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.


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.


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.


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


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


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.


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


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


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


Eigenvectors

λ0 = 1M = 100d = 2340

λ0 = 1M = 140d = 4417


Eigenvectors

λ0 = 1M = 200d = 8753

λ0 = 1M = 280d = 16609


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


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


Arnoldi method

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

λ

M = 280d = 16609n = 1500


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


Arnoldi method

λ0 = 1M = 400d = 33107n = 2000

λ0 = 1M = 560d = 63566n = 2000


Arnoldi method

λ0 = 1M = 800d = 127282n = 2000

λ0 = 1M = 1120d = 245968n = 2000


Arnoldi method

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


Arnoldi method

λ1 =0.99980431

M = 800d = 127282n = 2000

λ2 =0.99878108

M = 800d = 127282n = 2000


Arnoldi method

λ3 =0.99808332

M = 800d = 127282n = 2000

λ4 =0.99697284

M = 800d = 127282n = 2000


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 ?


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


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


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


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


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


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.


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


Strong chaos

λ0 = 1

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

λ1 =0.93874817

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


Strong chaos

λ2 =−0.49264273+i0.78912368

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

λ8 =0.87305253

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


Separatrix map

Separatrix map

p = p + sin(2πx)

x = x +Λ

2πln(|p|) (mod 1)

Λ = Λc = 3.1819316


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


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.


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.


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.


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