August 16, Alice in Stretch & SqueezeLand: 15 Knife Map. Chapter Summary-01. Overview-01. Overview-02. Rossler-01. Rossler-02.

Transcription

1 Summary- Overview- Rossler- August 16, 22 Logistic Knife

2 Abstract Summary- Overview- Rossler- Logistic What is the order of orbit creation in the Lorenz attractor? The attractor is created by a tearing and squeezing mechanism since g > 1. How are these orbits organized? For attractors created by a stretch and fold mechanism (g = 1) the logistic map x = f(x; a) = a x 2 provides useful insight. For attractors created by a tear and squeeze mechanism (g > 1) the knife map y = g(y; b) = b y 1/2 provides useful insight. The two maps share many similarities and exhibit important differences. Knife

3 Overview- Summary- Overview- Rossler- What We Did 1 Studied maps with 2 branches 2 L & R 3 Separated by a singularity 4 Models for Tearing Mechanism 5 Looked for universality 6 Searched for scaling Logistic Knife

4 Summary- Overview- Rossler- What We Found 1 Simple form: x = a x k 2 k = 2 folding; k = 1 2 tearing 3 Localized global attractor 4 Either chaos or pd. 1 fixed point 5 Orbits of periods 1 and 2 organize systematics 6 Explosions 7 Prime and compound orbits 8 Local and Global focus points Logistic Knife

5 Rössler Attractor Rössler Attractor Summary- Overview- Rossler- Logistic Knife

6 Rössler Attractor Rössler Attractor - Return Summary- Overview- Rossler- Logistic Knife

7 Lorenz Attractor Lorenz Attractor Summary- Overview- Rossler- Logistic Knife

8 Lorenz Attractor Return for Lorenz Attractor Summary- Overview- Rossler- Logistic Knife

9 Lorenz Attractor Image of Lorenz Return Summary- Overview- Rossler- Logistic Knife

10 Stability Regions Summary- Overview- Rossler- Logistic Knife

11 : Logistic Summary- Overview- Rossler- Logistic Knife

12 : Knife Summary- Overview- Rossler- Logistic Knife

13 02 Stability Regions Summary- Overview- Rossler- Logistic Knife

14 Logistic- Return - Rössler Attractor Summary- Overview- Rossler- Logistic Knife

15 Image Return - Lorenz Image Summary- Overview- Rossler- Logistic Knife

16 Logistic 02 Summary- Overview- Rossler- Logistic Return Approximations The Rossler return map is well approximated by the following maps: x = λx(1 x) x = a x 2 x = 1 µx 2 x = 1 x m w 2 Knife

17 Image Summary- Overview- Rossler- Image of Lorenz Return The image of the Lorenz return map is well approximated by the following maps: y = b y 1/2 y = 1 µ y 1/2 y = 1 y m w 1/2 Logistic Knife

18 Side by Side- Summary- Comparison: Logistic & Knife s Overview- Logistic M ap Knif e M ap Rossler- x = f(x; a) = a ( x ) 2 y = f(y; b) = b ( y ) 1/2 Logistic Knife

19 Logistic for several values of a Summary- Overview- Rossler- Logistic Knife

20 Image Lorenz-04 Knife Return s Summary- Overview- Rossler- Logistic Knife

21 Orbit Search- Second Return Summary- Overview- Rossler- Logistic Knife

22 Orbit Search-02 Period 1 & 2 Orbits - Logistic Summary- Overview- Rossler- Logistic Knife

23 Bifurcation- Bifurcation Diagram Summary- Overview- Rossler- Logistic Knife

24 Bifurcation-02.. Blow Up... with Caustics Summary- Overview- Rossler- Logistic Knife

25 Bifurcation-03 Knife - Bifurcation Diagram Summary- Overview- Rossler- Logistic Knife No windows! No caustics!

26 Bifurcation-04 Knife - Lyapunov Exponent Summary- Overview- Rossler- Logistic Knife

27 Orbit Search-03 Fixed Points (Knife) Summary- Overview- Rossler- Logistic Knife

28 Orbit Search-04 Second Iterates - Knife Summary- Overview- Rossler- Logistic Knife

29 Skeleton- Period-One & Period-Two Orbits Summary- Overview- Rossler- Logistic Knife

30 Skeleton-02 Attractor boundary (Knife) Summary- Overview- Rossler- Logistic Knife

31 Skeleton-03 Attractor Boundaries - Logistic Summary- Overview- Rossler- Logistic Knife

32 Rite of Passage- Summary- Overview- Rossler- Logistic Knife

33 Explosions- Knife Iterates Summary- Overview- Rossler- Logistic Knife

34 Explosions-02 Summary- Overview- Rossler- Table: Values M (p) of y where the pth iterate f (p) (y; b) has maxima. These locations are determined by a simple recursion relation (last line) where the indices s p = ±1 are incoherent. p Number Max. Coordinate Values ±b ±(b ± b 2 ) 2 p p M (p+1) = s p (b + M (p) ) 2 Logistic Knife

