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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:

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Dynamical Systems and Chaos

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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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