From Determinism to Stochasticity

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1 From Determinism to Stochasticity Reading for this lecture: (These) Lecture Notes. Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2

2 Cave: Sensory Immersive Visualization A video or two: Cave Dynamics Protein Manipulation Tour: This Thursday (27 May) Two Sessions: 4:-5: PM & 5: - : PM Where: Rm Earth & Physical Sciences Bldg Top floor, northwest corner. Monday, May 24, 2 2

3 Cave Tour Sign-Up Thursday 27 May Balanced Numbers Session One: 4:-5: PM Session Two: 5:-: PM Monday, May 24, 2 3

4 One-D Map Tool Example Code Now on course website See Homework Page Monday, May 24, 2 4

5 Probability Theory of Dynamical Systems: The Big Deal: Deterministic systems can be: Unpredictable Noisy Random Chaotic What is the role of probability? Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 5

6 Probability Theory of Dynamical Systems: Probability Theory Review: Continuous Random Variable: X Takes values over continuous space: X Cumulative distribution function: P (x) = Pr(X x) If continuous, then random variable is. Probability density function: p(x) =P (x) Normalized: dx p(x) = or Pr(X < ) = Support of distribution: suppx = {x : p(x) > } Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2

7 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 = [, ] Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 7

8 Probability Theory of Dynamical Systems: Probability Theory Review... Continuous random variable X: Gaussian: X = R Density: p(x) = (x µ) 2 σ 2π e 2σ 2 Distribution: P (x) = x dy p(y) Erf(x) Support: supp X = R Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 8

9 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) Model of measuring a system Evolve a density? p (x) T p (x) Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 9

10 Dynamical Evolution of Distributions... Conservation of probability: p (y)dy = p (x)dx y = T (x) Perron-Frobenius Operator: Locally: y = T (x) dy p (y)dy p n+ (y) = p n(x) T (x) Globally: p n+ (y) = x T (y) p n (x) T (x) p (x)dx dx Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2

11 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) y p(y) p(y) y = T (x) dx δ(x) = p(x) x Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2

12 Dynamical Evolution of Distributions... Example: Delta function initial distribution Map: x n+ = f(x n ) Initial condition: x R Initial distribution: p (y) = = p (x) =δ(x x ) dx p (x) δ(y f(x)) dx δ(x x ) δ(y f(x)) = δ(y f(x )) = δ(y x ) Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield. p n (y) = δ(y x n )... reduces to an orbit Monday, May 24, 2 2

13 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... Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 3

14 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 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 4

15 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 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield x x Monday, May 24, 2 5

16 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 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield x x Monday, May 24, 2

17 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 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield x x Monday, May 24, 2 7

18 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 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield x x Monday, May 24, 2 8

19 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??? Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 9

20 Invariant measures for D Maps: Probability distribution (density p (x)) that is invariant:. Distribution s support must be an invariant set: Λ = f(λ), Λ = supp p (x) ={x : p (x) > } I. Probabilities invariant : Distribution a fixed point of Frobenius-Perron Equation p (y) = dx p (x) δ(y f(x)) Functional equation: Find p ( ) that satisfies this. Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 2

21 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! Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield has density ( k ) (x x i ) i= Monday, May 24, 2 2

22 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 x p (y) =p (y)+p (y) p (x) =p (x) (y x) Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 22

23 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)δ(y f(x)) dx δ(y 2x) p (y) = = 2 2 dx p (x)δ(y f(x)) dx δ(y 2x) = 2 = 2 p (y) = p (y)+p (y) = p (x) (y x) Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 23

24 { ax n, x n Example: Tent map x n+ = 2 a( x n ), 2 <x n Fully two-onto-one: a =2 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) = 2 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield = 2 dx p (x)δ(y f(x)) dx δ(y (2 2x)) = 2 x n Monday, May 24, 2 24

25 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 ) Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 25

26 Example: Logistic map x n+ = rx n ( x n ) Fully two-onto-one: r =4 Invariant distribution? p(x) = π x( x) Exercise. Hint: Coordinate transform to tent map. Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 2

27 Numerical Example: Tent map Typical chaotic parameter: a =.75 Two bands merge to one: a = 2 Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 27

28 Numerical Example: Tent map Typical chaotic parameter: a =.75 Two bands merge to one: a = 2 x Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 27

29 Numerical Example: Tent map Typical chaotic parameter: a =.75 Two bands merge to one: a = 2 x x Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 27

30 Numerical Example: Logistic map x n+ = rx n ( x n ) Typical chaotic parameter: r =3.7 Two bands merge to one: r = Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 28

31 Numerical Example: Logistic map x n+ = rx n ( x n ) Typical chaotic parameter: r =3.7 Two bands merge to one: r = x Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 28

32 Numerical Example: Logistic map x n+ = rx n ( x n ) Typical chaotic parameter: r =3.7 Two bands merge to one: r = x x Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 28

33 Numerical Example: Cusp map (a, b) =(, /2) x n+ = a( 2x n b ) x n+ x n Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 29

34 Numerical Example: Cusp map (a, b) =(, /2) x n+ = a( 2x n b ) x n+ x n x Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 29

35 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. Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 3

36 Reading for next lecture: Lecture Notes. Lecture 8: Nonlinear Physics, Physics 5/25 (Spring 2); Jim Crutchfield Monday, May 24, 2 3

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