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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:

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Dynamical Systems and Chaos

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Dynamical systems and chaos l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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


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