Area-PReserving Dynamics

Transcription

1 Area-PReserving Dynamics James Meiss University of Colorado at Boulder NZMRI Summer Workshop Raglan, New Zealand, January 9 14, 2011

2 The Standard Map K(θ,t) Frictionless, horizontal Rotor θ I ω Moment of inertia I, angular momentum p I θ = K(θ,t) I θ = p p = K(θ,t) Kick, K, impulsively and periodically K(θ,t) = k(θ) j= δ(t jt)

3 The Standard Map Integrate from t = -ε to t = ε Then from t = ε to t = T ε θ(ε) = θ( ε) + 1 ε p(t)dt I ε p(ε) = p( ε) + k(θ(0)) θ(t ε) = θ(ε) + (T ε) p(ε) I p(t ε) = p(ε) put it together: p = p(-ε), p =p(τ ε), θ = θ + T I p p = p + k(θ)

4 The Standard Map Integrate from t = -ε to t = ε θ(ε) = θ( ε) + 0 p(ε) = p( ε) + k(θ(0)) Then from t = ε to t = T ε Let ε 0 θ(t ε) = θ(ε) + T I p(ε) p(t ε) = p(ε) put it together: p = p(-ε), p =p(τ ε), θ = θ + T I p p = p + k(θ)

5 Chirikov & Taylor Boris Chirikov: charged particles in electrostatic waves Brian Taylor: magnetic field lines in tokamaks Also: Frenkel-Kontorova model in solid-state physics Cyclotron

6 The Standard Map K sin(θ) Chirikov-Taylor Map Scale and rename variables x = x + y y = y k 2π sin(2πx) Split-step, Euler integrator for pendulum

7 ORbits x = x + y mod 1 y = y F(x) An orbit is a sequence: (x t,p t ), t integer If k = 0, orbits are simple x t = x 0 + y 0 t mod 1 y = y t 0 e.g. rigid rotation on circle if y is irrational, orbits are dense, otherwise periodic.

8 Phase Portraits k = 0.3 k = 0.971

9 Phase Portraits k = 2.0 k = 6.0

10 Islands Everywhere? Chaotic region is a fat fractal Umberger, D. K. and J. D. Farmer (1985). Fat Fractals on the Energy Surface. Physical Review Letters 55: Genericity of islands Duarte, P. (1994). Plenty of Elliptic Islands for the Standard Familty of Area Preserving Maps. Annales de L'institut Henri Poincare. Analyse 11(4): 359.

11 INtegrability x = x + y mod 1 y = y F(x) An integral is a function left invariant by f I = I f Orbits must lie on contours I(x,y) = constant Integrable standard maps were classified by Suris, e.g., F(x) = 2 π arctan asin(2π x 1+ acos(2π x)

12 Periodic ORbits x = x + y mod 1 y = y F(x) Lift map to, x not mod 1 2 Type (p,q) periodic orbit (x, y ) = f q (x, y ) = (x + p, y ) (x, y ) q q Rotation number ρ=p/q (0,1): Fixed points (x,y) = (0,0), stable (elliptic) if 0 < k < 2 (x,y) = (0.5,0), unstable (hyperbolic) if 0 < k

13 STability x = x + y mod 1 y = y F(x) A periodic orbit is a fixed point of f n (x) Stability is governed by the 2x2 matrix Df n (x) Df (x, y) = 1 k cos(2π x) 1 k cos(2π x) 1 Multipliers (eigenvalues) λ 1 stable (x,y) = (0,0), stable (elliptic, λ = 1) if 0 < k < 2 (x,y) = (0.5,0), unstable (hyperbolic, 0 < λ 1 < 1 < λ 2 ) if 0 < k

14 Stability: The Residue det(df n ) = 1 λ 2 τλ +1 = 0 τ = tr(df n ) Greene s Residue: R = 1 4 ( 2 τ ) Elliptic modulus one multipliers λ = 1 τ 2 0 R τ R Hyperbolic real, positive multipliers 0 < λ 1 < 1 < λ 2 τ > 2 R < 0 Reflection Hyperbolic negative multipliers λ 1 < 1 < λ 2 < 0 τ < 2 R > 1

15 quasiperiodic Orbits For k = 0, y irrational: orbit dense on circle rotational invariant circle ω = y is irrational winding number Kolmogorov-Arnold-Moser (KAM) theory circles with sufficiently irrational ω persist for k small. Poincaré-Birkhoff: rational circles destroyed

16 quasiperiodic Orbits KAM Theory requires Diophantine Condition: Twist Condition: D c,τ = {ω : nω m > c n τ } x y (x, y) = 0 Smoothness: f C 3+ε (R 2 )

17 TWist τ = x' y x = x + y mod 1 y = y F(x) Twist is the rate of change of rotation frequency with amplitude i.e., anharmonicity (x,y) f Y(x ) (x,y ) Df(v) Twist is essential for x KAM theory (preservation of circles) Aubry-Mather theory (variational principles)

18 Twist x = x + y mod 1 y = y F(x)

19 Diophantine D c,τ = {ω : nω m > c n τ } If τ > 1, D c,τ is a Cantor set with measure meas(d c,τ ) c 0 1 Examples: Golden mean ( ) ω = γ = Algebraic irrationals: ω 2 + aω + b = 0, a,b Z Constant type Bounded c.f. elements ω = a 0 + a a = [a o ; a 1, a 2, ]

20 Reversibilty x = x + y mod 1 y = y F(x) A map f is reversible if it is the same as its inverse upon a change of coordinates: f S = S f 1 For an odd force F(x)=-F(-x) S 0 (x, y) = ( x, y + F(x)) S is an involution: S 2 = I. f = ( f S ) S = S S thus f is the composition of two involutions

21 Reversibilty x = x + y mod 1 y = y F(x) Symmetric orbits have points on fixed sets Fix(S 0 ) = {(0, y)} "#$ Fix(S 1 ) = {( 1 2 y, y)} %&'(#$ " * periodic orbits: if n=2k " %&'(#$ ) * (x 0, y 0 ) Fix(S 0 ), %&'(# ) * (x k, y k ) Fix(S 0 R m ) = { m 2, y} 1D secant: "#" %&'(# " * G(y 0 ) = x k (0, y 0 ) m 2!"#$!"#$ "#"! "#$

22 Stable Manifolds W s { (Λ) (x, y) : f t } (x, y) Λ, as t "#$ Stable Manifolds for fixed points and periodic orbits " "#" Under reversor SW s (Λ) = W u (SΛ)!"#$!"#$ "#"! "#$ k = 1.2

23 Resonance Zones Resonance zones bounded by initial segments of W u and W s. f h 0 h 1 All area not in rotational invariant circles filled by (p,q)-resonance zones! u W s W h -1 (0,1) Resonance s W u W Turnstiles give escaping flux, Area/iteration k=1.5

24 Resonance Zones For k > k cr resonances fill entire phase space! Chen, Q. (1987). Area as a Devil s Staircase in Twist Maps. Phys. Lett. A 123: I.Or b / 0.6 po ~ 0.4~ 0,2 O.~ 0.4 A(~) ' I... I,' x k = 1.97 o', 0'2 0'3 o4' 05 It Area Devil s Staircase

25 On to Twist Maps!