Existence of Noise Induced Order, a computer aided proof. S. - - PowerPoint PPT Presentation

Existence of Noise Induced Order, a computer aided proof.

• S. Galatolo
• Dip. Mat, Univ. Pisa

CIRM 2019

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 1 / 24

Computer aided proofs and estimates in dynamics

Certified numerical estimates helped in several important results. Mostly on the topological-dynamical side. Few and more recent results in the ergodic-statistical side. But we know this, since almost all the people working on this are here.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 2 / 24

The Belosuv-Zabotinsky reaction.

A chemical reaction with a chaotic behavior (1951-1971). In 1983 Matsumoto and Tsuda discovered by numerical simulations that the behavior of a model of the reaction is less chaotic when a certain quantity of noise is added. (noise induced order) The discovery was confirmed by real experiments with the actual chemical reaction Such kind of phenomena were found in several other systems, also of biological origin.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 3 / 24

The model

Ta,b,c(x) = (a + (x − 1

8)1/3)e−x + b

(0 ≤ x ≤ 0.3), c(10xe−10x/3)19 + b (x > 0.3). a = 0.5060735690368223 (near to T (0.3−) = 0), b = 0.0232885279 (near to a preperiodic condition for the critical value), c = 0.1212056927389751 (near to T(0.3−) = T(0.3+)), x → T(x) + ω where ω is ind. unif. distributed noise on [−ξ, ξ]

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 4 / 24

The behavior of the model

Matsumoto and Tsuda made mumerical simulations obtaining this behavior for the Lyapunov exp λµ := lim

n→∞

1 n

n

∑

log |T (xi)| =

• log |T | dµ

where xi is a typical trajectory and µ is the stationary measure. Notice that getting information on µ allows to estimate the integral.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 5 / 24

The behavior of the model

Notice that Lyap. exp. λµ > 0 when the noise is very small and then it becomes << 0 for larger noise. That was called Noise Induced Order. Similar behavior was found in the empirical entropy and other chaos indicators.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 6 / 24

What we prove

We prove the existence of this transition: for some small noise size the system has a positive Lyapunov exponent, while for some larger

• nes it has a negative Lyapunov exponent.

This is based on rigorous (certified) estimates of the properties of the stationary measure of the system, and its convergence to equilibrium. The approach used is quite general and could be applied to any map

• f the interval, perturbed by additive noise.

(with M. Monge and I. Nisoli. arXiv:1702.07024)

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 7 / 24

Why this is not easy?

The mathematical appoach to the problem is complicated by the structure of the deterministic part of the dynamics. In the map, strongly expanding and strongly contracting regions cohexist. With zero noise the map seems to have a singular invariant measure. The global dynamics is made in a way that with small noise, the dynamics visit the expanding part often enough to have a positive exponent.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 8 / 24

The regularizing effect of the noise makes things possible.

The presence of the noise makes the transfer operator being a regularizing one. This replaces in some sense the Lasota Yorke inequality allowing to prove quantitative stability results, even if the deterministic part of the dynamics is not hyperbolic. We need a suitable computational framework where the numerical error can be certified (interval arithmetics e.g.), and a suitable approximation strategy.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 9 / 24

The transfer operator for a deterministic system

Let us consider a metric space X with a dynamics defined by T : X → X. Let us also consider the space PM(X) of probability measures on X. Define the function L : PM(X) → PM(X) in the following way: if µ ∈ PM(X) then: Lµ(A) = µ(T −1(A)) Considering signed measures (SM(X) ) or complex valued measures we have a vector space and L is linear.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 10 / 24

The transfer operator with noise

If after applying T we add some noise, the transfer operator is composed by a convolution. Let ξ > 0,be the radius of the noise ρξ(x) = 1

ξ ρ( 1 ξ x) noise kernel (with ρ ∈ BV and

ρ = 1) Nξf = ρξ ∗ f , Transfer operator Lξ : SM([0, 1]) → L1[0, 1] is defined by Lξ = NξL. Stationary measures are fixed points of Lξ. There are BV stationary measures.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 11 / 24

Summary of the general strategy

The system has an associated transfer operator Lξ. The stationary measure fξ is the unique fixed prob. meas of Lξ. General task

1

Find a good approximation of fξ (small explicit error in L1+ explicit information on the variance in given intervals)

2

With this, compute the Lyapunov exponent log(T ) dfξ For the approximation of fξ. 1.a Approximate (up to an explicit error) the transfer operator Lξ of the system by a finite rank Lξ,δ (big matrix to be computed in a certified way) 1.b Approximate (up to an explicit error) the fixed point fξ of Lξ with the fixed point fξ,δ of Lξ,δ (here a quantitative understanding of the stability is needed)

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 12 / 24

The approximated transfer operator

Space X discretized by a partition Iδ Let Fδ be the σ−algebra associated to Iδ, then consider πδ : SM(X) → L1(X) (1) πδ(g) = E(g|Fδ) (2) The approximated operator we are going to use is defined by a kind of Ulam discretization Lδ,ξ = πδNξπδLπδ The fixed points and other properties of Lξ will be approximated by the ones of Lδ,ξ.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 13 / 24

One argument for the approximation of the fixed point

Let us suppose fδ,ξ and fξ be fixed points of Lδ,ξ and Lξ fδ,ξ − fξL1 = Ln

δ,ξfδ,ξ − Ln ξfξL1

= Ln

δ,ξfδ,ξ − Ln δ,ξfξ + Ln δ,ξfξ − Ln ξfξL1

≤ Ln

δ,ξ(fδ,ξ − fξ)L1 + (Ln δ,ξ − Ln ξ)fξL1.

Therefore if (we verify computionally that) for some n ||Ln

δ,ξ|V ||L1→L1 ≤ α < 1

where V = {f ∈ BV , f dm = 0} (zero avg. measures) And...

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 14 / 24

Approximation of the fixed point

Let us suppose fδ,ξ and fξ be fixed points of Lδ,ξ and Lξ fδ,ξ − fξL1 ≤ Ln

δ,ξ(fδ,ξ − fξ)L1 + (Ln δ,ξ − Ln ξ)fξL1.

Therefore if (we verfy computionally that) ||Ln

δ,ξ|V ||L1→L1 ≤ α < 1

where V = {f ∈ BV , f dm = 0} (zero avg. measures). Then fδ,ξ − fξL1 ≤ αfδ,ξ − fξL1 + (Ln

δ,ξ − Ln ξ)fξL1.

and fξ − fδ,ξL1 ≤ 1 1 − α

• (Ln

δ,ξ − Ln ξ)fξ

• L1 .

(find computationally a good compromise between α and n )

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 15 / 24

Approximation of the fixed point

Expanding

• (Ln

δ,ξ − Ln ξ)fξ

• L1 in a telescopic sum and applying triangular

ineq.:

• (Ln

δ,ξ − Ln ξ)fξ

• L1 =
• (πδNξπδL)nπδ − (NξL)n

fξ

• L1

≤ (πδ − 1)NξLfξL1 +

• n−1

∑

i=0

• Lδ,ξ|i

V

• L1→L1
• ×

×

• Nξ(πδ − 1)Lfξ
• L1 +
• NξπδL(πδ − 1)fξ
• L1
• S.Galatolo

(Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 16 / 24

Estimate for

• (πδ − 1)NξLfξ
• L1 ,
• Nξ(πδ − 1)Lfξ
• L1 ,
• NξπδL(πδ − 1)NξLfξ
• L1 .

Lemma

We have Nξ(1 − πδ)L1→L1 ≤ 1 2δξ−1Var(ρ). (1 − πδ)NξL1→L1 ≤ 1 2δξ−1Var(ρ). Recalling that πδL1→L1 ≤ 1, NξL1→L1 ≤ 1 we obtain fξ − fξ,δL1 ≤ 1 + ∑n−1

i=0

• Lδ,ξ|i

V

• L1→L1

2(1 − α) δξ−1Var(ρ). (3)

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 17 / 24

The results of the computation

˜ n about 50-70, α about 0.2 − 0.5, δ up to 2−26, ξ down to 0.873 × 10−4.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 18 / 24

Noise induced order exists.

Theorem

Let λξ be the Lyapunov expontent of the system with noise of size ξ. for ξ1 = 0.873 × 10−4 and ξ2 = 0.860 × 10−2 it holds I1 the Lyapunov exponent λξ1 ∈ [8.365 × 10−2, 8.917 × 10−2], hence it is rigorously certified to be positive; I2 the Lyapunov exponent λξ2 ∈ [−6.03602 × 10−1, −6.03536 × 10−1], hence it is rigorously certified to be negative. Therefore, the system exibits Noise Induced Order.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 19 / 24

Robustness (again, quantitative statistical stability)

Theorem

It holds that λξ varies in a Lipschitz way as a function of ξ, when ξ ≥ 0.873 × 10−4. For ξi as in Theorem 2, denoting by µTa,b,c,ξi and µTa,b,c,ξ the stationary measures of the system with coefficients a, b, c, ξi or a, b, c, ξ, ∀ i ∈ {1, 2} ||µTa,b,c,ξi − µTa,b,c,ξ||L1 ≤ K(|a − a| + |b − b| + |c − c| + |ξi − ξ|). The Lyapunov exponent λξi is also Lipschitz stable to changes of b, c and ξ, while for the changes of the parameter a we can prove a δlogδ continuity modulus.

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 20 / 24

Thanks for the attention!

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 21 / 24

Some referencess

• W. Bahsoun, S. Galatolo, I. Nisoli, X. Niu Rigorous approximation of

diffusion coefficients for expanding maps. J. Stat Phys. (2016)

• S. Galatolo, I Nisoli Rigorous computation of invariant measures of

piecewise hyperbolic maps: 2D Lorenz like maps. Erg. Th. Dyn. Sys.(2015)

• S. Galatolo, I. Nisoli An elementary approach to the rigorous

approximation of invariant measures. SIAM J. Appl. Dyn. Sys (2014)

• W. Bahsoun, S. Galatolo, I. Nisoli, X. Niu A Rigorous Computational

Approach to Linear Response Nonlinearity (2017) S Galatolo, P Giulietti Linear Response for dynamical systems with additive noise Nonlinearity (2019) Stefano Galatolo, Maurizio Monge, Isaia Nisoli Existence of Noise Induced Order, a Computer Aided Proof arXiv:1702.07024

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 22 / 24

If I is a union of intervals of Πδ, we have: 1 − πδVar(I )→L1, 1 − πδL1(I )→W ≤ δ/2, and 1 − πδVar(I )→W ≤ δ2/8, while concening the noise operator NξL1→Var, NξW →L1 ≤ Var(ρ) ξ . As a consequence (1 − πδ)NξL1, Nξ(1 − πδ)L1 ≤ δVar(ρ) 2ξ , and (assuming L = 1) we would have Nξ(1 − πδ)NξL1 ≤ δ2Var(ρ) 8ξ2 .

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 23 / 24

Ingredient 3: local norms for L

Need to understand Nξ(1 − πδ)LNξL˜ f L1, NξL(1 − πδ)NξL˜ f L1, breaking NξL˜ f along the intervals of Πδ if necessary. We can prove that LW (I )→W ≤ T L∞(I ), (spoiler: W -norm is not increased too much, unless |T | is very big) and VarI (Lg) ≤∑

i

• VarT −1

i

(I )(g) ·

• 1

T

• L∞(T −1

i

(I ))

+ gL1(T −1

i

(I )) ·

• T

T 2

• L∞(T −1

i

(I ))

+

∑

y∈∂Dom(Ti ):T (y)∈I

• g(y)

T (y)

• (spoiler: variation is not increased too much, unless |T | is close to 0).

S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 24 / 24