From random Poincar e maps to stochastic mixed-mode-oscillation - PowerPoint PPT Presentation

From random Poincar´ e maps to stochastic mixed-mode-oscillation patterns Nils Berglund MAPMO, Universit´ e d’Orl´ eans CNRS, UMR 7349 & F´ ed´ eration Denis Poisson www.univ-orleans.fr/mapmo/membres/berglund nils.berglund@math.cnrs.fr Collaborators: Barbara Gentz (Bielefeld), Christian Kuehn (Vienna) Sixth Workshop on Random Dynamical Systems Bielefeld, October 31, 2013

Mixed-mode oscillations (MMOs) Belousov-Zhabotinsky reaction [Hudson 79] Stellate cells [Dickson 00] Mean temperature based on ice core measurements [Johnson et al 01] 1

Mixed-mode oscillations (MMOs) Belousov-Zhabotinsky reaction [Hudson 79] Stellate cells [Dickson 00] ⊲ Deterministic models reproducing these oscillations exist and have been abundantly studied They often involve singular perturbation theory ⊲ We want to understand the effect of noise on oscillatory patterns Noise may also induce oscillations not present in deterministic case 1-a

The deterministic Koper model x = f ( x, y, z ) = y − x 3 + 3 x ε ˙ y = g 1 ( x, y, z ) = kx − 2( y + λ ) + z ˙ z = g 2 ( x, y, z ) = ρ ( λ + y − z ) ˙ ⊲ 0 < ε ≪ 1 ⊲ k, λ, ρ ∈ R : control parameters 2

The deterministic Koper model x = f ( x, y, z ) = y − x 3 + 3 x ε ˙ y = g 1 ( x, y, z ) = kx − 2( y + λ ) + z ˙ z = g 2 ( x, y, z ) = ρ ( λ + y − z ) ˙ ⊲ 0 < ε ≪ 1 ⊲ k, λ, ρ ∈ R : control parameters ⊲ Critical manifold: C 0 = { f = 0 } = { y = x 3 − 3 x } ⊲ Folds: L = { f = 0 , ∂ x f = 0 } = { y = x 3 − 3 x, x = ± 1 } = L + ∪ L − 2-a

Critical manifold x ≫ 1 ˙ Fold Stable critical manifold Unstable critical manifold y Stable critical manifold x z x ≪ − 1 ˙ Fold 3

The deterministic Koper model x = f ( x, y, z ) = y − x 3 + 3 x ε ˙ y = g 1 ( x, y, z ) = kx − 2( y + λ ) + z ˙ z = g 2 ( x, y, z ) = ρ ( λ + y − z ) ˙ ⊲ 0 < ε ≪ 1 ⊲ k, λ, ρ ∈ R : control parameters ⊲ Critical manifold: C 0 = { f = 0 } = { y = x 3 − 3 x } 4

The deterministic Koper model x = f ( x, y, z ) = y − x 3 + 3 x ε ˙ y = g 1 ( x, y, z ) = kx − 2( y + λ ) + z ˙ z = g 2 ( x, y, z ) = ρ ( λ + y − z ) ˙ ⊲ 0 < ε ≪ 1 ⊲ k, λ, ρ ∈ R : control parameters ⊲ Critical manifold: C 0 = { f = 0 } = { y = x 3 − 3 x } ⊲ Reduced flow on C 0 (Fenichel theory) : eliminate y x = kx − 2( x 3 − 3 x + λ ) + z ˙ 3( x 2 − 1) z = ρ ( λ + x 3 − 3 x − z ) ˙ ⋉ Generic fold points: ˙ x diverges as x → ± 1 ⋊ ⋉ Folded node singularity: ˙ x finite, ⋊ (desingularized) system has a node 4-a

Folded node singularity Normal form [Beno ıt, Lobry ’82, Szmolyan, Wechselberger ’01] : ˆ x = y − x 2 ǫ ˙ y = − ( µ + 1) x − z ˙ ( + higher-order terms ) z = µ ˙ 2 5

Folded node singularity Normal form [Beno ıt, Lobry ’82, Szmolyan, Wechselberger ’01] : ˆ x = y − x 2 ǫ ˙ y = − ( µ + 1) x − z ˙ ( + higher-order terms ) z = µ ˙ 2 y C r 0 C a 0 L z x 5-a

Folded node singularity Theorem [Beno ıt, Lobry ’82, Szmolyan, Wechselberger ’01] : ˆ For 2 k + 1 < µ − 1 < 2 k + 3, the system admits k canard solutions The j th canard makes (2 j + 1) / 2 oscillations Mixed-mode oscillations (MMOs) Picture: Mathieu Desroches 6

Global dynamics Fold Stable critical manifold Canard Stable critical Folded node manifold Fold ⊲ Canard orbits track unstable manifold (for some time) Typical orbits may jump earlier 7

Global dynamics Fold Typical orbit Stable critical manifold Canard Stable critical Folded node manifold Fold ⊲ Canard orbits track unstable manifold (for some time) ⊲ Typical orbits may jump earlier to stable manifold 7-a

Poincar´ e map c.f. e.g. [Guckenheimer, Chaos, 2008] Fold Σ Stable critical Folded node manifold Fold ⊲ Poincar´ e map Π : Σ → Σ, invertible, 2-dimensional ⊲ Due to contraction along C 0 , close to 1d, non-invertible map 8

