Poincare Sections and the Uniqueness of ODE Solutions: A Study on Iterated Maps

1. Iterated Map • Poincare sections: ODEs → Map • Maps by themselves. • 1-D map: xn+1 = f(xn) • Map → ODEs ???

2. Poincare Sections Uniqueness of solutions of ODEs → existence of Poincare map Example of Reduction to 1-D map:

3. Another example: Dissipative system: IC cluster → 0 volume Poincare section of 3-D system Directions with M<<1 ( λ<<0 ) quickly contract. If system is sufficiently dissipative: IC cluster → curve → 1-D map

4. Essentially 1-D system: Diode circuit High Q: 2-D ? Low Q: 1-D

5. Effectively 1-D System: Pendulum D = 2/3, b = 1, F = 2.048

6. 1-D Iterated Map Maps have greater range of dynamical behavior than their ODE counterparts Logistic map: Fixed point: Logistic map: Linear stability analysis:

7. Bifurcations: Period Doubling Route to Chaos

8. 2-cycle: → Same stability at x1* and x2* S,U,S At bifurcation point ( A1 = 3 ) : x* = x1* = x2* In general  But not vice versa

9. (Un)stable f(x*)  (Un)stable f(n)(x*) Let xi(n)* be the ith fixed point of an n-cycle, then (Ex.5.4-4) for i = 1, …, n Prime period of a fixed point = smallest n  x* = f(n)(x*) Ex 5.4-6

10. f(n): f(2n): f(n) > -1 f(n) = -1 f(n) < -1 S S S, U, S

11. Lyapunov Exponents • Chaos: • Divergence of nearby trajectories. • x ~ exp(λt) n = Mn-1 1 with λ>0 & |M|>1 • Dissipative system: ergodic Consider 2 neighboring points on a chaotic attractor: n large, ε→ 0 xi on same trajectory Best choice of n: dn ~ X/2 An 1-D map is chaotic if λ > 0

12. Universality: The U Sequence • Unimodal map: • Piecewise C(1) function of [0,1] into itself with a single maximum at the critical point xc. • Monotonic otherwise. • By convention, f(0) = f(1) = 0 and xc = 0.5. Shift + scale to unimodal Non-unimodal

13. Metropolis, Stein & Stein’s U-Sequence Reprinted in P.Cvitanovic, “Universality in Chaos”, 2nd ed., Adam Hilger (84,89) • Supercycle of period n: • x1 = f(xc) x2 = f(x1) … xn-1 = f(xn-2) xc = f(xn-1) • Symbol (kneading) sequence: xi > xc → R, xi < xc → L → Supercycles are most stable

14. λ = .9895107

15. U-sequences beyond periodic doubling accumulation point ( A ~ 3.5699… for logistic map ) are in windows within chaotic regime. • Universality. • Non-uniqueness: there are other sequences not involving xc. (cf. Singer’s theorem ) • Comparison with experiments: favorable. • Exceptions: • oscillating chemical reactions • varactor diode circuit • Gaussian map Ex. 5.5-2

16. Some Generalizations • Single minimum: f(x) = 1 – Ax(1-x) • Ex. 5.5-3 • Scaling: f(x) = Bx(b-x) • u = x/b

17. Schwarzian derivative Introduced originally for complex analysis • Sf = 0 if f = (az+b)/(cz+d) = linear fractional transform. • Sf < 0, Sg < 0 → S(f。g) < 0 • Sf < 0 → Sf(n) < 0. • Sf < 0 restricts existence of inflection points: • Schwarzian min-max principle: Sf < 0 → • f has no positive local minimum or negative local maximum. Pf: f(x*) is min → f(x*) = 0 & f(x*) > 0  Sf < 0 → f(x*) < 0 A unimodal map can have an infinite series of pitchfork bifurcations only if Sf < 0

18. f < 0, f > 0 , f < 0 f < 0 f > 0, f < 0, f > 0 f > 0 Allowed: Pichfork bifurcation

19. Singer’s Theorem Let Dom(f)  Ran(f) and Sf < 0, then n critical points → at most n+2 attracting cycles Proof: D.Gulick, “Encounters with Chaos”, McGraw Hill (92) or, R.L.Devaney, “ An Introduction to Chaotic Dynamical Systems”, Benjamin (86)…Theorem 11.4. Corollary: unimodal map has at most 3 stable cycles ( usually only 1 with basin almost everywhere )

20. Sarkovskii’s Theorem Let f(x,A)  C0. If f has a prime period m at A0, then f(x,A0) has period n to the right of m in the following sequence: 3→5→7→ … → 2×3 → 2×5 → … → 22×3 → 22×5 → … … → 23 → 22 → 2 → 1 { 20(2n+1) } → { 21(2n+1) } → { 22(2n+1) } → … → { 2n ↓ } Note: Multiple appearances not counted. Only stable cycles computable. Pf: R.L.Devaney, “ An Introduction to Chaotic Dynamical Systems”, Benjamin (86) Period 3 → all periods Logistic map: A = 3.8319…. Li & Yorke, “ Period 3 implies Chaos? ” Logistic map: 1→2→4→ … Chaos → ( windows …→7→5→3 )

