Recorências e difusão anômala em sistemas Hamiltoneanos caóticos

Transcrição

1 Recorências e difusão anômala em sistemas Hamiltoneanos caóticos IFUSP - Março 2010 Eduardo. G. Altmann

2 Apresentação 1: Torus, mapa padrão, ilhas ao redor de ilhas

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8 Apresentação II: Recorrências para detectar rompimento de tori

9 Interesse em sistemas temporalmente reversíveis (dinâmica quasi-hamiltoneana pode aparecer) Se mais de uma simetria esta presente no sistema a condição de torção (twist) é violada: Exemplo: Osciladores acoplados N=4

10 4 osciladores de fase acoplados: Sistema não Hamiltoneano mas com dinâmica quasi- Hamiltoneana e torus não torcidos!

11 Reconexões típicas de sistemas não torcionais também ocorrem em sistemas não Hamiltoneanos

12 Para o transporte de trajetórias é importante determinar quais parâmetros (ε,ω) o torus existe (divide o espaço de fases). Método: Verificar se uma trajetória que tem de pertencer ao torus (IP) satisfaz o teorema de Slater (e.g., máximo 3 Ts distintos). Particularmente útil quando: - Grande número de parâmetros (ε,ω) tem de ser varridos. Sistema de tempos contínuo (difícil integração/sessão de Poincaré) Parâmetros (ε,ω) próximos ao rompimento são escolhidos. Nesse caso ilhas ( stickiness ) fazem demais métodos muito lentos Limitação: -N=4, i.e., mapas bi-dimensionais

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14 Rompimento do torus

15 Exemplo: mapa bi-dimensional Jacobiano: Simetrias:

16 Exemplo: mapa bi-dimensional

17 Apresentação III: mapa linear por partes espaço de fases hierárquico

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21 Caso genérico / hierárquico

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23 Figure 2.3: (Color online) Sticking time distribution ρ(τ) for 100 different standard maps (2.14) with a constant K added to the y equation: K [0.5, 0.6],K [0, 0.2]. The central green (gray) curve is the average [for fixed ρ(τ)] over all curves, and the red curve (axis on the right) corresponds to the standard deviation of the curves (for fixed ρ(τ) projected to the x-axis). The further parameters are equivalent to those of Fig. 6.1b below.

24 Apresentação 1V: Efeito de ruído branco e altas dimensões no aprisionamento de trajetórias

25 Coupled standard maps: 2.1 Motivation / model 2.2 Noise perturbation 2.3 High dimensional

26 Qual o problema? (do ponto de vista de Mec. Estatística)

27 Qual o problema? (do ponto de vista de Mec. Estatística) Violate the hypothesis of strong chaos: 1. Ergodicity, i.e., negligible measure of regular components 2. Strong mixing, i.e., fast decay of correlations

28 Qual o problema? (do ponto de vista de Mec. Estatística) Violate the hypothesis of strong chaos: 1. Ergodicity, i.e., negligible measure of regular components 2. Strong mixing, i.e., fast decay of correlations What happens for increasing phase space dimension?

29 Coupled symplectic maps model C

30 Coupled symplectic maps model C

31 Coupled symplectic maps model Map is symplectic iff:

32 Coupled symplectic maps model Map is symplectic iff:

33 Coupled standard maps: 2.1 Motivation / model 2.2 Noise perturbation 2.3 High dimensional

34 2.2 Noise perturbation

35 2.2 Noise perturbation

36 2.2 Noise perturbation

37 2.2 Noise perturbation

38 2.2 Noise perturbation

39 2.2 Noise perturbation

40 RW 2.2 theory Noise perturbation

41 RW 2.2 theory Noise perturbation

42 2.2 Noise perturbation RW theory

43 Coupled standard maps: 2.1 Motivation / model 2.2 Noise perturbation 2.3 High dimensional

44 Coupled symplectic maps model C

45 Coupled symplectic maps model Ergodicity? C

46 Coupled symplectic maps model Ergodicity? C

47 Coupled symplectic maps model Ergodicity?

48 Coupled symplectic maps model 1. Ergodicity, i.e., negligible measure of regular components e.g., zero measure sets on Bunimovich stadium Billiards 2. Strong mixing, i.e., fast decay of correlations? N=2-5 show power-law behavior [Kantz, Grassberger (1987), Ding, Bountis, Ott (1990)]

