Haris Skokos Physics Department, Aristotle University of Thessaloniki Thessaloniki, Greece

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1 The Smaller (SALI) and the Generalized (GALI) Alignment Index methods of chaos detection Haris Skokos Physics Department, Aristotle University of Thessaloniki Thessaloniki, Greece URL: Main contributors: Tassos Bountis, Chris Antonopoulos, Thanos Manos, Enrico Gerlach

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 Summary

3 Definition of Smaller Alignment Index (SALI) Consider the 2N-dimensional phase space of a conservative dynamical system (symplectic map or Hamiltonian flow). An orbit in that space with initial condition : P=(x 1, x 2,,x 2N ) and a deviation vector v=(δx 1, δx 2,, δx 2N ) 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)

4 Definition of SALI We follow the evolution in time of two different initial deviation vectors (v 1, v 2 ), and define SALI (Ch.S. 2001, J. Phys. A) 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

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 maximum Lyapunov exponent.

6 Behavior of SALI for chaotic motion For chaotic orbits the two initially different deviation vectors tend to coincide with the direction defined by the maximum Lyapunov exponent. P(t) P

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

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

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

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

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

12 Behavior of SALI for chaotic motion The evolution of a deviation vector can be approximated by: n i 1 2 σ t σ t σ t v (t) = c (1) e u ˆ c (1) e u ˆ + c (1) e uˆ 1 i i i=1 where σ 1 >σ 2 σ n are the Lyapunov exponents, and u j=1, 2,, 2N the corresponding eigendirections. In this approximation, we derive a leading order estimate of the ratio (1) σ1t (1) σ2t (1) v ˆ ˆ 1(t) c1 e u 1 + c2 e u2 c2 -(σ1 -σ 2 )t = ± uˆ + ˆ (1) σ 1 e u2 1t (1) v 1(t) c1 e c1 and an analogous expression for v 2 (2) σ1t (2) σ2t (2) v ˆ ˆ 2 (t) c1 e u 1 + c2 e u2 c2 -(σ1 -σ 2 )t = ± uˆ + ˆ (2) σ 1 e u2 1t (2) v 2 (t) c1 e c1 So we get: (1) (2) v 1(t) v 2 (t) v 1(t) v 2 (t) c 2 c ± 2 SALI(t) = min +, - e (1) (2) v 1(t) v 2 (t) v 1(t) v 2 (t) c1 c1 ˆ j -(σ -σ )t 1 2

13 Behavior of SALI for chaotic motion We test the validity of the approximation SALIµe -(σ1-σ2)t (Ch.S., Antonopoulos, Bountis, Vrahatis, 2004, J. Phys. A) for a chaotic orbit of the 3D Hamiltonian H = ω 3 i (q i + p i )+ q1q 2 + q1q3 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 ª0.011

14 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.

15 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.

16 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

17 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 P(t)

18 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. v ˆ 1 v ˆ 2 P P(t)

19 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. v ˆ 1 v ˆ 2 P P(t)

20 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 v ˆ 1 v ˆ 2 P(t) v ˆ 1 v ˆ 2

21 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 v ˆ 1 v ˆ 2 P(t) v ˆ 1 v ˆ 2

22 Applications Hénon-Heiles Heiles system As an example, we consider the 2D Hénon-Heiles system: For E=1/8 we consider the orbits with initial conditions: Regular 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

23 Applications Hénon-Heiles Heiles system p y y

24 Applications 4D Accelerator map We consider the 4D symplectic map x 1 cosω1 -sinω1 0 0 x x 2 sinω 1 cosω1 0 0 x 2 + x 1 - x 3 = x cosω2 -sinω 2 x 3 x sinω 2 cosω2 x 4-2x 1x 3 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 goal is to estimate the region of stability of the particle s motion, the socalled dynamic aperture of the beam (Bountis, Ch.S., 2006, Nucl. Inst Meth. Phys Res. A) and to increase its size using chaos control techniques (Boreaux, Carletti, Ch.S., Vittot, 2012, Com. Nonlin. Sci. Num. Sim. Boreaux, Carletti, Ch.S., Papaphilippou, Vittot, 2012, Int. J. Bif. Chaos).

25 4D Accelerator map "Global" study Regions of different values of the SALI on the subspace x 2 =x 4 =0, after 10 5 iterations (q x = q y =0.4152) 4D map Controlled 4D map log(sali)

26 4D Accelerator map "Global" study Increase of the dynamic aperture We evolve many orbits in 4D hyperspheres of radius r centered at x 1 =x 2 =x 3 =x 4 =0, for 10 5 iterations. 4D map Controlled 4D map Regular orbits Chaotic orbits

27 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: regular orbit A with initial conditions x 1 =2, x 2 =0. chaotic orbit B with initial conditions x 1 =3, x 2 = X 1

28 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

29 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?

