Presented by

Arkajit Dey, Matthew Low, Efrem Rensi, Eric Prawira Tan, Jason Thorsen, Michael Vartanian, Weitao Wu.

• I nt roduct ion• Transient Chaos• Simulat ion and Anim at ion• Return Map I• Return Map I I• Modified DHR Model• Fixed Points• Recap• Acknowledgem ent

Art ist ’s View of Neut ron Star (L) Accret ing Mat ter From Com panion Star (R)

Under such ext rem e condit ions, standard models break down, so ...

• Constant accretion into cells

• Diffusion from neighbors

• Cell “drips” when full

• Result: chaos

Original Model with Recent Observat ions

• Miller & Lam b “Effect of Radiat ion Forces on Accret ion”

• Outward radiat ion force causes t im e-varyingaccret ion

• Radiat ion drag force causes asym m etricdiffusion

Extended Model with Recent Observat ions

• Original model accounts for chaos and low- frequency oscillat ions in recent observat ions

• Our extended m odel m ay help explain high- frequency oscillat ions as well

• Scargle & Young: original model displays chaos only for limited (“transient”) times

• How does the power spectrum of our extended model evolve over long periods?

Chaot ic init ial spect rum at t = 1

Non-chaot ic Periodic spect rum at t = 50

Chaot ic spect rum with high- frequency oscillat ions at t = 1

Unchanged spect rum at t = 50

• “Transient Chaos” in the original model : Significant change in the power spect rum over a period of t im e

• “Perm anent Chaos” in the extended model: The power spect rum stays the sam e indefinitely - advantage

I nner edge of disk represented as cells,

Each cell having a state. “Density”

• Cells accrete m ass (state values increse)

• Diffusion occurs between cells

• Cell density resets at a threshold value

• “Return m ap” is a m isnom er.• Com pare m ass at a part icular t im e xn to

the m ass at a future t im e xn+k

– xn vs. xn+k

• Return m ap I : – Random init ial condit ions– n and k both fixed

• Return m ap I I : – Sam e init ial condit ion– n varies, k fixed.

• Mass at a certain t im e vs. one t im e step later• We don’t expect m uch change

• Variabilit y increases as t ime m oves forward

• Bands form in the lower- r ight -hand corner• Mass appears to “discret ize”

• Higher accret ion rate• Pat tern repeats itself once

• Where the dots are m ore concent rated, the cell’s mass is more likely to be “ located” in that area.

• After enough t im e, the m ass in a cell becom es “discret ized” , i.e., can only take on one of finitely m any values

• I t would be interest ing to exam ine raw ast ronom ical data to confirm these observat ions.

• Single cell’s m ass at t im e n vs. at t im e n+ 5• Going through cycles with sm all shifts

• Total m ass of the cells at t ime n vs. at t im e n+ 1.

• Showing fractals

• Adding onto Young & Scargle’s DHR m odel, we have the following discrete dynam ical system . The t im e variable is discrete.

–

–

–

• I n the extended m odel we added a constant > 0 to m odel dynam ic accret ion. Then the m odified m at r ix, A, is as shown above.

1 ( )n nX f X

: N Nf H H

( )f X AX b

• Each vector X has n coordinates all with values between 0 and 1 ( i.e. ) that is the density of the corresponding cell.

• One of the first ways to invest igate a dynam ical system is by finding eigenvalues. Adding the constant m akes the m odified eigenvalues . This guarantees that at least one eigenvalue is greater than 1 cont r ibut ing to perm anent chaos.

• The m odified m at r ix has the sam e eigenvectors as the or iginal m at r ix does.

NX H

NX H

i i

• A fixed point will sat isfy:

• The solut ion is:

I f m is an integer and every com ponent has value between 0 and 1. I f there is no t im e-varying accret ion, fixed points do not exist .

• Our extended model shows promiseof explaining recent observat ions

• Our visualizat ion and return map studiesgive valuable new ways of extracting info

• Our abstract study has given a deeperunderstanding of the underlyingdynam ics

Dr. J. Scargle (NASA)

Dr. S. Sim ic (SJSU Math)

The Woodward Fund