The Smaller (SALI) and the Generalized (GALI) Alignment Index Methods of Chaos Detection: Theory and Applications. Haris Skokos

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1 The Smaller (SALI) and the Generalized (GALI) Alignment Index Methods of Chaos Detection: Theory and Applications Haris Skokos Max Planck Institute for the Physics of Complex Systems Dresden, Germany URL:

2 Outline Smaller ALignment Index SALI Definition Behavior for chaotic and regular motion Applications Generalized ALignment Index GALI Definition - Relation to SALI Behavior for chaotic and regular motion Applications Global dynamics Motion on low-dimensional tori Conclusions 2

3 Definition of Smaller Alignment Index (SALI) Consider the n-dimensional phase space of a conservative dynamical system (symplectic map or Hamiltonian flow). An orbit in that space with initial condition : P(0)=(x 1 (0), x 2 (0),,x n (0)) and a deviation vector v(0)=(dx 1 (0), dx 2 (0),, dx n (0)) The evolution in time (in maps the time is discrete and is equal to the number N of the iterations) of a deviation vector is defined by: the variational equations (for Hamiltonian flows) and the equations of the tangent map (for mappings) 3

4 Definition of SALI We follow the evolution in time of two different initial deviation vectors (v 1 (0), v 2 (0)), and define SALI (Skokos, 2001, J. Phys. A, 34, 10029) as: SALI(t) = min where { v ˆ (t) + v ˆ (t), v ˆ (t) - v ˆ (t) } v (t) v(t)= v (t) 1 ˆ 1 1 When the two vectors become collinear SALI(t) 0 4

5 Behavior of SALI for chaotic motion For chaotic orbits the two initially different deviation vectors tend to coincide with the direction defined by the maximal Lyapunov exponent. v(t) ˆ 2 P(t) v ˆ 1 (t) v(t) 2 SALI(t) v(t) 1 v(0) ˆ 2 SALI(0) v(0) ˆ 1 P(0) Trajectory 5

6 Behavior of SALI for chaotic motion We test the validity of the approximation SALIμe -(σ1-σ2)t (Skokos et al., 2004, J. Phys. A, 37, 6269) for a chaotic orbit of the 3D Hamiltonian ω H= (q +p )+q q +q q 3 i i i i=1 2 with ω 1 =1, ω 2 =1.4142, ω 3 =1.7321, Η=0.09 σ 1 ª0.037 slope=-(σ 1 -σ 2 )/ln(10) σ 2 ª

7 Behavior of SALI for regular motion Regular motion occurs on a torus and two different initial deviation vectors become tangent to the torus, generally having different directions. P(0) v(0) ˆ 1 v(0) ˆ 2 P(t) v(0) ˆ 1 v(0) ˆ 2 7

8 Applications Hénon-Heiles system For E=1/8 we consider the orbits with initial conditions: Ordered orbit, x=0, y=0.55, p x =0.2417, p y =0 Chaotic orbit, x=0, y=-0.016, p x = , p y =0 Chaotic orbit, x=0, y= , p x = , p y =0 8

9 Applications Hénon-Heiles system 0-4 log(sali) -8 t= y 0-4 log(sali) -8 t= y 9

10 Applications Hénon-Heiles system p y y 10

11 Applications 4D map x 1 = x 1+ x2 x 2 = x 2 - ν sin(x 1+ x 2) - μ [1- cos(x 1+ x 2 + x 3 + x)] 4 x 3 = x 3 + x4 x = x - κ sin(x + x ) - μ [1- cos(x + x + x + x)] (mod 2π) For ν=0.5, κ=0.1, μ=0.1 we consider the orbits: ordered orbit C with initial conditions x 1 =0.5, x 2 =0, x 3 =0.5, x 4 =0. chaotic orbit D with initial conditions x 1 =3, x 2 =0, x 3 =0.5, x 4 = X 2 0 C D X 1-1 (a) 0 (d) -2 logl N -3-4 log(sali) logn logn 11

12 Applications 4D Accelerator map We consider the 4D symplectic map x 1 cosω1 -sinω1 0 0 x1 2 2 x2 sinω1 cosω1 0 0 x 2+x1 -x3 = x cosω2 -sinω2 x3 x4 0 0 sinω2 cosω2 x4-2x1x3 describing the instantaneous sextupole kicks experienced by a particle as it passes through an accelerator (Turchetti & Scandale 1991, Bountis & Tompaidis 1991, Vrahatis et al. 1996, 1997). x 1 and x 3 are the particle s deflections from the ideal circular orbit, in the horizontal and vertical directions respectively. x 2 and x 4 are the associated momenta ω 1, ω 2 are related to the accelerator s tunes q x, q y by ω 1 =2πq x, ω 2 =2πq y Our problem is to estimate the region of stability of the particle s motion, the so-called dynamic aperture of the beam (Bountis & Skokos, 2006, Nucl. Inst Meth. Phys Res. A, 561, 173). 12

