dynamical systems and chaos
Download
Skip this Video
Download Presentation
Dynamical Systems and Chaos

Loading in 2 Seconds...

play fullscreen
1 / 7

Dynamical Systems and Chaos - PowerPoint PPT Presentation


  • 117 Views
  • Uploaded on

Dynamical Systems and Chaos. Low Dimensional Dynamical Systems Bifurcation Theory Saddle-Node, Intermittency, Pitchfork, Hopf Normal Forms = Universality Classes Feigenbaum Period Doubling Transition from Quasiperiodicity to Chaos: Circle Maps Breakdown of the Last KAM Torus:

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Dynamical Systems and Chaos' - india


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
dynamical systems and chaos

Dynamical Systems and Chaos

  • Low Dimensional Dynamical Systems
  • Bifurcation Theory
    • Saddle-Node, Intermittency, Pitchfork, Hopf
    • Normal Forms = Universality Classes
  • Feigenbaum Period Doubling
  • Transition from Quasiperiodicity to Chaos:
  • Circle Maps
  • Breakdown of the Last KAM Torus:
  • Synchrotrons and the Solar System

Coarse-Graining in Time

Feigenbaum Period Doubling

Attractor vs. l

Onset of Chaos = Fractal

percolation

Percolation

  • Connectivity Transition
  • Punch Holes at Random,
  • Probability 1-P
  • Pc =1/2 Falls Apart
  • (2D, Square Lattice, Bond)
  • Static (Quenched) Disorder

Structure on All Scales

Largest Connected Cluster

P=Pc

P=0.49

P=0.51

self similarity

Self-Similarity

Self-similarity → Power Laws

Expand rulers by B=(1+e);

Up-spin cluster size S, probability distribution D(S)

D[S] = A D[S’/C]

=(1+ae) D[(1+ce) S’)]

a D = -cS’ dD/dS

D[S] = D0 S-a/c

Self-Universality on Different Scales

Random Walks

Ising Model at Tc

Universal critical exponentsc=df=1/sn, a/c=t : D0 system dependent

Ising Correlation C(x) ~ x-(d-2+h)at Tc,random walk x~t1/2

fractal dimension d f

Fractal Dimension Df

Logistic map:

Fractal at critical point

Cantor Set

Middle third

Base 3 without 1’s

Df = log(2)/log(3)

Mass ~ SizeDf

Percolation

Random Walk: generic scale invariance

critical point

universality shared critical behavior

Universality: Shared Critical Behavior

Same critical exponent b=0.332!

Ising Model and Liquid-Gas Critical Point

Liquid-Gas Critical Point

r-rc ~ (Tc-T)b

r Ar(T) = A r CO(BT)

Ising Critical Point

M(T) ~ (Tc-T)b

r Ar(T) = A(M(BT),T)

Universality: Same Behavior up to Change in Coordinates

A(M,T) = a1 M+ a2 + a3T

Nonanalytic behavior at critical point (not parabolic top)

All power-law singularities (c, cv,x) are shared by magnets, liquid/gas

the renormalization group

The Renormalization Group

  • Renormalization Group
  • Not a group
  • Renormalized parameters
  • (electron charge from QED)
  • Effect of coarse-graining
  • (shrink system, remove
  • short length DOF)
  • Fixed point S* self-similar
  • (coarse-grains to self)
  • Critical points flow to S*
  • Universality
  • Many methods (technical)
  • real-space, e-expansion, Monte Carlo, …
  • Critical exponents from
  • linearization near fixed point

Why Universal? Fixed Point under Coarse Graining

System Space Flows

Under Coarse-Graining

renormalization group

Dynamics = Map

f(x) = 4m x(1-x)

Renormalization Group

m

Coarse-Graining in Time

Renormalization Group

xn = f(xn-1)

x0,x1, x2,x3, x4,x5,…

New map: xn = f(f(xn-2))

x0,x2, x4,x6, x8,x10,…

Decimation by two!

Universality

fsin(x) = B sin(p x)

~B

ad