Chaotic behavior of disordered nonlinear lattices

Transcription

1 Chaotic behavior of disordered noninear attices Haris Skokos Department of Mathematics and Appied Mathematics, University of Cape Town Cape Town, South Africa E-mai: URL:

2 Outine Disordered attices: The quartic Kein-Gordon (KG) mode The disordered noninear Schrödinger equation (DNLS) Different dynamica behaviors Chaotic behavior of the KG mode Lyapunov exponents Deviation Vector Distributions Numerica methods Sympecic Integrators Tangent Map method Summary

3 Interpay of disorder and noninearity Waves in disordered media Anderson ocaization [Anderson, Phys. Rev. (1958)]. Experiments on BEC [Biy et a., Nature (008)] Waves in noninear disordered media ocaization or deocaization? Theoretica and/or numerica studies [Shepeyansky, PRL (1993) Moina, Phys. Rev. B (1998) Pikovsky & Shepeyansky, PRL (008) Kopidakis et a., PRL (008) Fach et a., PRL (009) S. et a., PRE (009) Muansky & Pikovsky, EPL (010) S. & Fach, PRE (010) Laptyeva et a., EPL (010) Muansky et a., PRE & J.Stat.Phys. (011) Bodyfet et a., PRE (011) Bodyfet et a., IJBC (011)] Experiments: propagation of ight in disordered 1d waveguide attices [Lahini et a., PRL (008)]

4 The Kein Gordon (KG) mode p ε 1 1 H = + u + N 4 K + u u+1 - u =1 4 W with fixed boundary conditions u 0 =p 0 =u N+1 =p N+1 =0. Typicay N= Parameters: W and the tota energy E. ε chosen uniformy from,. Linear case (negecting the term u 4 /4) Ansatz: u =A exp(iωt). Norma modes (NMs) A ν, - Eigenvaue probem: λ = Wω -W -, ε = W(ε - 1) λa = ε A - (A +1 + A -1 ) with The discrete noninear Schrödinger (DNLS) equation We aso consider the system: β H = ε ψ - ψ ψ + ψ ψ N 4 * * D + ψ =1 W W where ε chosen uniformy from, and is the noninear parameter. Conserved quantities: The energy and the norm S of the wave packet.

5 Distribution characterization We consider normaized energy distributions in norma mode (NM) space z ν Eν m E with E = A + ωa ν ν ν ν m of the νth NM (KG) or norm distributions (DNLS). Second moment: Participation number: 1 N ν ν=1, where A ν is the ampitude m = ν - ν z N with ν = νzν P= 1 N z ν=1 ν measures the number of stronger excited modes in z ν. Singe mode P=1. Equipartition of energy P=N. ν=1

6 Linear case: Δ K = 1+ W ω ν, + W 4 (Δ D = W + 4) Scaes, width of the squared frequency spectrum: Locaization voume of an eigenstate: V~ N =1 1 A 4 ν, Average spacing of squared eigenfrequencies of NMs within the range of a ocaization voume: d K Δ V K Noninearity induced squared frequency shift of a singe site osciator δ = 3E ε E The reation of the two scaes dk ΔK with the noninear frequency shift δ determines the packet evoution. (δ = β ψ )

7 Different Dynamica Regimes Three expected evoution regimes [Fach, Chem. Phys (010) - S. & Fach, PRE (010) - Laptyeva et a., EPL (010) - Bodyfet et a., PRE (011)] Δ: width of the frequency spectrum, d: average spacing of interacting modes, δ: noninear frequency shift. Weak Chaos Regime: δ<d, m ~t 1/3 Frequency shift is ess than the average spacing of interacting modes. NMs are weaky interacting with each other. [Moina, PRB (1998) Pikovsky, & Shepeyansky, PRL (008)]. Intermediate Strong Chaos Regime: d<δ<δ, m ~t 1/ m ~t 1/3 Amost a NMs in the packet are resonanty interacting. Wave packets initiay spread faster and eventuay enter the weak chaos regime. Seftrapping Regime: δ>δ Frequency shift exceeds the spectrum width. Frequencies of excited NMs are tuned out of resonances with the nonexcited ones, eading to seftrapping, whie a sma part of the wave packet subdiffuses [Kopidakis et a., PRL (008)].

