Chaotic behavior of disordered nonlinear systems

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1 Chaotic behavior of disordered noninear systems Haris Skokos Department of Mathematics and Appied Mathematics, University of Cape Town Cape Town, South Africa E-mai: URL: Work in coaboration with Sergej Fach, Joshua Bodyfet, Ioannis Gkoias, Dima Krimer, Stavros Komineas, Tanya Laptyeva, Bob Senyange, Tassos Bountis, Chris Antonopouos

2 Outine Disordered 1D attices: The quartic Kein-Gordon (KG) mode The disordered noninear Schrödinger equation (DNLS) Different dynamica behaviors Chaotic behavior of the KG mode q-gaussian distributions Lyapunov exponents Deviation Vector Distributions Integration techniques (Sympectic integrators and 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 (2008)] Waves in noninear disordered media ocaization or deocaization? Theoretica and/or numerica studies [Shepeyansky, PRL (1993) Moina, Phys. Rev. B (1998) Pikovsky & Shepeyansky, PRL (2008) Kopidakis et a., PRL (2008) Fach et a., PRL (2009) S. et a., PRE (2009) Muansky & Pikovsky, EPL (2010) S. & Fach, PRE (2010) Laptyeva et a., EPL (2010) Muansky et a., PRE & J.Stat.Phys. (2011) Bodyfet et a., PRE (2011) Bodyfet et a., IJBC (2011)] Experiments: propagation of ight in disordered 1d waveguide attices [Lahini et a., PRL (2008)]

4 The Kein Gordon (KG) mode p ε 1 1 H = + u + N K + u u+1 - u = 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,. 2 2 Linear case (negecting the term u 4 /4) Ansatz: u =A exp(iωt). Norma modes (NMs) A ν, - Eigenvaue probem: 2 λ =Wω -W - 2, 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 2 4 * * D + ψ =1 2 W W where ε chosen uniformy from, and is the noninear parameter Conserved quantities: The energy and the norm S of the wave packet. 2

5 Distribution characterization We consider normaized energy distributions in norma mode (NM) space z ν Eν m E with E = A + ωa ν 2 ν ν ν m of the νth NM (KG) or norm distributions (DNLS). Second moment: Participation number: 1 N 2 2 ν ν=1, where A ν is the ampitude m = ν - ν z N with ν = νzν P= 1 N 2 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: ω ν, W 4 Δ K = 1+ W (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 2ε E The reation of the two scaes d K ΔK with the noninear frequency shift δ determines the packet evoution. 2 (δ = β ψ )

7 Different Dynamica Regimes Three expected evoution regimes [Fach, Chem. Phys (2010) - S. & Fach, PRE (2010) - Laptyeva et a., EPL (2010) - Bodyfet et a., PRE (2011)] Δ: width of the frequency spectrum, d: average spacing of interacting modes, δ: noninear frequency shift. Weak Chaos Regime: δ<d, m 2 ~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 (2008)]. Intermediate Strong Chaos Regime: d<δ<δ, m 2 ~t 1/2 m 2 ~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 (2008)].

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 2 ~t α ) over 20 reaizations: sope 1/6 sope 1/6 α=0.33±0.05 (KG) α=0.33±0.02 (DLNS) Fach et a., PRL (2009) S. et a., PRE (2009)

9 KG: Different spreading regimes

10 KG: Different spreading regimes

11 Crossover from strong to weak chaos We consider compact initia wave packets of width L=V [Laptyeva et a., EPL (2010) - Bodyfet et a., PRE (2011)]. Time evoution DNLS KG

12 Crossover from strong to weak chaos (bock excitations) DNLS β= 0.04, 0.72, 3.6 KG E= 0.01, 0.2, 0.75 W=4 Average over 1000 reaizations! (og t) d og m dog t 2 α=1/2 α=1/3 Laptyeva et a., EPL (2010) Bodyfet et a., PRE (2011)

13 q-gaussian distributions We construct probabiity distribution functions (pdfs) of rescaed sums of M vaues of an observabe η(t i ), which depends ineary on positions u. We rescae them by their standard deviation and compare the resuting numericay computed pdfs with a q-gaussian [Tsais, Springer (2009)] q (entropic index) q=1: Gaussian pdf q 1: system is at the so-caed edge of chaos regime, characterized by the non-additive and generay non-extensive Tsais entropy.

