methode d’ulam et op´ erateur de perron-frobenius ... filemethode d’ulam et op´ erateur de...

Methode d’Ulam et op erateur de Perron-Frobenius

appliqu es a l’application standard de Chirikov

Klaus FrahmDima Shepelyansky Eur. Phys. J. B 76, 57 (2010)Quantware MIPS CenterUniversite Paul Sabatier

Laboratoire de Physique Theorique, UMR 5152, IRSAMC

3eme Journee LPT-IMT 2011, Toulouse, 4 Avril 2011

Chirikov Standard map

Chirikov Standard map

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

p

x

k=0.7

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

p

x

k=0.971635406

pn+1 = pn +k

2πsin(2π xn)

xn+1 = xn + pn+1

x and p are taken modulo 1and the symmetry(x, p) → (1− x, 1− p)allows to restrict:x ∈ [0, 1] and p ∈ [0, 0.5].

Transition to “global” chaos atkc = 0.971635406.

Klaus Frahm 2 Toulouse, 4 Avril 2011

Perron-Frobenius operators

Perron-Frobenius operators

discrete Markov process :

pi(t + 1) =∑

j

Aij pj(t)

with probabilities pi(t) ≥ 0 and the Perron-Frobenius matrix A suchthat: ∑

i

Aij = 1 , Aij ≥ 0 .

⇒ complex eigenvalues |λj| ≤ 1 and (at least) one eigenvalueλ0 = 1 and the right eigenvector for λ0 is the stationary distribution:limt→∞ pi(t). Recent interest in the context of the Google matrix :

A = αS + (1− α)1

neeT

S = column normalized link matrix, typically α = 0.85 andeigenvector for λ0 = is the PageRank.

Klaus Frahm 3 Toulouse, 4 Avril 2011

Perron-Frobenius matrix for chaotic maps

Perron-Frobenius matrix forchaotic maps

Ulam proposed in 1960 a method to construct a Perron-Frobeniusmatrix for chaotic maps by dividing the phase space in a finite numberof small cells i. The transition probabilities Aij from cell j to cell i aredetermined by one application of the map for many random initialconditions in the initial cell j.

⇒ This approach is problematic in the case of mixed phase spacewith chaotic regions and stable islands since the space discretizationproduces an effective noise allowing for diffusion from the chaotic tothe regular regions.

Klaus Frahm 4 Toulouse, 4 Avril 2011

A new variant of the Ulam Method

A new variant of the Ulam Methodto construct the Perron-Frobenius matrix for the case of mixedphase space:

• Subdivide x space in M cells and p space in M/2 cells with Mbeing an (even) integer number.

• Iterate (for a very long time: N ∼ 1011 − 1012) a classicaltrajectory and attribute a new number to each new cell which isentered. At the same time count the number of transitions fromcell i to cell j (⇒ Nji).

• Calculate the d× d matrix

Aji =Nji∑l Nli

of dimension d ≈ M 2/4 and which has similary properties as thegoogle matrix, i. e.: Aji ≥ 0,

∑j Aji = 1, Aji sparse.

Klaus Frahm 5 Toulouse, 4 Avril 2011

Illustration

Illustration

M = 10, N = 106 ⇒ d = 35

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

p

x

0 1

2

3

4

5

6

Klaus Frahm 6 Toulouse, 4 Avril 2011

Matrix structure

Matrix structure

for M = 10, N = 106 and d = 35

density plot of matrix elements

(blue=min=0, green=medium, red=max)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6

distribution of number of non-zero matrix elements percolumn

Klaus Frahm 7 Toulouse, 4 Avril 2011

Eigenvalues

Eigenvalues

for M = 10, N = 106 and d = 35

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

λ Phase space representation of theeigenvector for λ0 = 1.

Klaus Frahm 8 Toulouse, 4 Avril 2011

Eigenvalues

Eigenvalues

for M = 280, N = 1012 and d = 16609

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

λ-0.01

0

0.01

0.97 0.98 0.99 1

λ

Klaus Frahm 9 Toulouse, 4 Avril 2011

Complex density of states

Complex density of states

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

ρ(λ)

|λ|

M=280M=200M=140M=100

Klaus Frahm 10 Toulouse, 4 Avril 2011

Eigenvectors

Eigenvectors

λ0 = 1, M = 25, d = 177

λ0 = 1, M = 50, d = 641

λ0 = 1, M = 35, d = 332

λ0 = 1, M = 70, d = 1189

Klaus Frahm 11 Toulouse, 4 Avril 2011

Eigenvectors

λ0 = 1M = 100d = 2340

λ0 = 1M = 140d = 4417

Klaus Frahm 12 Toulouse, 4 Avril 2011

Eigenvectors

λ0 = 1M = 200d = 8753

λ0 = 1M = 280d = 16609

Klaus Frahm 13 Toulouse, 4 Avril 2011

Arnoldi method

Arnoldi method

to (partly) diagonalize large sparse non-symmetric d× d matrices:

