Handrail: An Atrophysical Accretion Elsevier/Academic Press, 2 ed., 2004. 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.

6. GRATUITOUS APPENDIX We would like to thank Dr. Jeﬀrey Scargle for his assistance and introducing us to this problem, Dr. Slobodan Simi´cfor advising us, and Dr. Tim Hsu for giving us the opportunity.