30 Definition of Generalized Alignment Index (GALI) SALI effectively measures the area of the parallelogram formed by the two deviation vectors.

31 Definition of Generalized Alignment Index (GALI) SALI effectively measures the area of the parallelogram formed by the two deviation vectors. v ˆ 2 (t) P(t) v ˆ 1 (t) v ˆ 2 v ˆ 1 P

32 Definition of Generalized Alignment Index (GALI) SALI effectively measures the area of the parallelogram formed by the two deviation vectors. v ˆ 2 (t) P(t) v ˆ 1 (t) v ˆ 2 v ˆ 1 P

33 Definition of Generalized Alignment Index (GALI) SALI effectively measures the area of the parallelogram formed by the two deviation vectors. v ˆ 2 (t) P(t) v ˆ 1 (t) v ˆ 2 v ˆ 1 P

34 Definition of Generalized Alignment Index (GALI) SALI effectively measures the area of the parallelogram formed by the two deviation vectors. v ˆ 2 (t) v ˆ 2 P v ˆ 1 P(t) v ˆ 1 (t) vˆ ˆ ˆ ˆ 1 - v2 v1 + v2 Area = vˆ ˆ 1 ^ v2 = = 2 { ˆ ˆ ˆ + ˆ } max v - v, v v SALI 2 Area SALI

35 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 (Ch.S., Bountis, Antonopoulos, 2007, Physica D) 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 1 1

36 Wedge product We consider as a basis of the 2N-dimensional tangent space of the system 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 = vˆ ˆ k vk1 vk2 vk 2N e2n 1 2 k v2 i v 1 2 i v 2 2 ik vˆ vˆ v ˆ = eˆ eˆ e ˆ 1 2 k i i i 1 i <i < <i 2N 1 2 k v v v 1 i 1 i 1 i v v v k i k i k i 1 2 k 1 2 k

37 Norm of wedge product We define as norm of the wedge product the quantity : 2 v 1 i v 1 1 i v 2 1 ik v2 i v 1 2 i v 2 2 ik vˆ ˆ ˆ 1 v2 v k = 1 i 1 <i 2 < <ik 2N v k i v 1 k i v 2 k ik 1/2

38 Computation of GALI - Example Let us compute GALI 3 in the case of 2D Hamiltonian system (4- dimensional phase space). eˆ 1 vˆ 1 v11 v12 v13 v14 ˆ ˆ = e2 v2 v21 v22 v23 v24 eˆ 3 vˆ 3 v31 v32 v33 v34 eˆ 4

39 Computation of GALI - Example Let us compute GALI 3 in the case of 2D Hamiltonian system (4- dimensional phase space). eˆ 1 vˆ 1 v11 v12 v13 v14 ˆ ˆ = e2 v2 v21 v22 v23 v24 eˆ 3 vˆ 3 v31 v32 v33 v34 eˆ 4 Columns v v v GALI = vˆ vˆ v ˆ = v v v v v v

40 Computation of GALI - Example Let us compute GALI 3 in the case of 2D Hamiltonian system (4- dimensional phase space). eˆ 1 vˆ 1 v11 v12 v13 v14 ˆ ˆ = e2 v2 v21 v22 v23 v24 eˆ 3 vˆ 3 v31 v32 v33 v34 eˆ 4 Columns v v v GALI = vˆ vˆ v ˆ = v v v v v v v v v v v v v v v

41 Computation of GALI - Example Let us compute GALI 3 in the case of 2D Hamiltonian system (4- dimensional phase space). eˆ 1 vˆ 1 v11 v12 v13 v14 ˆ ˆ = e2 v2 v21 v22 v23 v24 eˆ 3 vˆ 3 v31 v32 v33 v34 eˆ 4 Columns v v v GALI = vˆ vˆ v ˆ = v v v v v v v v v v v v v v v v v v v v v v v v

42 Computation of GALI - Example Let us compute GALI 3 in the case of 2D Hamiltonian system (4- dimensional phase space). eˆ 1 vˆ 1 v11 v12 v13 v14 ˆ ˆ = e2 v2 v21 v22 v23 v24 eˆ 3 vˆ 3 v31 v32 v33 v34 eˆ 4 Columns v v v GALI = vˆ vˆ v ˆ = v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v /2

43 Efficient computation of GALI For k deviation vectors: vˆ ˆ ˆ 1 v11 v12 v1 2N e1 e1 ˆ ˆ ˆ v2 v21 v22 v2 2N e2 e2 = = A vˆ 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

44 Efficient 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 (Ch.S., Bountis, Antonopoulos, 2008, Eur. Phys. J. Sp. Top): k k k i k i i=1 i=1 T ( ) ( ) T GALI = det(a A ) = w log GALI = log w

45 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 exponentsσ 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.