14 q-gaussian distributions Weak chaos case: E=0.4, W=4. Dotted curves: Gaussian pdf (q=1) η 1 =u 1 We defined chaos η 29 =u 486 +u u 513 +u 514 (29 centra partices) q 1 edge of chaos Antonopouos et a. Chaos (2014)

15 q-gaussian distributions Weak chaos case: E=0.4, W=4. Antonopouos et a. Chaos (2014)

16 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 2N-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 v (t) m L C E = λ 1 = im n t t v (0 ) λ 1 =0 Reguar motion (t -1 ) λ 1 0 Chaotic motion

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

18 KG: Weak Chaos (E=0.4)

19 KG: Weak Chaos (E=0.4)

20 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 (L=21 sites) E=0.21, W=4 Bock excitation (L=37 sites) E=0.37, W=3 L d og dog t S. et a., PRL (2013)

21 Deviation Vector Distributions (DVDs) Deviation vector: DVD: 2 2 v(t)=(δu 1 (t), δu 2 (t),, δu N (t), δp 1 (t), δp 2 (t),, δp N (t)) 2 2 u p w u p

22 Deviation Vector Distributions (DVDs) Deviation vector: DVD: 2 2 v(t)=(δu 1 (t), δu 2 (t),, δu N (t), δp 1 (t), δp 2 (t),, δp N (t)) 2 2 u p w u p

23 Deviation Vector Distributions (DVDs) Deviation vector: DVD: 2 2 v(t)=(δu 1 (t), δu 2 (t),, δu N (t), δp 1 (t), δp 2 (t),, δp N (t)) 2 2 u p w u p

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

25 Individua run, L=37, E=0.37, W=3 DVDs Weak chaos Singe site excitation E=0.4, W=4 Bock excitation (21 sites) E=0.21, W=4 Bock excitation (37 sites) E=0.37, W=3 Maximum absoute deviation of DVD s mean position M ( t) max w( t) w( t) [ t, tt]

26 KG: Strong chaos Energy DVD Individua run L=83, E=8.3, W=3

27 Lyapunov exponent KG: Strong chaos Bock excitation (37 sites) E=7.4, W=3 Bock excitation (83 sites) E=8.3, W=3 Bock excitation (330 sites) E=33.0, W=1 Characteristics of DVD

28 Weak and Strong chaos Same disordered reaization, L=37, W=3, E=0.37 and E=7.4 Energy DVD Energy DVD

29 Weak and Strong chaos: LEs

30 Weak and Strong chaos: DVDs For both cases the DVD s participation number remains practicay constant.

31 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:

32 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 τl H 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.

33 Sympectic Integrator SABA 2 C e L H The operator can be approximated by the sympectic integrator [Laskar & Robute, Ce. Mech. Dyn. Astr. (2001)]: S A B A = e c 1 L A e d 1 L B e c 2 L A e d 1 L B e c 1 L A with c 1 = -, c 2 =, d 1 = The integrator has ony sma positive steps and its error is of order 2. 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 C = e 2-3 with c =. 24 Thus the fu integrator scheme becomes: SABAC 2 = C (SABA 2 ) C and its error is of order 4. 2

34 Tangent Map (TM) Method Use sympectic integration schemes for the whoe set of equations (S. & Gerach, PRE (2010) We appy the SABAC 2 integrator scheme to the Hénon-Heies system 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:

35 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.

36 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 (2010) Gerach & S., Discr. Cont. Dyn. Sys. (2011) Gerach et a., IJBC (2012)].

37 The KG mode We appy the SABAC 2 integrator scheme to the KG Hamitonian by using the spitting: H K = N = 1 p ε 1 1 u + u + u - u 2 4 2W A B with a corrector term which corresponds to the Hamitonian function: C= N 2 A,B,B = u ( ε u ) ( u -1 + u+1-2u ). =1 1 W 2

38 Summary We presented three different dynamica behaviors for wave packet spreading in 1d noninear disordered attices (KG and DNLS modes): Weak Chaos Regime: δ<d, m 2 ~t 1/3 Intermediate Strong Chaos Regime: d<δ<δ, m 2 ~t 1/2 m 2 ~t 1/3 Seftrapping Regime: δ>δ KG mode Lyapunov exponent computations show that: Chaos not ony exists, but aso persists. Sowing down of chaos does not cross over to reguar dynamics. mles and DVDs show different behaviors for the weak and the strong chaos regimes. Chaotic hot spots meander through the system, supporting a homogeneity of chaos inside the wave packet. The behavior of DVDs can provide important information about the chaotic behavior of a dynamica system.

39 A shameess promotion Contents 1. Paritz: Estimating Lyapunov Exponents from Time Series 2. Lega, Guzzo, Froesché: Theory and Appications of the Fast Lyapunov Indicator (FLI) Method 3. Barrio: Theory and Appications of the Orthogona Fast Lyapunov Indicator (OFLI and OFLI2) Methods 4. Cincotta, Giordano: Theory and Appications of the Mean Exponentia Growth Factor of Nearby Orbits (MEGNO) Method 5. Ch.S., Manos: The Smaer (SALI) and the Generaized (GALI) Aignment Indices: Efficient Methods of Chaos Detection 6. Sándor, Maffione: The Reative Lyapunov Indicators: Theory and Appication to Dynamica Astronomy 7. Gottwad, Mebourne: The 0-1 Test for Chaos: A Review 8. Siegert, Kantz: Prediction of Compex Dynamics: Who Cares About Chaos? 2016, Lect. Notes Phys., 915, Springer

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