Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Randomness and determinism in dynamical systems Vaughn Climenhaga University of Houston October 5, 2012 Vaughn Climenhaga (University of Houston) October 5, 2012 1 / 19 Chaos Examples Doubling map Logistic map Bifurcation diagram Summary The talk in one slide Deterministic systems can exhibit stochastic behaviour Phenomenon over long time scales Known Mechanism driving this is phase space expansion Examples Lorenz equations, expanding maps, logistic map Research What happens when expansion is non-uniform? Vaughn Climenhaga (University of Houston) October 5, 2012 2 / 19 Predictions rely on finding f n(x) given x. initial error ) must compare f n(x) and f n(y) when x ∼ y Distinct problem from accounting for discrepancy between model and real-world system, or for numerical error in computation of f n(x). Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Predictions in dynamical systems Key objects: X = phase space for a dynamical system. Points in X correspond to configurations of the system. f : X describes evolution of the state of the system over a single time step. Can also consider continuous-time systems. Standing assumptions: n X ⊂ R f is continuous Vaughn Climenhaga (University of Houston) October 5, 2012 3 / 19 Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Predictions in dynamical systems Key objects: X = phase space for a dynamical system. Points in X correspond to configurations of the system. f : X describes evolution of the state of the system over a single time step. Can also consider continuous-time systems. Standing assumptions: n X ⊂ R f is continuous Predictions rely on finding f n(x) given x. initial error ) must compare f n(x) and f n(y) when x ∼ y Distinct problem from accounting for discrepancy between model and real-world system, or for numerical error in computation of f n(x). Vaughn Climenhaga (University of Houston) October 5, 2012 3 / 19 Chaos Examples Doubling map Logistic map Bifurcation diagram Summary A mechanism for stochastic behaviour Fix x ∼ y. Two extremes: Stable behaviour: d(f nx; f ny) ! 0 Even better: there is p = f (p) such that f nx ! p for all x Unstable behaviour: d(f nx; f ny) grows quickly In \chaotic" systems, unstable behaviour is prevalent: initial error grows exponentially fast prediction f n(x) quickly diverges from reality Another perspective: U ⊂ X a small neighbourhood, consider f n(U). In chaotic systems, diameter of iterates f n(U) becomes large (exponentially quickly) no matter how small U is. Vaughn Climenhaga (University of Houston) October 5, 2012 4 / 19 f(x) 1 1 ix 2 i(2x) Doubling map f : S , S ⊂ C, z = e 7! z = e + N Full shift Σ2 = f0; 1g , f = σ : x0x1x2 ::: 7! x1x2x3 ::: 0 1 x f(x) Logistic map fλ : [0; 1] ; x 7! λx(1 − x), λ 2 [0; 4] Code trajectories with 0s and 1s, but don't get full shift. x Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Lorenz equations (1963) { atmospheric dynamics x_ = σ(y − x) σ = 10 y_ = x(ρ − z) − y ρ = 28 z_ = xy − βz β = 8=3 Vaughn Climenhaga (University of Houston) October 5, 2012 5 / 19 f(x) 0 1 x f(x) Logistic map fλ : [0; 1] ; x 7! λx(1 − x), λ 2 [0; 4] Code trajectories with 0s and 1s, but don't get full shift. x Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Lorenz equations (1963) { atmospheric dynamics x_ = σ(y − x) σ = 10 y_ = x(ρ − z) − y ρ = 28 z_ = xy − βz β = 8=3 1 1 ix 2 i(2x) Doubling map f : S , S ⊂ C, z = e 7! z = e + N Full shift Σ2 = f0; 1g , f = σ : x0x1x2 ::: 7! x1x2x3 ::: Vaughn Climenhaga (University of Houston) October 5, 2012 5 / 19 f(x) 0 1 x f(x) x Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Lorenz equations (1963) { atmospheric dynamics x_ = σ(y − x) σ = 10 y_ = x(ρ − z) − y ρ = 28 z_ = xy − βz β = 8=3 1 1 ix 2 i(2x) Doubling map f : S , S ⊂ C, z = e 7! z = e + N Full shift Σ2 = f0; 1g , f = σ : x0x1x2 ::: 7! x1x2x3 ::: Logistic map fλ : [0; 1] ; x 7! λx(1 − x), λ 2 [0; 4] Code trajectories with 0s and 1s, but don't get full shift. Vaughn Climenhaga (University of Houston) October 5, 2012 5 / 19 law of large numbers Birkhoff ergodic theorem: for every ' 2 L1(S1) and ν-a.e. z 2 S1, n−1 1 X Z lim '(f k z) = '(y) dν(y) n!1 n 1 k=0 S f(x) 0 1 x Predictions are easy: Lebesgue measure ν on the circle is f -invariant ν(f −1E) = νfz j f (z) 2 Eg = ν(E) for every measurable E ⊂ S1 It is also ergodic: if f −1(E) = E then ν(E) = 0 or 1. Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Predictions for the doubling map 1 1 ix 2 i(2x) Doubling map f : S , S ⊂ C, z = e 7! z = e + N Full shift Σ2 = f0; 1g , f = σ : x0x1x2 ::: 7! x1x2x3 ::: Predictions are impossible: If initial error is then error at time n is 2n. Lengthening prediction by time 1 requires doubling initial accuracy. Vaughn Climenhaga (University of Houston) October 5, 2012 6 / 19 f(x) 0 1 x law of large numbers Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Predictions for the doubling map 1 1 ix 2 i(2x) Doubling map f : S , S ⊂ C, z = e 7! z = e + N Full shift Σ2 = f0; 1g , f = σ : x0x1x2 ::: 7! x1x2x3 ::: Predictions are easy: Lebesgue measure ν on the circle is f -invariant ν(f −1E) = νfz j f (z) 2 Eg = ν(E) for every measurable E ⊂ S1 It is also ergodic: if f −1(E) = E then ν(E) = 0 or 1. Birkhoff ergodic theorem: for every ' 2 L1(S1) and ν-a.e. z 2 S1, n−1 1 X Z lim '(f k z) = '(y) dν(y) n!1 n 1 k=0 S Vaughn Climenhaga (University of Houston) October 5, 2012 6 / 19 f(x) 0 1 x Chaos Examples Doubling map Logistic map Bifurcation diagram Summary Predictions for the doubling map 1 1 ix 2 i(2x) Doubling map f : S , S ⊂ C, z = e 7! z = e + N Full shift Σ2 = f0; 1g , f = σ : x0x1x2 ::: 7! x1x2x3 ::: Predictions are easy: Lebesgue measure ν on the circle is f -invariant ν(f −1E) = νfz j f (z) 2 Eg = ν(E) for every measurable E ⊂ S1 It is also ergodic: if f −1(E) = E then ν(E) = 0 or 1. Birkhoff ergodic theorem: for every ' 2 L1(S1) and ν-a.e. z 2 S1, n−1 1 X Z lim '(f k z) = '(y) dν(y) law of large n!1 n 1 k=0 S numbers Vaughn Climenhaga (University of Houston) October 5, 2012 6 / 19 These are IID (Bernoulli process). p1 Pn Central limit theorem: n k=1(Xk − EX ) converges to Gaussian 1 Pn Large deviations: Estimates on P(j n k=1 Xk − EX j > δ) Pn (X − X ) Law of the iterated logarithm: pk=1 k E converges to zero in n log log n probability but not almost surely . and so on . This works for any \nice enough" ', and all this happens even though the dynamical system is deterministic. Chaos Examples Doubling map Logistic map Bifurcation diagram Summary A Bernoulli process + N Lebesgue measure on the circle passes to a measure µ on Σ2 = f0; 1g + 1 P1 −k µ(E) = ν(π(E)), where π :Σ2 ! S ; x 7! exp(πi k=0 xk 2 ) + Define ':Σ2 ! R by '(x) = x0. This gives a sequence of random + n variables on (Σ2 ; µ) by Xn = '(f x). Vaughn Climenhaga (University of Houston) October 5, 2012 7 / 19 This works for any \nice enough" ', and all this happens even though the dynamical system is deterministic. Chaos Examples Doubling map Logistic map Bifurcation diagram Summary A Bernoulli process + N Lebesgue measure on the circle passes to a measure µ on Σ2 = f0; 1g + 1 P1 −k µ(E) = ν(π(E)), where π :Σ2 ! S ; x 7! exp(πi k=0 xk 2 ) + Define ':Σ2 ! R by '(x) = x0. This gives a sequence of random + n variables on (Σ2 ; µ) by Xn = '(f x). These are IID (Bernoulli process). p1 Pn Central limit theorem: n k=1(Xk − EX ) converges to Gaussian 1 Pn Large deviations: Estimates on P(j n k=1 Xk − EX j > δ) Pn (X − X ) Law of the iterated logarithm: pk=1 k E converges to zero in n log log n probability but not almost surely . and so on . Vaughn Climenhaga (University of Houston) October 5, 2012 7 / 19 Chaos Examples Doubling map Logistic map Bifurcation diagram Summary A Bernoulli process + N Lebesgue measure on the circle passes to a measure µ on Σ2 = f0; 1g + 1 P1 −k µ(E) = ν(π(E)), where π :Σ2 ! S ; x 7! exp(πi k=0 xk 2 ) + Define ':Σ2 ! R by '(x) = x0. This gives a sequence of random + n variables on (Σ2 ; µ) by Xn = '(f x). These are IID (Bernoulli process). p1 Pn Central limit theorem: n k=1(Xk − EX ) converges to Gaussian 1 Pn Large deviations: Estimates on P(j n k=1 Xk − EX j > δ) Pn (X − X ) Law of the iterated logarithm: pk=1 k E converges to zero in n log log n probability but not almost surely .