13 4D Accelerator map "Global" study Regions of different values of the SALI on the subspace x 2 (0)=x 4 (0)=0, after 10 4 iterations (q x = q y =0.4152) 13

14 4D Accelerator map "Global" study We consider 1,922,833 orbits by varying all x 1, x 2, x 3, x 4 within spherical shells of width 0.01 in a hypersphere of radius 1. (q x = q y =0.4152) 14

15 Applications 2D map 3 x = x + x x = x - ν sin(x + x ) (mod 2π) 2 1 X 2 0 A B For ν=0.5 we consider the orbits: ordered orbit A with initial conditions x 1 =2, x 2 =0. chaotic orbit B with initial conditions x 1 =3, x 2 = X 1 15

16 Behavior of SALI 2D maps SALI 0 both for regular and chaotic orbits following, however, completely different time rates which allows us to distinguish between the two cases. Hamiltonian flows and multidimensional maps SALI 0 for chaotic orbits SALI constant 0 for regular orbits 16

17 Questions Can we generalize SALI so that the new index: Can rapidly reveal the nature of chaotic orbits with σ 1 ªσ 2 (SALIμe -(σ1-σ2)t )? Depends on several Lyapunov exponents for chaotic orbits? Exhibits power-law decay for regular orbits depending on the dimensionality of the tangent space of the reference orbit as for 2D maps? 17

18 Definition of Generalized Alignment Index (GALI) SALI effectively measures the area of the parallelogram formed by the two deviation vectors. v(t) ˆ 2 v(0) ˆ 2 P(0) Trajectory v(0) ˆ 1 Area = P(t) vˆ 1 v ˆ 1 (t) ^ vˆ max SALI 2 = { vˆ - vˆ, vˆ + vˆ } Area 1 vˆ 1 - vˆ SALI 1 2 vˆ 1 + vˆ 2 2 = 18

19 Definition of GALI In the case of an N degree of freedom Hamiltonian system or a 2N symplectic map we follow the evolution of k deviation vectors with 2 k 2N, and define (Skokos et al., 2007, Physica D, 231, 30) the Generalized Alignment Index (GALI) of order k : GALI (t) = v ˆ (t) v ˆ (t)... v ˆ (t) k 1 2 k where v (t) v(t)= v (t) 1 ˆ

20 Wedge product We consider as a basis of the 2N-dimensional tangent space of the Hamiltonian flow the usual set of orthonormal vectors: e ˆ ˆ ˆ 1 = (1,0,0,...,0), e 2 = (0,1,0,...,0),..., e 2N = (0,0,0,...,1) Then for k deviation vectors we have: vˆ ˆ 1 v11 v12 v1 2N e1 ˆ ˆ v2 v21 v22 v2 2N e2 = v ˆ ˆ k vk1 vk2 vk 2N e2n v v v 1 i 1 i 1 i 1 2 k v2 i v 1 2 i v 2 2 ik vˆ vˆ v ˆ = eˆ eˆ e ˆ 1 i 1<i 2< <ik 2N v v v 1 2 k i i i k i k i k i 1 2 k 1 2 k 20

21 For k deviation vectors: Computation of GALI vˆ ˆ ˆ 1 v11 v12 v1 2N e1 e1 ˆ ˆ ˆ v2 v21 v22 v2 2N e2 e2 = = A v ˆ ˆ ˆ k vk1 vk2 vk 2N e2n e2n the norm of the wedge product is given by: 2 v 1 i v 1 1 i v 2 1 ik v2 i v 1 2 i v 2 2 ik T vˆ ˆ ˆ 1 v2 v k = = det(a A ) 1 i 1<i 2< <ik 2N v k i v 1 k i v 2 k ik 1/2 21

22 Computation of GALI From Singular Value Decomposition (SVD) of A T we get: A T = U W V where U is a column-orthogonal 2N k matrix (U T U=I), V T is a k k orthogonal matrix (V V T =I), and W is a diagonal k k matrix with positive or zero elements, the so-called singular values. So, we get: T T T T T det(a A ) = det(v W U U W V ) = det(v W I W V ) = k 2 T T k i i=1 det(v W V ) = det(v diag(w, w, w ) V ) = w Thus, GALI k is computed by: k k k i k i i=1 i=1 T ( ) ( ) T GALI = det(a A ) = w log GALI = log w 22

23 Behavior of GALI k for chaotic motion GALI k (2 k 2N) tends exponentially to zero with exponents that involve the values of the first k largest Lyapunov exponents σ 1, σ 2,, σ k : GALI (t) e k [ ] k -(σ -σ )+(σ -σ )+...+(σ -σ )t The above relation is valid even if some Lyapunov exponents are equal, or very close to each other. 23