49 Coupled symplectic maps model Strong mixing?

50 Coupled symplectic maps model Strong mixing?

51 Coupled symplectic maps model Strong mixing?

52 Coupled symplectic maps model Strong mixing?

53 Coupled symplectic maps model

54 Coupled symplectic maps model

55 Coupled symplectic maps model

56 Coupled symplectic maps model

57 Coupled symplectic maps model Strong mixing?

58 Coupled symplectic maps model Strong mixing? ξ =0.05 N=2,3,4,5 ξ =0.05 N=2,3,...,15

59 Coupled symplectic maps model Strong mixing?

60 Coupled symplectic maps 1. Ergodicity, i.e., negligible measure of regular components 2. Strong mixing, i.e., fast decay of correlations Non-exponential decay, but sufficiently fast power-law

61 Apresentação V: fluído incompressível

62 Passive scalar field θ( x, t) (contaminant), advected by a flow with velocity field given by v(x, t) [Aref,1984] θ t +.( vθ) =D m 2 θ, where D m is the molecular diffusion coefficient. The motion of fluid elements (Lagrangian description) is written as d x dt = v( x, t)+η(t), where η i (t)η j (t ) =2D m δ i,j δ(t t ). Consider an incompressible. v =02-D fluid x =(x, y). In this case there exist a stream function ψ(x, y, t) such that dx dt = v x = ψ y and dy dt = v y = ψ x.

63 I- One model with generic features Consider a fluid channel infinite in the x direction having the following two flows: Laminar regime: ψ 1 (x, y) = v 1 sin(πy); Vortex regime: ψ 2 (x, y) =v 2 cos(2x)(1 y 2 ) 2 Doktorandentag 12/12/2006 p.4/11

64 I- One model with generic features Alternating periodically between the two regimes in a period t 0 and mapping the evolution from nt 0 (n + 1)t 0 one gets x n = x n+1 + λ sin(πy n ) 2ρ π y n(1 yn) 2 cos[2π(x n + 1)] + ξδ n, y n+1 = y n ρ(1 yn) 2 2 sin[2πx n+1 ]+ξδ n. I- One model with generic features ρ = πv 2 t 0 /2 intensity of the vortex regime; x n x n+1 + λ sin(πy n ) 2ρ π y n(1 yn) 2 λ = v 1 t 0 /2 intensity of cos[2π(x n + 1)] + ξδ n, y n+1 = y n Alternating ρ(1 periodically yn) 2 the 2 laminar regime; ξ intensity of the white noise sin[2πx variable between n+1 δ (ξ the ]+ξδ two Dregimes n. m ); in a period t 0 and mapping the evolution from nt 0 (n + 1)t 0 one gets The value ρ =0.6 x n will = xbe n+1 fixed + λand sin(πy initially n ) 2ρ the π y n(1 deterministic yn) 2 cos[2π(x case n + ξ =0 1)] + ξδ n, is considered y for two n+1 = values y n ρ(1 of λ yn) 2 2 sin[2πx n+1 ]+ξδ n. ρ = πv 2 t 0 /2 intensity of the vortex regime; λ = v 1 t 0 /2 intensity of the laminar regime; ξ intensity of the white noise variable δ (ξ D m ); Doktorandentag 12/12/2006 p.5/11 The value ρ =0.6 will be fixed and initially the deterministic case ξ =0 is considered for two values of λ

65 Alternating periodically between the two regimes in a period t 0 and mapping the evolution from nt 0 (n + 1)t 0 one gets Espaço misto para dois parâmetros de controle x n = x n+1 + λ sin(πy n ) 2ρ π y n(1 yn) 2 cos[2π(x n + 1)] + ξδ n, y n+1 = y n ρ(1 yn) 2 2 sin[2πx n+1 ]+ξδ n. ρ = πv 2 t 0 /2 intensity of the vortex regime; λ = v 1 t 0 /2 intensity of the laminar regime; ξ intensity of the white noise variable δ (ξ D m ); The value ρ =0.6 will be fixed and initially the deterministic case ξ =0 is considered for two values of λ Doktorandentag 12/12/2006 p.5/

66 Transporte super-difusivo!x "=0.25 "=1 #=1 (normal diffusion) #=2 (ballistic motion) t

67 x flying flying trapping flying!=1!=0.25 y y flying flying trapping flying number of iterations - n

68 Estatísitca de aprisionamento e voo Flights #=1 Flights #=0.25 Traps #=0.25 "(!) !

69 Efeito da difusão molecular no aprisionamento $=1 flies $=0.25 flies $=0.25 traps #=0 #=0.005 #= "(!) 10-5 "(!) 10-5 "(!) ! ! ! Tempo final do regime de super-aprisionamento t~ 1/ξ 2

70 Coeficiente de difusão como função do tempo (a) #=0.6 $=1 D=!x 2 /t "=0 "= "=0.001 "=0.002 "=0.01 "=0.1 λ= e+05 t

71 Coeficiente de difusão como função do tempo λ=0.25

72 Difusão total (advecção+molecular) como função da difusão molecular

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