From Determinism to Stochasticity Reading for this lecture: (These) Lecture Notes. Outline of next few lectures: Probability theory Stochastic processes Measurement theory
You Are Here A B C... α γ β δ System Instrument Process Modeller The Learning Channel 2
A B C... α γ β δ System Instrument Process Modeller The Learning Channel 2
You Will Soon Be Here A B C... α γ β δ System Instrument Process Modeller The Learning Channel 2
Probability Theory of Dynamical Systems: Probability Theory Review: Discrete Random Variable (RV): X Events (Alphabet): X = {, 2,...,k} Realization: x X Probability mass function ( distribution ): Pr(x) = Pr{X = x} apple Pr(x) apple, x 2X Normalized: Pr(x) = x X 3
Probability Theory of Dynamical Systems: Probability Theory Review... Discrete random variables:. Biased coin: X = {H, T} Pr(H) =/3 Pr(T )=2/3 2. Sequence: No pairs of s X = {,,,,,,, } Pr(s 3 )=,, 3 otherwise 4
Probability Theory of Dynamical Systems: Probability Theory Review... Continuous Random Variable: X Takes values over continuous event space: X Cumulative distribution function: P (x) = Pr(X x) apple Pr(x) apple, x 2X If continuous, then random variable is. Probability density function: p(x) =P (x) apple p(x), x 2X p(x)dx =Pr(X<x+ dx) Normalization: (e.g., if x 2 R) Pr(X < ) = or Pr(X <x) dx p(x) = Support of distribution: suppx = {x : p(x) > } 5
Probability Theory of Dynamical Systems: Probability Theory Review... Continuous random variable X: Uniform distribution on interval: X = R { Density: p(x) =, x, otherwise x< Distribution: Pr(x) = x x x> Support: supp X = [, ]
Probability Theory of Dynamical Systems: Probability Theory Review... Continuous random variable X: Gaussian: Density: X = R p(x) = (x µ) 2 σ 2π e 2σ 2 Distribution: P (x) = x dy p(y) Erf(x) Support: supp X = R 7
Probability Theory of Dynamical Systems: Probability Theory Review... Discrete RVs: X over X & Y over Y Joint distribution: Pr(X, Y ) Marginal distributions: Pr(X) = y Y Pr(X, y) Pr(Y )= x X Pr(x, Y ) 8
Probability Theory of Dynamical Systems: Probability Theory Review... Factor joint distribution: Pr(X, Y ) = Pr(X Y )Pr(Y ) Pr(X, Y ) = Pr(Y X)Pr(X) Conditional distributions: Pr(Y X) = Pr(X, Y ) Pr(X) Pr(X Y )= Pr(X, Y ) Pr(Y ), Pr(X) =, Pr(Y ) = 9
Probability Theory of Dynamical Systems: Probability Theory Review... Statistical independence: X Y Pr(X, Y ) = Pr(X)Pr(Y ) Conditional independence ( shielding ): X Z Y Pr(X, Y Z) = Pr(X Z)Pr(Y Z)
Dynamical Evolution of Distributions: Dynamical system: {X, T} State density: p(x) x X Can evolve individual states and sets: T : x x Initial density: p (x) E.g., model of measuring a system Evolve a density? p (x) T p (x)
Dynamical Evolution of Distributions... Conservation of probability: p (y)dy = p (x)dx y = T (x) Perron-Frobenius Operator: Locally: y = T (x) p n+ (y) = p n(x) T (x) y + dy y p (y)dy Globally: p n+ (y) = x T (y) p n (x) T (x) x p (x)dx x + dx 2
Dynamical Evolution of Distributions... Frobenius-Perron Equation: p n+ (y) = dx p n (x)δ(y T(x)) Dirac delta-function: {, x = δ(x) =, x dx δ(x c)f(x) =f(c) dx (x) = y p(y) p(y) y = T (x) p(x) x 3
Dynamical Evolution of Distributions... Example: Delta function initial distribution Map: x n+ = f(x n ) Initial condition: x R Initial distribution: p (x) =δ(x x ) p (y) = dx p (x) (y f(x)) = dx (x x ) (y f(x)) = (y f(x )) = (y x ). p n (y) = (y x n )... reduces to an orbit 4
Dynamical Evolution of Distributions... Delta function IC: The easy case and expected result. What happens when the IC has finite support? p (x) = { 2, x /3.25, otherwise Consider a set of increasingly more complicated systems and how they evolve distributions... 5
Dynamical Evolution of Distributions... t = t = t = 2 Example: Linear circle map x n+ =.+x n (mod ) t = 3 t = 4 t = 5 x n+ t = t = 7 t = 8 x n f (x) = x x x
