# Example Chaotic Maps (that you can analyze)

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1 Example Chaotic Maps (that you can analyze) Reading for this lecture: NDAC, Sections Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4

2 Example D Maps... Shift Map: x n [, ] x n+ = f(x n ) = 2x n (mod ) x n+ Fixed Point: x = Unstable: f (x )=2> Period-2 Orbit: {x } = {/3, 2/3} Unstable: (f 2 ) (x )=4> All periodic orbits unstable x n x n+2 x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 2

3 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

4 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ Solvable! x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

5 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ Solvable! x n =2 n x (mod ) x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

6 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ Solvable! x n =2 n x (mod ) Chaotic mechanism: shift up least significant digits x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

7 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ Solvable! x n =2 n x (mod ) Chaotic mechanism: shift up least significant digits x n x =... x =... Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

8 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ Solvable! x n =2 n x (mod ) Chaotic mechanism: shift up least significant digits x n x =... x =... Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

9 Example D Maps... Shift Map: x n [, ] x n+ =2x n (mod ) x n+ Solvable! x n =2 n x (mod ) Chaotic mechanism: shift up least significant digits x n x =... x =... Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

10 Example D Maps... Lyapunov Characteristic Exponent for D Maps: x n+ = f(x n ) f(x) f (x ) 2 f (x ) 2 f (x )f (x )... δ N δ e λ N f(x + δ ) f(x) δ δ x x + δ or, Definition: LCE λ = lim N N log 2 δ N δ Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 4

11 Example D Maps... Lyapunov Characteristic Exponent for D Maps... λ = lim N N log 2 f N (x + δ ) f N (x ) (x + δ ) x δ λ = lim N N log 2 (f N ) (x ) (f N ) (x )=f (x N )(f N ) (x )=f (x )f (x ) f (x N ) λ = lim N N N n= log 2 f (x n ) λ< stable λ> unstable Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 5

12 Example D Maps... Back to Shift Map: Its LCE... x n+ = f(x n ) = 2x n (mod ) λ = lim N N N n= log 2 f (x n ) 2 x Independent of state: f (x) =2 x Amplification per step (or bits of resolution lost): λ = Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 6

13 Example D Maps... Tent Map: x n [, ] { ax n, x n x n+ = 2 a( x n ), 2 <x n Slope: a [, 2] Height at max: a 2 x n+ x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 7

14 Example D Maps... Tent Map Bifurcation Diagram: {x n }.5. a.5 2. Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 8

15 Example D Maps... Tent Map... Stable fixed point: x =, a< a Unstable fixed points: {, +a }, a 2 All periodic orbits unstable: p> No periodic windows x n+2 x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 9

16 Example D Maps... Tent Map LCE: λ = lim N N N n= log 2 f (x n ) Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4

17 Example D Maps... Tent Map LCE: λ = lim N N N n= log 2 f (x n ) λ = lim N N N n= log 2 ±a = log 2 a Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4

18 Example D Maps... Tent Map LCE: λ = lim N N N n= log 2 ±a = log 2 a.5 Chaotic -.5 Periodic: P Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4

19 Example D Maps... Logistic map: State space: Parameter (height): x n+ = rx n ( x n ) x n [, ] r [, 4] x n+ x n Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 2

20 Example D Maps... Logistic map bifurcation diagram.... {x n }. 3. r 4. Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 3

21 Example D Maps... Logistic map LCE: Local stability depends on state: λ = lim N λ = lim N N N N n= N n= log 2 f (x n ) log 2 r( 2x n ) f (x) =r( 2x) Period : x =, r f (x ) = r λ = log 2 r Period : x = r, r 3 λ = log r 2 2 r Superstable: f (x i )= λ r =2 Bifurcations: λ = Onset of chaos: λ = Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 4

22 Example D Maps... LCE for D Maps... an aside on the Ergodic Theorem Rather than time average: λ = lim N N N n= log 2 f (x n ) Average over attractor s distribution: Pr(x), x Λ Invariant distribution: Pr(x) =f Pr(x) State-space averaged LCE: λ = Λ dx Pr(x) log 2 f (x) Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 5

23 Example D Maps... Logistic map LCE: r =4 Invariant distribution: Pr(x) = π x( x) f(x) log 2 Pr(x) λ = dx log 2 4 8x π x( x) λ = bit per step x Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 6

24 Example D Maps... LCE view of period-doubling route to chaos: r B r 2B B.5 r c λ = r 2 4 λ = Superstable Period-2 λ Superstable Period-4 r =3.83 Period-3 Window Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 7

25 Example D Maps... Logistic map bifurcation diagram self-similarity. {x n }. 3. r {x n } r {x n } Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield r Monday, January 27, 4 8

26 Example D Maps... Logistic map bifurcation diagram self-similarity. {x n }. 3. r {x n } r {x n } Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield r Monday, January 27, 4 8

27 Example D Maps... Bifurcation Theory of D Maps... Scaling analysis of period-doubling cascade:. {x n } d d 3 x =/2 d r 2ss r r 2 r 4ss r 8ss Universal constants: Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield δ = lim n α = lim n r n = r n+ d n = d n+ Monday, January 27, 4 9

28 Example D Maps... Bifurcation Theory of D Maps... Renormalization group analysis of period-doubling: r r f 2 r (x) f r (x) αy y αx x > <, flip Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 2

29 Example D Maps... Bifurcation Theory of D Maps... Renormalization group analysis of period-doubling... f(x, r ) αf 2 ( x α,r ) f 2 ( x α,r ) α 2 f 4 ( x α 2,r 2)... f(x, r ) α n f (2n) ( x α n,r n) Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 2

30 Example D Maps... Bifurcation Theory of D Maps... Renormalization group analysis of period-doubling... Universal Map: g (x) = lim n αn f (2n) ( x α n,r n) for x x max Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 22

31 Example D Maps... Bifurcation Theory of D Maps... Renormalization group analysis of period-doubling... r : f(x, r ) αf 2 ( x α,r ) for x x max Limiting functional equation: ( x max = ) ( ) g(x) =αg 2 x α Boundary conditions: g() = & g () = Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 23

32 Example D Maps... Bifurcation Theory of D Maps... Renormalization group analysis of period-doubling... How to solve? g(x) =αg 2 ( x α ) with g() = & g () = Taylor expansion: g(x) =a + bx 2 + cx 4 + g(x) even Find: α = Parameter rescaling: (more work) Find: δ = Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 24

33 Example D Maps... Reading for next lecture: Lecture Notes. Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January 27, 4 25

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