8 Singe site excitations DNLS W=4, β= 0.1, 1, 4.5 KG W = 4, E = 0.05, 0.4, 1.5 No strong chaos regime sope 1/3 sope 1/3 In weak chaos regime we averaged the measured exponent α (m ~t α ) over 0 reaizations: sope 1/6 sope 1/6 α=0.33±0.05 (KG) α=0.33±0.0 (DLNS) Fach et a., PRL (009) S. et a., PRE (009)

9 KG: Different spreading regimes

10 Crossover from strong to weak chaos We consider compact initia wave packets of width L=V [Laptyeva et a., EPL (010) - Bodyfet et a., PRE (011)]. Time evoution DNLS KG

11 Crossover from strong to weak chaos (bock excitations) DNLS β= 0.04, 0.7, 3.6 KG E= 0.01, 0., 0.75 W=4 Average over 1000 reaizations! (og t) d og m dog t α=1/ α=1/3 Laptyeva et a., EPL (010) Bodyfet et a., PRE (011)

12 Lyapunov Exponents (LEs) Roughy speaking, the Lyapunov exponents of a given orbit characterize the mean exponentia rate of divergence of trajectories surrounding it. Consider an orbit in the N-dimensiona phase space with initia condition x(0) and an initia deviation vector from it v(0). Then the mean exponentia rate of divergence is: 1 mlce = λ 1 = im n t t v(t) v(0) λ 1 =0 Reguar motion (t -1 ) λ 1 0 Chaotic motion

13 KG: LEs for singe site excitations (E=0.4)

14 KG: Weak Chaos (E=0.4)

15 KG: Weak Chaos Individua runs Linear case E=0.4, W=4 sope -1 sope -1 α L = -1/4 Average over 50 reaizations Singe site excitation E=0.4, W=4 Bock excitation (1 sites) E=0.1, W=4 Bock excitation (37 sites) E=0.37, W=3 L d og dog t S. et a. PRL (013)

16 Deviation Vector Distributions (DVDs) Deviation vector: DVD: v(t)=(δu 1 (t), δu (t),, δu N (t), δp 1 (t), δp (t),, δp N (t)) u p w u p

17 Deviation Vector Distributions (DVDs) Individua run E=0.4, W=4 Chaotic hot spots meander through the system, supporting a homogeneity of chaos inside the wave packet.

18 Integration scheme Consider an N degree of freedom autonomous Hamitonian system having a Hamitonian function of the form: positions momenta H(q 1,q,,q N, p 1,p,,p N ) The time evoution of an orbit (trajectory) with initia condition P(0)=(q 1 (0), q (0),,q N (0), p 1 (0), p (0),,p N (0)) is governed by the Hamiton s equations of motion dpi H dqi H = -, = dt q dt p i i

19 Autonomous Hamitonian systems Let us consider an N degree of freedom autonomous Hamitonian systems of the form: As an exampe, we consider the Hénon-Heies system: Hamiton equations of motion: Variationa equations:

20 Sympectic Integrators (SIs) Formay the soution of the Hamiton equations of motion can be written as: n dx t n tlh = H, X = LHX X(t) = LHX = e X dt n! where X is the fu coordinate vector and L H the Poisson operator: N H f H f LH f = - j=1 p j q j q j p j If the Hamitonian H can be spit into two integrabe parts as H=A+B, a sympectic scheme for integrating the equations of motion from time t to time t+τ consists of approximating the operator by j n0 e τl H τlh τ(l A +L B ) ciτla diτlb n+1 e = e e e + O(τ ) i=1 for appropriate vaues of constants c i, d i. This is an integrator of order n. So the dynamics over an integration time step τ is described by a series of successive acts of Hamitonians A and B.

