Mechanisms of Chaos: Stable Instability Reading for this lecture: NDAC, Sec. 2.-2.3, 9.3, and.5.
Unpredictability: Orbit complicated: difficult to follow Repeatedly convergent and divergent Net amplification of small variations What geometry produces this? Stretch and fold: Flow stretches state space But to be stable (i.e., have an attractor): Must be done in a compact region So flow must fold back into region 2
Baker s transformation: kneading state space Shrink Stretch /2 2 Cut & Stack Repeat 3
Baker s transformation... kneading state space Shrink Stretch /2 2 Cut & Stack Repeat Lecture 6: Natural Computation & Self-Organization, Physics 25 (Winter 28); Jim Crutchfield 4
Baker s transformation... 2D Baker s Map: (x n,y n ) [, ] [, ] x n+ = 2x n (mod ) { y n+ = 2 y n, x n 2 2 + 2 y n, x n > 2 5
Baker s transformation... Stability? 2 A = 2 Calculate: λ =2 Stretch v =(, ) Only horizontal λ 2 =/2 Shrink v 2 =(, ) Only vertical Det(A) = Independent of x Area preserving: No attractor per se But confined to compact region 6
Dissipative Baker s Map: y y a x x a 7
Dissipative Baker s Map... again! y y 2 a 2 a 2 x x 2 a 2 a 2 Lecture 6: Natural Computation & Self-Organization, Physics 25 (Winter 28); Jim Crutchfield 8
Dissipative Baker s Map... x n+ = 2x n (mod ) { y n+ = ay n, x n 2 2 + ay n, x n > 2 a [, 2 ] 9
Dissipative Baker s Map... 2 Stability? A = a Calculate: λ =2 v =(, ) λ 2 = a v 2 =(, ) Independent of x Det(A) =2a Dissipative: a</2 Volume contraction Attractor!
Dissipative Baker s Map Simulation: a =.3 y n x n
Dissipative Baker s Map... Stability? (x, y) versus (x + ɛ, y + δ) x = 2(x + ) 2x = 2 y = a(y + δ) ay = aδ x n =2 n ɛ y n = a n δ Exponential Growth of Errors Exponential Stability 2
Dimension of a Set: Number of boxes to cover set at measurement resolution : = = 2 N = 4 N = 6 = N = 64 4 8 N() 2 Generalizing N(ɛ) ɛ d n = n =2 n =3 N( = 2 n ) = 2 n 2 = 2 2n Or (Definition) dimension: log N(ɛ) d = lim ɛ log ɛ 3
Dimension of Dissipative Baker s Attractor... At iteration n: 2 n strips of thickness a n How many boxes at resolution ɛ? Take: = a n N() to cover attractor Number of boxes for each strip: a n y n ɛ ɛ N(ɛ) =a n 2 n = ( a 2 ) n x n 4
Dimension of Dissipative Baker s Attractor... Dimension: log N() d = lim log ɛ ɛ = lim n log(a/2) n log a n y n = + log 2 log a x n a =.3 d =.576... < 2! Area preserving: as a 2, d 2 5
Cat map (aka Toral automorphism): (x, y) T 2 Intrinsic stretch/shrink directions C D A B B A D C ( ) xn+ y n+ = ( 2 )( xn y n ) (mod ) Fixed point: x =(, ) 6
Cat map (aka Toral automorphism)... ( ) xn+ y n+ = ( 2 )( xn y n ) (mod ) Calculate: λ = 3+ 5 2 λ 2 = 3 5 2 > < stretch shrink v =( + 5 2 v 2 =( 5 2, ), ) Det(A) = area preserving Independent of x v 2 v 7
Poincare stretch demo: ~/Dynamical Systems/Dynamics Demos/ 8
Hénon map: (x, y) R 2 x n+ = y n + ax 2 n y n+ = bx n Stretch and fold depend on location 9
Henon map... Stretch & fold depend on location: Jacobian: ( ) 2axn A = b Dissipative when b < (and orientation reversing): Det(A) = b 2
Henon Attractor: Control parameters: (a, b) =(.4,.3) 2
