From random Poincaré maps to stochastic mixed-mode-oscillation patterns Nils Berglund MAPMO, Université d’Orléans CNRS, UMR 7349 & Fédération 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 εẋ = f (x, y, z) = y − x3 + 3x ẏ = g1(x, y, z) = kx − 2(y + λ) + z ż = g2(x, y, z) = ρ(λ + y − z) .0<ε1 . k, λ, ρ ∈ R : control parameters 2 The deterministic Koper model εẋ = f (x, y, z) = y − x3 + 3x ẏ = g1(x, y, z) = kx − 2(y + λ) + z ż = g2(x, y, z) = ρ(λ + y − z) .0<ε1 . k, λ, ρ ∈ R : control parameters . Critical manifold: C0 = {f = 0} = {y = x3 − 3x} . Folds: L = {f = 0, ∂xf = 0} = {y = x3 − 3x, x = ±1} = L+ ∪ L− 2-a Critical manifold Fold ẋ 1 Stable critical manifold Unstable critical manifold y Stable critical manifold x ẋ −1 z Fold 3 The deterministic Koper model εẋ = f (x, y, z) = y − x3 + 3x ẏ = g1(x, y, z) = kx − 2(y + λ) + z ż = g2(x, y, z) = ρ(λ + y − z) .0<ε1 . k, λ, ρ ∈ R : control parameters . Critical manifold: C0 = {f = 0} = {y = x3 − 3x} 4 The deterministic Koper model εẋ = f (x, y, z) = y − x3 + 3x ẏ = g1(x, y, z) = kx − 2(y + λ) + z ż = g2(x, y, z) = ρ(λ + y − z) .0<ε1 . k, λ, ρ ∈ R : control parameters . Critical manifold: C0 = {f = 0} = {y = x3 − 3x} . Reduced flow on C0 (Fenichel theory): eliminate y kx − 2(x3 − 3x + λ) + z ẋ = 3(x2 − 1) ż = ρ(λ + x3 − 3x − z) n o Generic fold points: ẋ diverges as x → ±1 n o Folded node singularity: ẋ finite, (desingularized) system has a node 4-a Folded node singularity Normal form [Benoı̂t, Lobry ’82, Szmolyan, Wechselberger ’01]: ẋ = y − x2 ẏ = −(µ + 1)x − z µ ż = 2 (+ higher-order terms) 5 Folded node singularity Normal form [Benoı̂t, Lobry ’82, Szmolyan, Wechselberger ’01]: ẋ = y − x2 ẏ = −(µ + 1)x − z µ ż = 2 (+ higher-order terms) y C0r C0a L x z 5-a Folded node singularity Theorem [Benoı̂t, Lobry ’82, Szmolyan, Wechselberger ’01]: For 2k + 1 < µ−1 < 2k + 3, the system admits k canard solutions The j th canard makes (2j + 1)/2 oscillations Mixed-mode oscillations (MMOs) Picture: Mathieu Desroches 6 Global dynamics Fold Stable critical manifold Canard Stable critical manifold Folded node 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 manifold Folded node Fold . Canard orbits track unstable manifold (for some time) . Typical orbits may jump earlier to stable manifold 7-a Poincaré map c.f. e.g. [Guckenheimer, Chaos, 2008] Fold Σ Stable critical manifold Folded node Fold . Poincaré map Π : Σ → Σ, invertible, 2-dimensional . Due to contraction along C0, close to 1d, non-invertible map 8 Poincaré map zn 7→ zn+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 1 σ dxt = f (xt, yt, zt) dt + √ F (xt, yt, zt) dWt ε ε dyt = g1(xt, yt, zt) dt + σ 0G1(xt, yt, zt) dWt dzt = g2(xt, yt, zt) dt + σ 0G2(xt, yt, zt) dWt . Wt: k-dimensional Brownian motion . σ, σ 0: small parameters (may depend on ε) 10 The stochastic Koper model 1 σ dxt = f (xt, yt, zt) dt + √ F (xt, yt, zt) dWt ε ε dyt = g1(xt, yt, zt) dt + σ 0G1(xt, yt, zt) dWt dzt = g2(xt, yt, zt) dt + σ 0G2(xt, yt, zt) dWt (a) y z L− L+ L− (b) C0a− C0a+ x x L+ z x (c) s 10-a The stochastic Koper model 1 σ dxt = f (xt, yt, zt) dt + √ F (xt, yt, zt) dWt ε ε dyt = g1(xt, yt, zt) dt + σ 0G1(xt, yt, zt) dWt dzt = g2(xt, yt, zt) dt + σ 0G2(xt, yt, zt) dWt Random Poincaré map In appropriate coordinates dϕt = fˆ(ϕt, Xt) dt + σ̂ Fb (ϕt, Xt) dWt b dXt = ĝ(ϕt, Xt) dt + σ̂ G(ϕ t , Xt ) dWt ϕ∈R X∈E⊂Σ . all functions periodic in ϕ (say period 1) . fˆ > c > 0 and σ̂ small ⇒ ϕt likely to increase . process may be killed when X leaves E 10-b Random Poincaré map X E X1 X2 X0 ϕ 1 2 . X0, X1, . . . form (substochastic) Markov chain 11 Random Poincaré map X E X1 X2 X0 ϕ 1 2 . X0, X1, . . . form (substochastic) Markov chain . τ : first-exit time of Zt = (ϕt, Xt) from D = (−M, 1) × E . µZ (A) = P Z {Zτ ∈ A}: harmonic measure (wrt generator L) . [Ben Arous, Kusuoka, Stroock ’84]: under hypoellipticity cond, µZ admits (smooth) density h(Z, Y ) wrt Lebesgue on ∂D . For B ⊂ E Borel set P X0 {X1 ∈ B} = K(X0, B) := Z B K(X0, dy) where K(x, dy) = h((0, x), (1, y)) dy =: k(x, y) dy 11-a Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 0 -8.5 -8.4 -8.3 12 Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 2 · 10−7 -8.4 -8.3 12-a Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 2 · 10−6 -8.4 -8.3 12-b Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 2 · 10−5 -8.4 -8.3 12-c Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 2 · 10−4 -8.4 -8.3 12-d Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 2 · 10−3 -8.4 -8.3 12-e Poincaré map zn 7→ zn+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 k = −10, λ = −7.6, ρ = 0.7, ε = 0.01, σ = σ 0 = 10−2 -8.4 -8.3 12-f Random Poincaré map Observations: . Size of fluctuations depends on noise intensity and canard number k: high order canards are more sensitive . Saturation effect: constant distribution of zn+1 for k > kc(σ, σ 0) ∗ , number of SAOs increases . Consequence: if kc < kdet 13 Random Poincaré map Observations: . Size of fluctuations depends on noise intensity and canard number k: high order canards are more sensitive . Saturation effect: constant distribution of zn+1 for k > kc(σ, σ 0) ∗ , number of SAOs increases . Consequence: if kc < kdet Questions: . Prove saturation effect . How does kc depend on σ, σ 0? . How does size of fluctuations depend on σ, σ 0 and canard number k? . In particular, size of fluctuations for k > kc? 13-a Size of noise-induced fluctuations det det ζt = (xt, yt, zt) − (xdet t , yt , z t ) 1 σ 1 dζt = A(t)ζt dt + √ F (ζt, t) dWt + b(ζt, t) dt ε ε ε | {z }2 Z t σ ζt = √ ε 0 U (t, s)F (ζs, s) dWs + Z 1 t ε 0 =O(kζt k ) U (t, s)b(ζs, s) ds where U (t, s) principal solution of εζ̇ = A(t)ζ. 14 Size of noise-induced fluctuations det det ζt = (xt, yt, zt) − (xdet t , yt , z t ) 1 σ 1 dζt = A(t)ζt dt + √ F (ζt, t) dWt + b(ζt, t) dt ε ε ε | {z }2 Z t σ ζt = √ ε 0 U (t, s)F (ζs, s) dWs + Z 1 t ε 0 =O(kζt k ) U (t, s)b(ζs, s) ds where U (t, s) principal solution of εζ̇ = A(t)ζ. Lemma (Bernstein-type estimate): ) ) ( 2 Z s h P sup G(ζu, u) dWu > h 6 2n exp − 2V (t) 06s6t 0 Z s where G(ζu, u)G(ζu, u)T du 6 V (s) and n = 3 0 ( 14-a Size of noise-induced fluctuations det det ζt = (xt, yt, zt) − (xdet t , yt , z t ) 1 σ 1 dζt = A(t)ζt dt + √ F (ζt, t) dWt + b(ζt, t) dt ε ε ε | {z }2 Z t σ ζt = √ ε 0 U (t, s)F (ζs, s) dWs + Z 1 t ε 0 =O(kζt k ) U (t, s)b(ζs, s) ds where U (t, s) principal solution of εζ̇ = A(t)ζ. Lemma (Bernstein-type estimate): ) ) ( 2 Z s h P sup G(ζu, u) dWu > h 6 2n exp − 2V (t) 06s6t 0 Z s where G(ζu, u)G(ζu, u)T du 6 V (s) and n = 3 0 ( Remark: more precise results using ODE for covariance matrix of t σ 0 ζt = √ U (t, s)F (0, s) dWs ε 0 Z 14-b Regular fold Σ5 C0a+ Σ′4 Σ4 Σ6 C0r Σ1 Σ3 Folded node Σ2 C0a− Transition Σ2 → Σ3 Σ3 → Σ4 Σ4 → Σ04 Σ04 → Σ5 Σ5 → Σ6 Σ6 → Σ1 Σ1 → Σ01 Σ01 → Σ001 √ if z = O( µ) Σ001 → Σ2 ∆x σ + σ0 σ + σ0 σ σ0 + 1/3 ε1/6 ε σ + σ0 σ + σ0 Σ′′1 Σ′1 ∆y (σ + σ 0 )ε1/4 (σ + σ 0 )(ε/µ)1/4 ∆z √ σ ε + σ0 √ σ ε + σ0 p σ ε|log ε| + σ 0 √ σ ε + σ 0 ε1/6 √ σ ε + σ0 √ σ ε + σ0 σ0 σ 0 (ε/µ)1/4 (σ + σ 0 )ε1/4 σ 0 ε1/4 √ σ ε + σ 0 ε1/6 15 Example: Analysis near the regular fold (δ0 , y ∗ , z ∗ ) Σ∗n (x0 , y0 , z0 ) Σ′4 c1 ǫ2/3 Σ∗n+1 Σ5 ǫ1/3 2n C0a− y C0r x z Proposition: For h1 = O(ε2/3), ( P k(yτΣ , zτΣ ) − (y ∗, z ∗)k > h1 5 5 ) κh2 C|log ε| 1 exp − 6 ε σ 2ε + (σ 0)2ε1/3 ( √ 0 Useful if σ, σ ε ) ( κε + exp − 2 σ + (σ 0)2ε )! 16 The global return map Theorem [B, Gentz, Kuehn, 2013] P2 = (x∗2, y2∗ , z2∗ ) ∈ Σ2 (x∗1, y1∗ , z1∗ ) deterministic first-hitting point of Σ1 (x1, y1∗ , z1) stochastic first-hitting point of Σ1 o n ∗ ∗ P P 2 |x1 − x1| > h or |z1 − z1| > h1 ( ) 2 C|log ε| κh 6 ε exp − 2 σ + (σ 0)2 ( κh2 1 ) + exp − 2 σ ε|log ε| + (σ 0)2 ( )! κε + exp − σ 2 + (σ 0)2ε−1/3 17 The global return map Theorem [B, Gentz, Kuehn, 2013] P2 = (x∗2, y2∗ , z2∗ ) ∈ Σ2 (x∗1, y1∗ , z1∗ ) deterministic first-hitting point of Σ1 (x1, y1∗ , z1) stochastic first-hitting point of Σ1 o n ∗ ∗ P P 2 |x1 − x1| > h or |z1 − z1| > h1 ( ) 2 C|log ε| κh 6 ε exp − 2 σ + (σ 0)2 ( κh2 1 ) + exp − 2 σ ε|log ε| + (σ 0)2 ( )! κε + exp − σ 2 + (σ 0)2ε−1/3 √ . Useful for σ ε, σ 0 ε2/3 . ∆x σq + σ0 . ∆z σ ε|log ε| + σ 0 17-a Local analysis near the folded node [B, Gentz, Kuehn, JDE 2012] Thm 1: (Canard spacing) For z = 0, the kth canard lies at dist. √ −c(2k+1)2µ εe from primary canard Thm 2: Size of fluctuations √ (σ + σ 0)(ε/µ)1/4 up to z = εµ 2 √ (σ + σ 0)(ε/µ)1/4 ez /(εµ) for z > εµ Thm 3: (Early escape) Prob. to stay near primary canard 2 0 6 C|log(σ + σ 0)|γ e−κz /(εµ|log(σ+σ )|) √ kµ ε √ µ ε x (+z) √ ε √ ε e−cµ √ z ε e−c(2k+1) 18 2 µ Local analysis near the folded node [B, Gentz, Kuehn, JDE 2012] Thm 1: (Canard spacing) For z = 0, the kth canard lies at dist. √ −c(2k+1)2µ εe from primary canard Thm 2: Size of fluctuations √ (σ + σ 0)(ε/µ)1/4 up to z = εµ 2 √ (σ + σ 0)(ε/µ)1/4 ez /(εµ) for z > εµ Thm 3: (Early escape) Prob. to stay near primary canard 2 0 6 C|log(σ + σ 0)|γ e−κz /(εµ|log(σ+σ )|) √ kµ ε √ µ ε x (+z) √ ε √ z µε (σ + σ 0 )(ε/µ)1/4 18-a Local analysis near the folded node [B, Gentz, Kuehn, JDE 2012] Thm 1: (Canard spacing) For z = 0, the kth canard lies at dist. √ −c(2k+1)2µ εe from primary canard Thm 2: Size of fluctuations √ (σ + σ 0)(ε/µ)1/4 up to z = εµ 2 √ (σ + σ 0)(ε/µ)1/4 ez /(εµ) for z > εµ Thm 3: (Early escape) Prob. to stay near det. solution 2 0 6 C|log(σ + σ 0)|γ e−κz /(εµ|log(σ+σ )|) √ kµ ε √ µ ε x (+z) √ ε √ z µε (σ + σ 0 )(ε/µ)1/4 18-b Local analysis near the folded node [B, Gentz, Kuehn, JDE 2012] Thm 1: (Canard spacing) For z = 0, the kth canard lies at dist. √ −c(2k+1)2µ εe from primary canard √ kµ ε √ µ ε Thm 2: Size of fluctuations √ (σ + σ 0)(ε/µ)1/4 up to z = εµ 2 √ (σ + σ 0)(ε/µ)1/4 ez /(εµ) for z > εµ x (+z) √ ε √ Thm 3: (Early escape) Prob. to stay near det. solution 2 0 6 C|log(σ + σ 0)|γ e−κz /(εµ|log(σ+σ )|) z µε (σ + σ 0 )(ε/µ)1/4 Consequence: Dichotomy . Canards with k 6 q . Canards with k > q q 1/µ: ∆z σ ε|log ε| + σ 0 |log(σ + σ 0)|/µ: ∆z 6 O (assuming ε 6 µ) 0 εµ|log(σ + σ )| q 18-c Local analysis near the folded node: early escapes (a) z ΣJ η PτD Pτz x̄ + z̄ y √ D µ p x (b) x = −0.3 1,200 p(y, z) pdet 800 P1 200 γw 0 0.1 0.075 z̄ Σ′′1 0.05 y −0.005 0.025 0.015 0.035 0 0.055 z 19 Summary . q 1/µ < kc 6 q |log(σ + σ 0)|/µ q . For k 6 kc, dispersion ∆z σ ε|log ε| + σ 0 . For k > kc, dispersion ∆z 6 O q εµ|log(σ + σ 0)| . If the deterministic system has MMO pattern with k∗ SAOs and k∗ < kc then noise increases number of SAOs -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 20 Further ways to analyse random Poincaré map . Theory of singularly perturbed Markov chains ε 1 1 4 ε 1 ε 2 ε 1 3 1 ε 5 1 ε2 21 Further ways to analyse random Poincaré map . Theory of singularly perturbed Markov chains ε 1 1−ε 4 ε ε 1−ε 2 1−ε ε 5 1−ε ε 3 1 − 2ε − ε2 ε2 21-a Further ways to analyse random Poincaré map . Theory of singularly perturbed Markov chains ε 1 1−ε 4 ε ε 1−ε 2 1−ε ε 5 1−ε ε 3 1 − 2ε − ε2 ε2 . For coexisting stable periodic orbits: Metastable transitions 21-b Thanks for your attention – Further reading N.B. and Barbara Gentz, Noise-induced phenomena in slow-fast dynamical systems, A sample-paths approach, Springer, Probability and its Applications (2006) N.B. and Barbara Gentz, Stochastic dynamic bifurcations and excitability, in C. Laing and G. Lord, (Eds.), Stochastic methods in Neuroscience, p. 65-93, Oxford University Press (2009) N.B., Stochastic dynamical systems in neuroscience, Oberwolfach Reports 8:2290–2293 (2011) N.B., Barbara Gentz and Christian Kuehn, Hunting French Ducks in a Noisy Environment, J. Differential Equations 252:4786–4841 (2012). arXiv:1011.3193 N.B. and Damien Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh–Nagumo model, Nonlinearity 25:2303– 2335 (2012). arXiv:1105.1278 N.B. and Barbara Gentz, On the noise-induced passage through an unstable periodic orbit II: General case, preprint arXiv:1208.2557 www.univ-orleans.fr/mapmo/membres/berglund 22