35 Explosions-03 Summary- Overview- Rossler- As p, with all s j = +1, the abscissa of the rightmost point goes to a limit. The quadratic equation for this limit gives: ( ) 1 1 y(b) = 2 b 4 b At b = 1 4 the bounding box is a square beyond that the diagonal fails to intersect all the zig - zags. Orbits begin to get pruned away in singular saddle node bifurcations. Logistic Knife

36 Explosions-04 Structural Stability: 0 < b < 1 4 Summary- Overview- Rossler- Logistic Knife

37 Rite of Passage-02 End Play - Near b = 1 Summary- Overview- Rossler- Logistic Knife

38 Rite of Passage-03 Iterates Near b = 1 Summary- Overview- Rossler- Logistic Knife implosion1

39 Renormalization- Note Scaling Relations Summary- Overview- Rossler- Logistic Knife

40 Explosions-05 Structural Stability: 3 4 < b < 1 Summary- Overview- Rossler- Logistic Knife

41 Orbit Search-05 Hunt for Saddle-Node Bifurcations Caustic Crossings Summary- Overview- Rossler- Logistic Knife

42 Orbit Search-06 Hunt for Singular SNBs Summary- Overview- Rossler- Logistic Knife

43 Orbit Search-07 Anti Caustic Crossings Summary- Overview- Rossler- Logistic Knife

44 Orbit Search-08 Anti Caustic Crossings: Expansion Summary- Overview- Rossler- Logistic Knife

45 Orbit Search-09 Period Three Singular SNB Summary- Overview- Rossler- Logistic Knife

46 Renormalization-02 Summary- Overview- Rossler- Local expression near y = 0 for the period-three explosion: h(y; b) = f (3) (y; b) = b b b y h(b 3 + ɛ; y) ( b3 ) b 3 b 3 + ( 2 ) ( ) b y ɛ + b3 b 3 b3 4 b3 b 3 b3 Logistic Knife

47 Renormalization-03 Renormalization for the period-three explosion. y = h(y; b 3 + ɛ) (b b 3 ) + α y = Summary- Overview- Rossler (b b 3 ) y z = ( /α 2 )(b 3 b) z Logistic Knife

48 Renormalization-04 Summary- Overview- Rossler- Logistic Renormalization Algorithm: K10* 1 Write down the symbol sequence for the primary period-p orbit: K10 = Kσ 1 σ 2 σ p 1. 2 Make the identification σ = +1 s = +1, σ = 0 s = 1. 3 Construct f (p) (b; y) b s p 1 (b s 2 (b s 1 (b y)) ) 4 Taylor expand this function to terms linear in b and y and determine the value of b for which the constant term vanishes. Knife

49 Renormalization-05 Summary- Overview- Rossler- Equations: K10* For the saddle node pair 5 2 = K10 this algorithm gives b (+1)(b ( 1)(b ( 1)(b (+1)(b y)))) The constant term vanishes for b = , and for this value of b y = (b b 52 ) + α y = b y Logistic Knife

50 Renormalization-06 Results: K10* to Period 6 y = (b b c ) + α y y, y 0 Summary- Overview- Rossler- Logistic Orbit Symbolics b c α 3 1 K K K K K K K K Knife

51 Renormalization-07 Renormalization for the final period-two explosion. f (2) (1 ɛ, y) ɛ 2 + ( ɛ 4) y (1) Summary- Overview- Rossler- Logistic Knife

52 Orbit Search-05 Hunt for Saddle-Node Bifurcations Summary- Overview- Rossler- Logistic Knife

53 Orbit Search-08 Hunt for S. Saddle-Node Bifurcations Summary- Overview- Rossler- Logistic Knife

54 Important Markers Summary- Overview- Rossler- Breakpoints Table: Important parameter values for global stability and unstable periodic orbit behavior. Global Stability Unstable Orbits 0.0 1/4 1/ / Logistic Knife

55 U Sequence Summary- Overview- Rossler- Logistic Knife

56 Endplay- Symbol Exchange Near Endplay Summary- Overview- Rossler- Logistic Knife

57 Endplay-02 Summary- Overview- Rossler- Symbol Exchange Near Endplay Symbols 0, 1 created at b = 0 New orbit, (11), created at b = 3 4 Symbol pair , replaced by - (11) - as b 1 Implosions begin at b = , end at midpoint Explosions begin at midpoint, end at b = Implosions and explosions symmetrically matched Logistic Knife