1. Dynamical Systems and Chaos Coarse-Graining in Time 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 Feigenbaum Period Doubling Attractor vs. Onset of Chaos = Fractal

2. Percolation Structure on All Scales Connectivity Transition Punch Holes at Random, Probability 1-P P c =1/2 Falls Apart (2D, Square Lattice, Bond) Static (Quenched) Disorder Largest Connected Cluster P=P c P=0.51P=0.49

3. Self-Similarity Self-Universality on Different Scales Ising Model at T c Random Walks Self-similarity → Power Laws Expand rulers by B=(1+  ); Up-spin cluster size S, probability distribution D(S) D[S] = A D[S’/C] =(1+a  ) D[(1+c  S’)] a D = -cS’ dD/dS D[S] = D 0 S -a/c Universal critical exponents c=d f =1/ , a/c=  : D 0 system dependent Ising Correlation C(x) ~ x -(d-   at T c  random walk x~t 1/2

4. Fractal Dimension D f Mass ~ Size D f Percolation Cantor Set Middle third Base 3 without 1’s D f = log(2)/log(3) Logistic map: Fractal at critical point Random Walk: generic scale invariance critical point

5. Universality: Shared Critical Behavior Ising Model and Liquid-Gas Critical Point Liquid-Gas Critical Point  c ~ (T c -T)   Ar (T) = A  CO (BT) Ising Critical Point M(T) ~ (T c -T)   Ar (T) = A(M(BT),T) Same critical exponent  ! Universality: Same Behavior up to Change in Coordinates A(M,T) = a 1 M+ a 2 + a 3 T Nonanalytic behavior at critical point (not parabolic top) All power-law singularities (  c v,  are shared by magnets, liquid/gas

6. The Renormalization Group Why Universal? Fixed Point under Coarse Graining System Space Flows Under Coarse-Graining 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,  -expansion, Monte Carlo, … Critical exponents from linearization near fixed point

7. Renormalization Group Coarse-Graining in Time Renormalization Group x n = f(x n-1 ) x 0, x 1, x 2, x 3, x 4, x 5,… New map: x n = f(f(x n-2 )) x 0, x 2, x 4, x 6, x 8, x 10,… Decimation by two! Universality f sin (x) = B sin(  x)  ~B  Dynamics = Map f(x) = 4  x(1-x)