Dynamical Evolution of Distributions... t = t = t = 2 Example: Shift map t = 3 t = 4 t = 5 x n+ x n t = t = 7 t = 8 Spreading: f (x) =2 x x x 7
Dynamical Evolution of Distributions... t = t = t = 2 Example: Tent map a = 2. t = 3 t = 4 t = 5 x n+ x n t = t = 7 t = 8 Spreading: f (x) =2 x x x 8
Dynamical Evolution of Distributions... t = t = t = 2 Example: Logistic map r = 4 x n+ t = 3 t = 4 t = 5 x n t = t = 7 t = 8 f (x) =4( 2x) Spreading: x<3/8 orx>5/8 Contraction: 3/8 <x<5/8 x x x 9
Dynamical Evolution of Distributions... Example: Logistic map r = 3.7 t = t = t = 2 t = 3 t = 4 t = 5 x n+ x n t = 8 t = 2 t = 2 Peaks in distribution are images of maximum x x x 2
Time-asymptotic distribution: What we observe How to characterize? Invariant measure: A distribution that maps onto itself Analog of invariant sets Stable invariant measures: Stable in what sense? Robust to noise or parameters or??? 2
Invariant measures for D Maps: Probability distribution (density p (x)) that is invariant:. Distribution s support must be an invariant set: = f( ), = supp p (x) ={x : p (x) > } I. Probabilities invariant : Distribution a fixed point of Frobenius-Perron Equation p (y) = dx p (x) (y f(x)) Functional equation: Find p ( ) that satisfies this. 22
Example: Periodic-k orbit {x,x 2,...,x k } p(x) =δ p (y) = dx p(x)δ(y f(x)) ( k ) = dx δ (x x i ) δ(y f(x)) i= ( k ) = δ (y f(x i )) i= ( k ) = δ (y x (i+)mod k ) i= ( k ) = δ (y x i ) Is it invariant? i= Yes! has density ( k ) (x x i ) i= 23
Example: Shift map invariant distribution x n+ Uniform distribution: p(x) =,x [, ] x n p (y) p (y) p (y) p (y) /2 /2 y /2 y y /2 y p (y) =p (y)+p (y) x p (x) =p (x) (y x) 24
Example: Shift map invariant distribution x n+ Uniform distribution: p(x) =,x [, ] x n Via Frobenius-Perron Equation: Two cases A: x /2 B: /2 <x p (y) = = 2 2 dx p (x) (y dx (y 2x) f(x)) p (y) = = 2 2 dx p (x) (y dx (y 2x) f(x)) = 2 = 2 p (y) = p (y)+p (y) = p (x) (y x) 25
( ax n, apple x n apple Example: Tent map x n+ = 2 a( x n ), Fully two-onto-one: a =2 2 <x n apple Uniform distribution is invariant: p(x) =,x [, ] Proof from FP Equation: Two cases First case: exactly that of shift map x n+ Second case: slope is all that s important /2 <x p (y) = Z 2 Z = 2 dx p (x) (y f(x)) dx (y (2 2x)) = 2 x n 2
Example: Tent map where two bands merge to one: Invariant distribution: p, x max = a/2 p(x) = x min = a( a/2) x = a/( + a) Equal areas: x min x x p, x <x x max, otherwise p(x) p (x x min )=p (x max x ) a = 2 p p x min 2 x x max Normalization: p (x x min )+p (x max x )= p = 2(x x min ) p = 2(x max x ) 27
Example: Logistic map x n+ = rx n ( x n ) Fully two-onto-one: r =4 Invariant distribution? p(x) = Exercise. π x( x) 28
Numerical Example: Tent map Typical chaotic parameter: a =.75 Two bands merge to one: a = 2 29
Numerical Example: Tent map Typical chaotic parameter: a =.75 Two bands merge to one: a = 2 x 29
Numerical Example: Tent map Typical chaotic parameter: a =.75 Two bands merge to one: a = 2 x x 29
Numerical Example: Logistic map x n+ = rx n ( x n ) Typical chaotic parameter: r =3.7 Two bands merge to one: r =3.785735428329 3
Numerical Example: Logistic map x n+ = rx n ( x n ) Typical chaotic parameter: r =3.7 Two bands merge to one: r =3.785735428329 x 3
Numerical Example: Logistic map x n+ = rx n ( x n ) Typical chaotic parameter: r =3.7 Two bands merge to one: r =3.785735428329 x x 3
Numerical Example: Cusp map (a, b) =(, /2) x n+ = a( 2x n b ) x n+ x n 3
Numerical Example: Cusp map (a, b) =(, /2) x n+ = a( 2x n b ) x n+ x n x 3
Issue: Many invariant measures in chaos: An infinite number of unstable periodic orbits: Each has one. But none of these are what one sees, one sees the aperiodic orbits. How to exclude periodic orbit measures? Add noise and take noise level to zero; which measures are left? Robust invariant measures. 32
Reading for next lecture: Lecture Notes. 33