21 Sympectic Integrator SABA C e L H The operator can be approximated by the sympectic integrator [Laskar & Robute, Ce. Mech. Dyn. Astr. (001)]: c L d L c L d L c L SABA = e e e e e A 1 B A 1 B 1 A with c 1 = -, c =, d 1 =. 6 3 The integrator has ony sma positive steps and its error is of order. In the case where A is quadratic in the momenta and B depends ony on the positions the method can be improved by introducing a corrector C, having a sma negative step: 3 c - L A,B,B with - 3 c =. 4 C = e Thus the fu integrator scheme becomes: SABAC = C (SABA ) C and its error is of order 4.

22 Tangent Map (TM) Method Use sympectic integration schemes for the whoe set of equations (S. & Gerach, PRE (010) We appy the SABAC integrator scheme to the Hénon-Heies system (with ε=1) by using the spitting: with a corrector term which corresponds to the Hamitonian function: We approximate the dynamics by the act of Hamitonians A, B and C, which correspond to the sympectic maps:

23 Tangent Map (TM) Method Let The system of the Hamiton s equations of motion and the variationa equations is spit into two integrabe systems which correspond to Hamitonians A and B.

24 Tangent Map (TM) Method Any sympectic integration scheme used for soving the Hamiton equations of motion, which invoves the act of Hamitonians A and B, can be extended in order to integrate simutaneousy the variationa equations [S. & Gerach, PRE (010) Gerach & S., Discr. Cont. Dyn. Sys. (011) Gerach et a., IJBC (01)].

25 The KG mode We appy the SABAC integrator scheme to the KG Hamitonian by using the spitting: H K = N = 1 p + ε 1 1 u + u + u - u 4 W A B with a corrector term which corresponds to the Hamitonian function: C= N A,B,B = u ( ε u ) ( u -1 + u+1 - u ). =1 1 W

26 The DNLS mode A nd order SABA Sympectic Integrator with 5 steps, combined with approximate soution for the B part (Fourier Transform): SIFT β 4 1 H = ε ψ + ψ - ψ ψ + ψ ψ ψ = q + ip ε β D q + p + q + p - qnqn+1 -pnpn+ 1 8 H = * * +1, D +1 A B

27 The DNLS mode Sympectic Integrators produced by Successive Spits (SS) H = D ε A β q - q q -p 8 + p + q + p B n n+1 n n+1 p B 1 B Using the SABA integrator we get a nd order integrator with 13 steps, SS : (3-3) (3-3) τl A τ 3τ τ L LA L τ L 6 B B 3 6 SS = e e e e e A ' = / (3-3) (3-3) τ' L B τ' 3τ' τ' τ' L 6 LB L B L 1 B B1 e e e e e (3-3) (3-3) τ' L B τ' 3τ' τ' τ' L 6 LB L B L 1 B B1 e e e e e

28 Three part spit sympectic integrators for the DNLS mode Three part spit sympectic integrator of order, with 5 steps: ABC H = D ε β q - q q -p 8 + p + q + p n n+1 n n+1 A B C τ τ τ τ L L L L τl A B B A C ABC = e e e e e p This ow order integrator has aready been used by e.g. Chambers, MNRAS (1999) Goździewski et a., MNRAS (008).

29 Composition Methods: 4 th order SIs Starting from any nd order sympectic integrator S nd, we can construct a 4 th order integrator S 4th using the composition method proposed by Yoshida [Phys. Lett. A (1990)]: 1/3 4th nd nd nd /3 1 1/3 S (τ) = S (x τ) S (x τ) S (x τ), x = -, x = - - In this way, starting with the nd order integrators SS, SIFT and ABC we construct the 4 th order integrators: SS 4 with 37 steps SIFT 4 with 13 steps ABC 4 [Y]with 13 steps Composition method proposed by Suzuki [Phys. Lett. A (1990)]: S (τ) = S (p τ) S (p τ) S ((1-4p )τ) S (p τ) S (p τ) 4th nd nd nd nd nd 1/3 1 4 p =, 1-4p 1/3 = - 1/ Starting with the nd order integrators ABC we construct the 4 th order integrator: ABC 4 [S] with 1 steps.