Henon Attractor... Self-similar: Self-similar attractor = Dissipation + Instability 22
Chaotic Mechanisms in ODEs:. Rössler attractor 2. Lorenz attractor 3. How to quantify chaos & stability? Lyapunov characteristic exponents 23
Chaotic Mechanisms in ODEs: Rössler Chaotic Attractor: Branched manifold: 24
Chaotic Mechanisms in ODEs... Rössler Chaotic Attractor... B B A A B Branched manifold unfurled: B B B A A A B B B A 25
Chaotic Mechanisms in ODEs... Rössler stability + instability: Dot spreading demo (ds) Integration step =.2 IC = (,-6,) Remembered trajectory = 6 Orient 5 e nens = K IC = (,-6,) radius =.,, 26
Chaotic Mechanisms in ODEs... Lorenz stability + instability: Dot spreading demo (ds) Remembered trajectory = 6 Orient e nens = 5K IC = (5,5,5) radius =.5,, 27
Quantifying instability: Lyapunov Characteristic Exponent (LCE): δ(t) δ() e λt λ t ln δ(t) δ() δ() x() x(t) δ(t) λ = lim lim δ() t t log 2 δ(t) δ() Exponential rate of growth of errors. Note: δ(t) aligns with most unstable direction! 28
Measurement Resolution: ɛ Number of scale factors to locate initial state: I = log 2 ɛ Prediction horizon: t unpredict I λ Resolution loss rate (bits per second): λ Error doubles each second λ = bit/second 29
Prediction horizon... Examples: Loss rate = Factor of 2 each second: λ = bit/second. Measurement resolution: ɛ = 3 I = bits t unpredict = seconds 2. Thousand times higher resolution: ɛ = 6 I = 2 bits t unpredict = 2 seconds 3
Quantifying instability and stability... Lyapunov Characteristic Exponent Spectrum: χ = {λ,λ 2,...,λ n },λ i λ i+ λ i = lim lim δ() t t log 2 δx i (t) δx i () {δ x,δ x 2,...,δ x n },δ x i δ x j =,i j δ x (t) δ x 3 () δ x 2 () δ x () δ x 3 (t) δ x 2 (t) x(t) x() 3
Quantifying instability and stability... LCE Spectrum and Submanifolds: λ i < stable manifold λ i > unstable manifold λ j > λ i < x(t) LCE Spectrum: Key to characterizing attractors 32
Divergence of vector field: F (x) = n i= f i x i Local volume change. = Tr(A(x)) x Dissipation rate: D = lim T T T dt F ( x (t)) n Theorem: D = λ i i= 33
LCE Spectrum Attractor Classification: Attractor: D < Need net volume contraction for global stability 34
Fractal dimension of attractor: j i= λ i d = j + λ j+ j j largest integer such that λ i i= (Conjectured) 35
Metric entropy of attractor: h µ (Λ) = λ i > λ i Total information production rate: [bits per second] 36
LCE Spectrum Attractor Classification... Dimension LCE Spectrum Attractor (-) Fixed Point 2 (-,-) Fixed Point 2 (,-) Limit Cycle 3 (-,-,-) Fixed Point 3 (,-,-) Limit Cycle 3 (,,-) Torus 3 (+,,-) Chaotic 4 (,,,-) 3-Torus 4 (+,,,-) Chaotic 2-Torus 4 (+,+,,-) Hyperchaos 37
Definition of chaotic attractor: () Attractor: Λ X (a) Invariant set: Λ=φ T (Λ) (b) Attracts an open set U X: Λ U &Λ T φ T (U) (c) Minimal: no proper subset is also (a) & (b) (2) Aperiodic long-term behavior of a deterministic system with exponential amplification (2 ) Positive maximum LCE: λ max (Λ) > (2 ) Positive metric entropy: h µ (Λ) > 38
Reading for next lecture: NDAC, Sections.5-.7. 39