• choose an initial normalized vector ξ0 (random or “otherwise”)

• determine the Krylov space of dimension n (typically:1 � n � d ) spanned by the vectors: ξ0, A ξ0, . . . , An−1ξ0

• determine by Gram-Schmidt orthogonalization an orthonormalbasis {ξ0, . . . , ξn−1} and the representation of A in this basis:

A ξk =

k+1∑j=0

Hjk ξj

Klaus Frahm 14 Toulouse, 4 Avril 2011

Arnoldi method

• diagonalize the Arnoldi matrix H which has Hessenberg form:

H =

∗ ∗ · · · ∗ ∗∗ ∗ · · · ∗ ∗0 ∗ · · · ∗ ∗... ... . . . ... ...0 0 · · · ∗ ∗0 0 · · · 0 ∗

which provides the Ritz eigenvalues that are very good

aproximations to the “largest” eigenvalues of A.

1

10-5

10-10

10-15

0 500 1000 1500

|λj-λ j

(Ritz

) |

j

M = 280, d = 16609,

n = 1500

Klaus Frahm 15 Toulouse, 4 Avril 2011

Arnoldi method

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

λ

M = 280d = 16609n = 1500

Klaus Frahm 16 Toulouse, 4 Avril 2011

Arnoldi method

complex density of states:

0

0.1

0.2

0.3

0.6 0.8 1

ρ(λ)

|λ|

M=280M=400M=560M=800

M=1120M=1600

Klaus Frahm 17 Toulouse, 4 Avril 2011

Arnoldi method

λ0 = 1M = 400d = 33107n = 2000

λ0 = 1M = 560d = 63566n = 2000

Klaus Frahm 18 Toulouse, 4 Avril 2011

Arnoldi method

λ0 = 1M = 800d = 127282n = 2000

λ0 = 1M = 1120d = 245968n = 2000

Klaus Frahm 19 Toulouse, 4 Avril 2011

Arnoldi method

λ0 = 1, M = 1600, d = 494964, n = 3000

Klaus Frahm 20 Toulouse, 4 Avril 2011

Arnoldi method

λ1 =0.99980431

M = 800d = 127282n = 2000

λ2 =0.99878108

M = 800d = 127282n = 2000

Klaus Frahm 21 Toulouse, 4 Avril 2011

Arnoldi method

λ3 =0.99808332

M = 800d = 127282n = 2000

λ4 =0.99697284

M = 800d = 127282n = 2000

Klaus Frahm 22 Toulouse, 4 Avril 2011

Diffuson modes

Diffuson modes

0

0.01

0.02

0.03

0 5 10 15 20

γ j

j

γ1 j2

γj = −2 ln(|λj|)

γj ≈ γ1 j2

for j ≤ 5.

What about eigenvectors for complex or real negative λj ?

Klaus Frahm 23 Toulouse, 4 Avril 2011

Diffuson modes

λ6 =−0.49699831+i 0.86089756≈ |λ6| ei 2π/3

M = 800d = 127282n = 2000

λ19 =−0.71213331+i 0.67961609≈ |λ19| ei 2π(3/8)

M = 800d = 127282n = 2000

Klaus Frahm 24 Toulouse, 4 Avril 2011

Diffuson modes

λ8 =0.00024596+i 0.99239222≈ |λ8| ei 2π/4

M = 800d = 127282n = 2000

λ10 =−0.99187524≈ |λ10| ei 2π(2/4)

M = 800d = 127282n = 2000

Klaus Frahm 25 Toulouse, 4 Avril 2011

Diffuson modes

λ13 =0.30580631+i 0.94120900≈ |λ13| ei 2π/5

M = 800d = 127282n = 2000

λ16 =−0.79927624+i 0.58085184≈ |λ16| ei 2π(2/5)

M = 800d = 127282n = 2000

Klaus Frahm 26 Toulouse, 4 Avril 2011

Eigenphase distribution

Eigenphase distribution

0

0.1

0.2

0.3

0.4

0.5

0 200 400 600 800 1000

ϕ j /(

2π)

j

0

1/81/7

1/61/5

1/42/7

1/3

3/82/5

3/7

1/2

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5

p(ϕ)

