Introduction Transient Chaos Simulation and Animation Return - - PowerPoint PPT Presentation

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

• Introduction
• Transient Chaos
• Simulation and Animation
• Return Map I
• Return Map II
• Modified DHR Model
• Fixed Points
• Recap
• Acknowledgement

Artist’s View of Neutron Star (L) Accreting Matter From Companion Star (R)

Under such extreme conditions, standard models break down, so ...

• Constant

accretion into cells

• Diffusion from

neighbors

• Cell “drips” when

full

• Result: chaos

Original Model with Recent Observations

• Miller & Lamb “Effect of

Radiation Forces on Accretion”

• Outward radiation force

causes time-varying accretion

• Radiation drag force

causes asymmetric diffusion

Extended Model with Recent Observations

• Original model accounts for chaos

and low-frequency oscillations in recent observations

• Our extended model may help

explain high-frequency oscillations 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?

Chaotic initial spectrum at t = 1 Non-chaotic Periodic spectrum at t = 50

Chaotic spectrum with high-frequency

• scillations at t = 1

Unchanged spectrum at t = 50

• “Transient Chaos” in the original

model : Significant change in the power spectrum over a period of time

• “Permanent Chaos” in the extended

model: The power spectrum stays the same indefinitely - advantage

Inner edge of disk represented as cells, Each cell having a state. “Density”

• Cells accrete mass (state values increse)
• Diffusion occurs between cells
• Cell density resets at a threshold value

• “Return map” is a misnomer.
• Compare mass at a particular time xn to

the mass at a future time xn+k – xn vs. xn+k

• Return map I:

– Random initial conditions – n and k both fixed

• Return map II:

– Same initial condition – n varies, k fixed.

• Mass at a certain time vs. one time step later
• We don’t expect much change

• Variability increases as time moves forward

• Bands form in the lower-right-hand corner
• Mass appears to “discretize”

• Higher accretion rate
• Pattern repeats itself once

• Where the dots are more

concentrated, the cell’s mass is more likely to be “located” in that area.

• After enough time, the mass in a cell

becomes “discretized”, i.e., can only take on one of finitely many values

• It would be interesting to examine

raw astronomical data to confirm these observations.

• Single cell’s mass at time n vs. at time n+ 5
• Going through cycles with small shifts

• Total mass of the cells at time n vs. at time

n+ 1.

• Showing fractals

• Adding onto Young & Scargle’s DHR model, we have

the following discrete dynamical system. The time variable is discrete.

– – –

• In the extended model we added a constant

> 0 to model dynamic accretion. Then the modified matrix, A, is as shown above.

1

( )

n n

X f X :

N N

f 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 investigate a

dynamical system is by finding

• eigenvalues. Adding the constant

makes the modified eigenvalues . This guarantees that at least one eigenvalue is greater than 1 contributing to permanent chaos.

• The modified matrix has the same

eigenvectors as the original matrix does.

N

X H

N

X H

i i

• A fixed point will satisfy:
• The solution is:

If m is an integer and every component has value between 0 and 1. If there is no time- varying accretion, fixed points do not exist.

• Our extended model shows promise
• f explaining recent observations
• Our visualization and return map studies

give valuable new ways of extracting info

• Our abstract study has given a deeper

understanding of the underlying dynamics

• Dr. J. Scargle

(NASA)

• Dr. S. Simic

(SJSU Math) The Woodward Fund