Deterministic Fractals

1

Seyed Mohammad Sadegh Movahed !

Department of Physics-Shahid Beheshti University, Tehran IRAN Computational Cosmology Group (CCG-SBU)

School of Physics (IPM), Tehran IRAN www.smovahed.ir

http://faculties.sbu.ac.ir/~movahed !

ObservationsTheoretical Models Measures

Simulations Mak

e ne

w stra

tegy

for

obse

rvat

ion

Data modelling

My road map Model - Observation & Experiments - Statistical analysis

Make new strategy for

theory

Classification of processes

i

x(i

)

250 500 750 1000

-5

0

5

10

15

• Deterministic processes

• Chaotic processes

• Stochastic Processes1) Purely Random processes2) Dependent processes 3) Markov processes

r

Xn

0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xn+1 = 4rxn (1− xn )x(t + Δt) = x(t) − γ x(t)Δt +η(t) Δt

A brief History on Fractals!

~1700 A.D. Gottfried leibniz

~1872 A.D. Karl Weierstrass

~1904 A.D. Helge von Koch

~1915 A.D. Waclaw Seirpinski

~1951 A.D. H. E. Hurst

~1968 A.D. B. B. Mandelbrot

Cloud structure

http://www.brockmann-consult.de/

Natural time seriesGeophysics time series: temperature, precipitation, water runoff, seismic events, climate dynamics and so on.

Medical and physiological time series: Heartbeat, blood pressure, glucose level, gene expression data and so on.

Astrophysical time series: X-ray and cosmic ray, sunspot, CMB (actually not time series),....

Social and technical time series: Traffic, internet, Finance, language characteristics, chemistry a petroleum, ....

Physics data: surface roughness, spectroscopy, ....

Fractional Gaussian Noise

Fractional Brownian Motion

می خواهیم یاد بگیریم که چگونه می توان چنین اشکالی ترسیم کرد؟

See Computational Physics, Rubin H. Landau et. al., 2011

با تشکر

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