ϕ

↓0

↓

1-2

↓

1-3

↓

1-4

↓

1-5

↓

2-5

↓

1-6

↓

1-7

↓

1-8

↓

2-7

↓

3-8

↓

3-7

0.32

0.34

0.36

0.38

650 700 750 800

ϕ j /(

2π)

j

1/3

3/8

0.2

0.22

0.24

0.26

400 450 500 550

ϕ j /(

2π)

j

1/5

1/4

Klaus Frahm 27 Toulouse, 4 Avril 2011

Extrapolation

Extrapolation

of γ1(M) in the limit M →∞:

0.1

0.01

0.001

10-4

100 1000

γ 1(M

)

M

f(M)

2.36 M -1.30

f (M) =D

M

1 + CM

1 + BM

D = 0.245B = 13.1C = 258

Klaus Frahm 28 Toulouse, 4 Avril 2011

Extrapolation

Extrapolation

of γ6(M) in the limit M →∞:

0.01

0.1

1600800400200

γ 6(M

)

M

389 M -1.55

γ6(M) ≈ 389 M−1.55 for M ≥ 400.

Klaus Frahm 29 Toulouse, 4 Avril 2011

Strong chaos

Strong chaos

at k = 7, M = 140, N = 1011 and d = 9800

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

λ

0

0.1

0.2

0.3

0 0.005 0.01γ 1

(M)

M -1

d+e M -1

a (1+b M -1) c

γclass.

=0.0866

a ≈ 0.0857± 0.0036, d ≈ 0.0994

limM→∞

γ1(M) > 0

Klaus Frahm 30 Toulouse, 4 Avril 2011

Strong chaos

λ0 = 1

k = 7M = 800d = 319978n = 1500

λ1 =0.93874817

k = 7M = 800d = 319978n = 1500

Klaus Frahm 31 Toulouse, 4 Avril 2011

Strong chaos

λ2 =−0.49264273+i0.78912368

k = 7M = 800d = 319978n = 1500

λ8 =0.87305253

k = 7M = 800d = 319978n = 1500

Klaus Frahm 32 Toulouse, 4 Avril 2011

Separatrix map

Separatrix map

p = p + sin(2πx)

x = x +Λ

2πln(|p|) (mod 1)

Λ = Λc = 3.1819316

Klaus Frahm 33 Toulouse, 4 Avril 2011

Poincare recurrences

Poincar e recurrences

0.01

10-3

10-4

0.01 0.1

j/N

d

γj

∼ γ 1.5203

0.01

10-3

10-4

10-5

0.01 0.1

j/N

d

γj

∼ γ 1.4995

ρΣ(γ) ∼ γβ , β ≈ 1.5 .

P (t) ∼∫ 1

0

dρΣ(γ)

dγexp(−γt) dγ ∼ 1

tβ

Klaus Frahm 34 Toulouse, 4 Avril 2011

Conclusion

Conclusion• Ulam Method and numerical diagonalization of the

Perron-Frobenius Operator (PFO) provide a new tool to studycertain properties of chaotic maps: decay time, subtlephase-space structure of particular modes, ...

• Physical questions: Signification of diffuson modes ? Possiblepower law time decay in the initial map versus exponential decayin the PFO ? ...

• Comparison with realistic google matrices:

- PFO: typically numerically well conditioned, no degeneracies,not defective, (at least small) gap between λ0 = 1 and|λ1| < 1.

- GM: a lot of degeneracies, defective (large Jordan blocks),requires the α shift transformation.

Klaus Frahm 35 Toulouse, 4 Avril 2011

Technical aspects of the Arnoldi method

Technical aspects of the Arnoldimethod

• Requires only modest resources:

memory ∼ d n (d 2)

computation time ∼ d n2 (d 3).

• In theory: AM provides only one eigenvector per degenerateeigenvalue but it may provide big parts of (or full) Jordan blocks.

• Rounding errors “may help” ⇒ new random start vector if theArnoldi iteration has fully explored an A-invariant subspace.

• In theory: choice of initial vector may allow to study different partsof “phase or web-space” but rounding errors complicate things.

Klaus Frahm 36 Toulouse, 4 Avril 2011

Technical aspects of the Arnoldi method

• AM produces a Hessenberg matrix and therefore no initialHouseholder transformation is needed before applying theQR-algorithm to compute the eigenvalues.

• Eigenvectors can be computed one by one efficiently (∼ n2 + n dper eigenvector) and in a numerical stable way. However, thereare subtle problems in case of Jordan blocks.

Klaus Frahm 37 Toulouse, 4 Avril 2011