Poincar´ e map z n �→ z n +1 -8.0 -8.1 -8.2 -8.3 -8.4 -8.5 -8.6 -8.7 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 -8.2 -8.1 -8.0 -7.9 -7.8 k = − 10 , λ = − 7 . 35 , ρ = 0 . 7 , ε = 0 . 01 9

The stochastic Koper model d x t = 1 εf ( x t , y t , z t ) d t + σ √ εF ( x t , y t , z t ) d W t d y t = g 1 ( x t , y t , z t ) d t + σ ′ G 1 ( x t , y t , z t ) d W t d z t = g 2 ( x t , y t , z t ) d t + σ ′ G 2 ( x t , y t , z t ) d W t ⊲ W t : k -dimensional Brownian motion ⊲ σ, σ ′ : small parameters (may depend on ε ) 10

The stochastic Koper model d x t = 1 εf ( x t , y t , z t ) d t + σ √ εF ( x t , y t , z t ) d W t d y t = g 1 ( x t , y t , z t ) d t + σ ′ G 1 ( x t , y t , z t ) d W t d z t = g 2 ( x t , y t , z t ) d t + σ ′ G 2 ( x t , y t , z t ) d W t L − L + z (a) y L − (b) C a − 0 x C a + 0 x L + z x (c) s 10-a

The stochastic Koper model d x t = 1 εf ( x t , y t , z t ) d t + σ √ εF ( x t , y t , z t ) d W t d y t = g 1 ( x t , y t , z t ) d t + σ ′ G 1 ( x t , y t , z t ) d W t d z t = g 2 ( x t , y t , z t ) d t + σ ′ G 2 ( x t , y t , z t ) d W t Random Poincar´ e map In appropriate coordinates σ � d ϕ t = ˆ f ( ϕ t , X t ) d t + ˆ F ( ϕ t , X t ) d W t ϕ ∈ R σ � d X t = ˆ g ( ϕ t , X t ) d t + ˆ G ( ϕ t , X t ) d W t X ∈ E ⊂ Σ ⊲ all functions periodic in ϕ (say period 1) ⊲ ˆ f � c > 0 and ˆ σ small ⇒ ϕ t likely to increase ⊲ process may be killed when X leaves E 10-b

Random Poincar´ e map X E X 1 X 2 X 0 ϕ 1 2 ⊲ X 0 , X 1 , . . . form (substochastic) Markov chain 11

Random Poincar´ e map X E X 1 X 2 X 0 ϕ 1 2 ⊲ X 0 , X 1 , . . . form (substochastic) Markov chain ⊲ τ : first-exit time of Z t = ( ϕ t , X t ) from D = ( − M, 1) × E ⊲ µ Z ( A ) = P Z { Z τ ∈ A } : harmonic measure (wrt generator L ) ⊲ [Ben Arous, Kusuoka, Stroock ’84] : under hypoellipticity cond, µ Z admits (smooth) density h ( Z, Y ) wrt Lebesgue on ∂ D ⊲ For B ⊂ E Borel set � P X 0 { X 1 ∈ B } = K ( X 0 , B ) := B K ( X 0 , d y ) where K ( x, d y ) = h ((0 , x ) , (1 , y )) d y =: k ( x, y ) d y 11-a

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 0 12

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 2 · 10 − 7 12-a

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 2 · 10 − 6 12-b

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 2 · 10 − 5 12-c

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 2 · 10 − 4 12-d

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 2 · 10 − 3 12-e

Poincar´ e map z n �→ z n +1 -8.5 -8.6 -8.7 -8.8 -8.9 -9.0 -9.1 -9.2 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3 k = − 10 , λ = − 7 . 6 , ρ = 0 . 7 , ε = 0 . 01 , σ = σ ′ = 10 − 2 12-f

Random Poincar´ e map Observations: ⊲ Size of fluctuations depends on noise intensity and canard number k : high order canards are more sensitive ⊲ Saturation effect: constant distribution of z n +1 for k > k c ( σ, σ ′ ) ⊲ Consequence: if k c < k ∗ det , number of SAOs increases 13

Random Poincar´ e map Observations: ⊲ Size of fluctuations depends on noise intensity and canard number k : high order canards are more sensitive ⊲ Saturation effect: constant distribution of z n +1 for k > k c ( σ, σ ′ ) ⊲ Consequence: if k c < k ∗ det , number of SAOs increases Questions: ⊲ Prove saturation effect ⊲ How does k c depend on σ, σ ′ ? ⊲ How does size of fluctuations depend on σ, σ ′ and canard number k ? ⊲ In particular, size of fluctuations for k > k c ? 13-a

Size of noise-induced fluctuations ζ t = ( x t , y t , z t ) − ( x det , y det , z det ) t t t d ζ t = 1 εA ( t ) ζ t d t + σ √ ε F ( ζ t , t ) d W t + 1 b ( ζ t , t ) d t � �� � ε = O ( � ζ t � 2 ) � t � t 0 U ( t, s ) F ( ζ s , s ) d W s + 1 ζ t = σ 0 U ( t, s ) b ( ζ s , s ) d s √ ε ε where U ( t, s ) principal solution of ε ˙ ζ = A ( t ) ζ . 14