24 Behavior of GALI k for chaotic motion 2D Hamiltonian (Hénon-Heiles system) σ 1 ª

25 Behavior of GALI k for chaotic motion 3D system: 3 i (q i +p i )+q1q 2+q1q3 i=1 2 H = ω with ω 1 =1, ω 2 = 2, ω 3 = 3, Η 3 =0.09. σ 1 ª0.03 σ 2 ª

26 Behavior of GALI k for chaotic motion N particles Fermi-Pasta-Ulam (FPU) system: N N β H = p + q -q + q -q 2 i=1 i=0 2 4 ( ) ( ) 2 4 i i+1 i i+1 i with fixed boundary conditions, N=8 and β=

27 Behavior of GALI k for regular motion If the motion occurs on an s-dimensional torus with s N then the behavior of GALI k is given by (Skokos et al., 2008, EPJ-ST, 165, 5): constant if 2 k s 1 GALI k (t) if s < k 2N - s k-s t 1 if 2N - s < k 2N 2(k-N) t while in the common case with s=n we have : constant if 2 k N GALI k (t) 1 if N < k 2N 2(k-N) t 27

28 Behavior of GALI k for regular motion 3D Hamiltonian 28

29 Behavior of GALI k for regular motion N=8 FPU system: The unperturbed Hamiltonian (β=0) is written as a sum of the so-called harmonic energies E i : with: E i = ( P i +ωiq i ), i = 1,..., N 2 N N 2 kiπ 2 kiπ iπ Q i = qsin i, P i = psin i, ω i =2sin N+1i=1 N+1 N+1i=1 N+1 2(N+1) 29

30 Global dynamics GALI 2 (practically equivalent to the use of SALI) GALI N Chaotic motion: GALI N Æ0 (exponential decay) Regular motion: GALI N Æconstantπ0 3D Hamiltonian Subspace q 3 =p 3 =0, p 2 0 for t=

31 Global dynamics GALI k with k>n The index tends to zero both for regular and chaotic orbits but with completely different time rates: Chaotic motion: exponential decay Regular motion: power law 2D Hamiltonian (Hénon-Heiles) Time needed for GALI 4 <

32 Regular motion on low-dimensional tori A regular orbit lying on a 2-dimensional torus for the N=8 FPU system. 32

33 Regular motion on low-dimensional tori A regular orbit lying on a 4-dimensional torus for the N=8 FPU system. 33

34 Low-dimensional tori x = x + x tori - 6D map { } K1 B x = x + sin(2πx )- sin[2π(x - x )]+ sin[2π(x - x )] 2 2 2π 1 2π x = x + x { } K2 B x = x + sin(2πx )- sin[2π(x - x )]+ sin[2π(x - x )] 4 4 2π 3 2π x = x + x K3 B x = x + sin(2πx )- sin[2π(x - x 6 6 2π 5 2π 3 5 { )]+ sin[2π(x - x )]} 1 5 (mod 1) 3D torus 2D torus 34

35 Behavior of GALI k GALI (t) e k Chaotic motion: GALI k 0 exponential decay [ ] k -(σ -σ )+(σ -σ )+...+(σ -σ )t Regular motion: GALI k constant 0 or GALI k 0 power law decay constant if 2 k s 1 GALI k(t) if s < k 2N- s k-s t 1 if 2N-s < k 2N 2(k-N) t 35

36 Conclusions Generalizing the SALI method we define the Generalized ALignment Index of order k (GALI k ) as the volume of the generalized parallelepiped, whose edges are k unit deviation vectors. GALI k is computed as the product of the singular values of a matrix (SVD algorithm). Behaviour of GALI k : Chaotic motion: it tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. Reguler motion: it fluctuates around non-zero values for 2 k s and goes to zero for s<k 2N following power-laws, with s being the dimensionality of the torus. GALI k indices : can distinguish rapidly and with certainty between regular and chaotic motion can be used to characterize individual orbits as well as "chart" chaotic and regular domains in phase space. are perfectly suited for studying the global dynamics of multidimentonal systems can identify regular motion in low dimensional tori 36

37 References SALI Skokos Ch. (2001) J. Phys. A, 34, Skokos Ch., Antonopoulos Ch., Bountis T. C. & Vrahatis M. N. (2003) Prog. Theor. Phys. Supp., 150, 439 Skokos Ch., Antonopoulos Ch., Bountis T. C. & Vrahatis M. N. (2004) J. Phys. A, 37, 6269 GALI Skokos Ch., Bountis T. C. & Antonopoulos Ch. (2007) Physica D, 231, Skokos Ch., Bountis T. C. & Antonopoulos Ch. (2008) Eur. Phys. J. Sp. Top., 165,