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
• 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
• 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?
• “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
• 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.
• 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.
• 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