30 More 4 th order SIs We construct few more integration schemes by considering the 4 th order sympectic integrators ABA864, ABA1064, ABAH864 and ABAH1064 introduced by Banes et a., App. Num. Math. (013) and Farrés et a., Ce. Mech. Dyn. Astr. (013). Approximating the soution of the B part by a Fourier Transform we construct the 4 th order integrators: SIFT with 43 steps SIFT with 49 steps Using successive spits for the B part and impementing the SABA integrator for its integartion, we construct the 4 th order integrators: SS with 49 steps SS with 55 steps

31 4 th order integrators: Numerica resuts (I) SIFT 4 τ=0.15 SIFT τ=0.05 ABC 4 [S] τ=0.1 SS 4 τ=0.1 ABC 4 [Y] τ=0.05 E r : reative energy error S r : reative norm error T c : CPU time (sec) S. et a., Phys. Lett. A (014)

32 4 th order integrators: Numerica resuts (II) SIFT τ=0.5 ABC 4 [Y] τ=0.05 SIFT τ=0.5 SS τ=0.5 SS τ=0.5 E r : reative energy error S r : reative norm error T c : CPU time (sec) S. et a., Phys. Lett. A (014)

33 Summary (I) We presented three different dynamica behaviors for wave packet spreading in 1d noninear disordered attices: Weak Chaos Regime: δ<d, m ~t 1/3 Intermediate Strong Chaos Regime: d<δ<δ, m ~t 1/ m ~t 1/3 Seftrapping Regime: δ>δ Generaity of resuts: Two different modes: KD and DNLS, Predictions made for DNLS are verified for both modes. Lyapunov exponent computations show that: Chaos not ony exists, but aso persists. Sowing down of chaos does not cross over to reguar dynamics. Chaotic hot spots meander through the system, supporting a homogeneity of chaos inside the wave packet. Our resuts suggest that Anderson ocaization is eventuay destroyed by noninearity, since spreading does not show any sign of sowing down.

34 Summary (II) We presented severa efficient integration methods suitabe for the integration of the DNLS mode, which are based on sympectic integration techniques. The construction of sympectic schemes based on 3 part spit of the Hamitonian was emphasized (ABC methods). Agorithms based on the integration of the B part of Hamitonian via Fourier transforms, i.e. methods SIFT, SIFT 4, SIFT and SIFT succeeded in keeping the reative norm error S r very ow. Drawback: they require the number of attice sites to be k, k. We hope that our resuts wi initiate future research both for the theoretica deveopment of new, improved 3 part spit integrators, as we as for their appications to different dynamica systems.

35 References Fach, Krimer, S. (009) PRL, 10, S., Krimer, Komineas, Fach (009) PRE, 79, S., Fach (010) PRE, 8, Laptyeva, Bodyfet, Krimer, S., Fach (010) EPL, 91, Bodyfet, Laptyeva, S., Krimer, Fach (011) PRE, 84, Bodyfet, Laptyeva, Gigoric, S., Krimer, Fach (011) Int. J. Bifurc. Chaos, 1, 107 S., Gkoias, Fach (013) PRL, 111, Tieeman, S., Lazarides (014) EPL, 105, 0001 S., Gerach (010) PRE, 8, Gerach, S. (011) Discr.Cont. Dyn. Sys.-Supp. 011, 475 Gerach, Egg, S. (01) Int. J. Bifurc. Chaos,, Gerach, Egg, S., Bodyfet, Papamikos (013) nin.cd/ S., Gerach, Bodyfet, Papamikos, Egg (014) Phys. Lett. A, 378, 1809 Thank you for your attention