46 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=1.5.

47 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 (Ch.S., Bountis, Antonopoulos, 2008, Eur. Phys. J. Sp. Top.): 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

E i : with: 1 ( 2 2 2 E ) i = P i +ωi Q i, i = 1,.

48 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: 1 ( E ) i = P i +ωi Q i, i = 1,..., N 2 N N 2 kiπ 2 kiπ iπ Q i = qksin, P i = pksin, ω i = 2sin N +1 k =1 N +1 N +1 k =1 N +1 2(N +1)

49 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=1000.

50 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 <10-12

51 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

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

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

54 Locating low-dimensional tori Orbits with q 1 =q 2 =0.1, p 1 =p 2 =p 3 =0, H= for the N=4 FPU system (Gerlach, Eggl, Ch.S., 2012, Int. J. Bif. Chaos).

Locating low-dimensional tori Orbits with q 1 =q 2 =0.1, p 1 =p 2 =p 3 =0, H=0.

55 Locating low-dimensional tori Orbits with q 1 =q 2 =0.1, p 1 =p 2 =p 3 =0, H= for the N=4 FPU system (Gerlach, Eggl, Ch.S., 2012, Int. J. Bif. Chaos).

56 Locating low-dimensional tori Orbits with q 1 =q 2 =0.1, p 1 =p 2 =p 3 =0, H= for the N=4 FPU system (Gerlach, Eggl, Ch.S., 2012, Int. J. Bif. Chaos).

57 Locating low-dimensional tori Orbits with q 1 =q 2 =0.1, p 1 =p 2 =p 3 =0, H= for the N=4 FPU system (Gerlach, Eggl, Ch.S., 2012, Int. J. Bif. Chaos).

58 GALI k indices : Summary 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 on low dimensional tori SALI/GALI methods have been successfully applied to a variety of conservative dynamical systems of Celestial Mechanics (e.g. Széll et al., 2004, MNRAS - Soulis et al., 2008, Cel. Mech. Dyn. Astr. - Libert et al., 2011, MNRAS) Galactic Dynamics (e.g. Capuzzo-Dolcetta et al., 2007, Astroph. J. - Carpintero, 2008, MNRAS - Manos & Athanassoula, 2011, MNRAS) Nuclear Physics (e.g. Macek et al., 2007, Phys. Rev. C - Stránský et al., 2007, Phys. Atom. Nucl. - Stránský et al., 2009, Phys. Rev. E) Statistical Physics (e.g. Manos & Ruffo, 2010, Trans. Theory Stat. Phys.)

59 Main references Bountis T.C. & Ch.S. (2012) Complex Hamiltonian Dynamics, Chapter 5, Springer Series in Synergetics SALI Ch.S. (2001) J. Phys. A, 34, Ch.S., Antonopoulos Ch., Bountis T. C. & Vrahatis M. N. (2003) Prog. Theor. Phys. Supp., 150, 439 Ch.S., Antonopoulos Ch., Bountis T. C. & Vrahatis M. N. (2004) J. Phys. A, 37, 6269 Bountis T. & Ch.S. (2006) Nucl. Inst Meth. Phys Res. A, 561, 173 Boreaux J., Carletti T., Ch.S. &Vittot M. (2012) Com. Nonlin. Sci. Num. Sim., 17, 1725 Boreaux J., Carletti T., Ch.S., Papaphilippou Y. & Vittot M. (2012) Int. J. Bif. Chaos, 22, GALI Ch.S., Bountis T. C. & Antonopoulos Ch. (2007) Physica D, 231, Ch.S., Bountis T. C. & Antonopoulos Ch. (2008) Eur. Phys. J. Sp. Top., 165, 5-14 Gerlach E., Eggl S. & Ch.S. (2012) Int. J. Bif. Chaos, 22, Manos T., Ch.S. & Antonopoulos Ch. (2012) Int. J. Bif. Chaos, 22, [Behavior of GALI for periodic orbits] Manos T., Bountis T. & Ch.S. (2013) J. Phys. A, 46, [Behavior of GALI for time dependent Hamiltonians]-> Talk of T. Manos on Thursday 20 June

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

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 E-mail: