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 � ( a + ( x − 1 8 ) 1 / 3 ) e − x + b ( 0 ≤ x ≤ 0 . 3 ) , T a , b , c ( x ) = c ( 10 xe − 10 x / 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 � n 1 log | T � ( x i ) | = log | T � | d µ ∑ λ µ : = lim n n → ∞ 0 where x i 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. λ µ > 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 ones 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 of 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: L µ ( 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 ξ x ) noise kernel (with ρ ∈ BV and � ρ ξ ( x ) = 1 ξ ρ ( 1 ρ = 1) N ξ f = ρ ξ ∗ f , Transfer operator L ξ : SM ([ 0 , 1 ]) → L 1 [ 0 , 1 ] is defined by L ξ = N ξ L . Stationary measures are fixed points of L ξ . 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 L ξ . The stationary measure f ξ is the unique fixed prob. meas of L ξ . General task Find a good approximation of f ξ (small explicit error in L 1 + 1 explicit information on the variance in given intervals) With this, compute the Lyapunov exponent � log ( T � ) df ξ 2 For the approximation of f ξ . 1.a Approximate (up to an explicit error) the transfer operator L ξ of the system by a finite rank L ξ , δ (big matrix to be computed in a certified way) 1.b Approximate (up to an explicit error) the fixed point f ξ of L ξ with the fixed point f ξ , δ of L ξ , 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 I δ Let F δ be the σ − algebra associated to I δ , then consider SM ( X ) → L 1 ( X ) π δ : (1) π δ ( g ) = E ( g | F δ ) (2) The approximated operator we are going to use is defined by a kind of Ulam discretization L δ , ξ = π δ N ξ π δ L π δ The fixed points and other properties of L ξ will be approximated by the ones of L δ , 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 f δ , ξ and f ξ be fixed points of L δ , ξ and L ξ � L n δ , ξ f δ , ξ − L n � f δ , ξ − f ξ � L 1 = ξ f ξ � L 1 � L n δ , ξ f δ , ξ − L n δ , ξ f ξ + L n δ , ξ f ξ − L n = ξ f ξ � L 1 � L n δ , ξ ( f δ , ξ − f ξ ) � L 1 + � ( L n δ , ξ − L n ≤ ξ ) f ξ � L 1 . Therefore if (we verify computionally that) for some n || L n δ , ξ | V || L 1 → L 1 ≤ 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 f δ , ξ and f ξ be fixed points of L δ , ξ and L ξ � f δ , ξ − f ξ � L 1 ≤ � L n δ , ξ ( f δ , ξ − f ξ ) � L 1 + � ( L n δ , ξ − L n ξ ) f ξ � L 1 . Therefore if (we verfy computionally that) || L n δ , ξ | V || L 1 → L 1 ≤ α < 1 where V = { f ∈ BV , � f dm = 0 } (zero avg. measures). Then � f δ , ξ − f ξ � L 1 ≤ α � f δ , ξ − f ξ � L 1 + � ( L n δ , ξ − L n ξ ) f ξ � L 1 . and � � 1 � � � ( L n δ , ξ − L n � f ξ − f δ , ξ � L 1 ≤ ξ ) f ξ � L 1 . 1 − α (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 � � � � � ( L n δ , ξ − L n ξ ) f ξ Expanding � L 1 in a telescopic sum and applying triangular ineq.: � � � � �� ( π δ N ξ π δ L ) n π δ − ( N ξ L ) n � � � � � ( L n δ , ξ − L n ξ ) f ξ � L 1 = f ξ L 1 � � n − 1 � � � L δ , ξ | i � ∑ ≤ � ( π δ − 1 ) N ξ Lf ξ � L 1 + × V L 1 → L 1 i = 0 �� � � � � � N ξ ( π δ − 1 ) Lf ξ � � N ξ π δ L ( π δ − 1 ) f ξ � × L 1 + L 1 S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 16 / 24
Estimate for � � � � � � � ( π δ − 1 ) N ξ Lf ξ � � N ξ ( π δ − 1 ) Lf ξ � � N ξ π δ L ( π δ − 1 ) N ξ Lf ξ � L 1 , L 1 , L 1 . Lemma We have � N ξ ( 1 − π δ ) � L 1 → L 1 ≤ 1 2 δξ − 1 Var ( ρ ) . � ( 1 − π δ ) N ξ � L 1 → L 1 ≤ 1 2 δξ − 1 Var ( ρ ) . Recalling that � π δ � L 1 → L 1 ≤ 1, � N ξ � L 1 → L 1 ≤ 1 we obtain � � � L δ , ξ | i � � f ξ − f ξ , δ � L 1 ≤ 1 + ∑ n − 1 i = 0 V L 1 → L 1 δξ − 1 Var ( ρ ) . (3) 2 ( 1 − α ) S.Galatolo (Dip. Mat, Univ. Pisa ) Existence of Noise Induced Order, a computer aided proof. CIRM 2019 17 / 24
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