21. First 8 images of xc. B.D.: only 10 points plotted. Organization of Chaos Logistic map: bifurcation • Heavy lines: images of xc since f’(xc) = 0. • If xc is within basin of attractor: • xmax = f(xc) xmin = f(2)(xc) • f(n)(xc): boundaries of interior bands • windows: merging of images of xc. • At Ac, images of xc = supercycle. • Sarkovskii’s sequence Misiureweiz point: Ex 5.5-7

22. Feigenbaum Universality Details in Appendices F & H. See operator approach in H.G.Schuster, “Deterministic Chaos” Distance from xc to its nearest neighbor (at half-way) in the 2n supercycle: Feigenbaum α:

23. Existence of α implies that of a universal function g  Proof: Define  Ansatz: →

24. Calculating α & g Universality classes of maps are determined by the form of g(y→0). Different class, different α Quadratic class: Maximum at y = 0 with  y Constant term: y2 term:  Better values obtained by considering higher order terms.

25. Tent ( Triangle ) Map Piecewise linear r Fixed points x* < ½ : Fixed points x* > ½ :

26. 1  r > ½ : chaotic Ex 5.7-3: <λ> > 0 f(4) All fixed points of all the f(n)s are unstable. No period-doubling route 1 unstable f.p. → 2 stable f.p.’s. Period-doubling requires f  C1

27. Shift Maps & Symbolic Dynamics Decimal shift map: x mod[1] = x (mod 1) = mod(x,1) = fractional part of x xn is left-shifted by 1 digit to give xn+1. x0 Q  fixed point / periodic cycle. x0 = 3/8 → x* = 0 x0 = 22/70 → 6-cycle x0 irrational  trajectory ergodic There’re infinitely many A numbers between any two B numbers ( A,B = rat/irrat )  Sensitive to I.C. Chaotic: <λ> > 0 R is uncountably infinite Q is countably infinite → measure 0 Computer: no irrat numbers

28. Bernoulli Shift Fractional binary numbers: (7/8)10 = (0.875000)10 = (0.111000)2 .5 + .25 + .125 = .875 Bernoulli Shift Ex. 5.8-1: <λ> = ln 2 Irrational number : digits = randomn sequence Symbolic dynamics C.f. U-sequence Ref: Devaney

29. Logistic Map (A = 4) ~ Bernoulli Shift <λ> = ln2 Use as random number generator

30. The Gaussian Map f  C1.  period-doubling sequence Inflection points at → |f|<< 1 for large |x| → stable fixed point at –x << -1 → period-halving sequence ( bubbling ) U-seq violated: diode, osc chem reaction… b = 7.5, x0 = 0

31. b = 7.5, x0 = 0 b = 4, x0 = 0 b = 7.5, x0 = 0.7 3 fixed points Another violation of U-seq: Antimonotonicity ~ dimple near xc -- e.g., Diode.

32. 2-D Maps Henon Map: C > 0; x,y  R 3-term recurrence: Invertible: For 0 <|B|< 1, y-spread → 0 B = 0.3, C = 1 f(sq) = dots

33. Bifurcation diagrams (B = 0.3) : {x0,y0} = {0,0.5} {x0,y0} = {0.5,0.5} combined 2 attractors → hysteresis crisis

34. Smale Horseshoe Map Ref: D.Gulick, “Encounters with Chaos”, McGraw Hill (92) T = [0,1]  [0,1] B, E = semi-circles S = T∪B∪E • Shrinks vertically by a < 1/3. • Expands horizontally by b = 3. • Reshape M(B) & M(E) into semicircles. • Fold to fit back in S ( boundaries of M(D) = semicircles ) 

35. M(n)(S) = 2n-1 connected horseshoes each of width ~ an. • M has a unique fixed point p in B to which iterates of all points in B & E converge. • All points in S migrate either to p or C+, where = intersect of nested sequence of strips Cijk… i,j,k,… = 0,1 vCijk…  vCi M(v)Cj M(2)(v)Ck… v ~ z = ˙ijk…Forward sequence Cantor set

36. Inverse mapping: Vi = M(Ci) M(-1)(Vi) = Ci v Vi j = M2(Ci j)  M(-1)(v) M(Ci j) Cj M(-2)(v)Ci j  Ci v ~ z = ij˙ backward sequence

37. h ←→ h-1 → h-1 h ← z-2˙z-1 z0˙z1 ˙z0 z-2z-1˙ z-1˙ ˙z0z1 Vij → ij˙ Cij → ˙ij

38. 2-sided sequence: … z-3z-2z-1˙z1 z2 z3… Invariant set: C* = C+ ∩ C- • M is strongly chaotic on C*: • Stretch & fold mechanisms. • Sensitive on I.C. • (Unstable) Periodic points: z  Q. • Ergodic ( dense orbits ).