Mathematics and Statistics Colloquium Department of Mathematics, University of Texas at Arlington 2 March 2007 The effect of noise on slow-fast systems Barbara Gentz , University of Bielefeld http://www.math.uni-bielefeld.de/ ˜ gentz Joint work with Nils Berglund, CPT, Marseille
Overview Introduction: Classical results for autonomous systems ⊲ Small random perturbations of dynamical systems ⊲ Exponential asymptotics for first-exit times (Wentzell–Freidlin) ⊲ Subexponential asymptotics Slowly time-dependent systems and stochastic resonance ⊲ The motion of a Brownian particle in a double-well potential ⊲ Simulations ⊲ Rigorous results ⊲ Deterministic dynamics ⊲ Stochastic dynamics for noise intensities below threshold ⊲ Stochastic dynamics for noise intensities above threshold General slow–fast systems ⊲ Dynamics near slow manifolds ⊲ Bifurcations and reduced dynamics 1
Autonomous dynamical systems: ODEs Deterministic ODE x det = f ( x det x det ∈ R d ˙ ) , t t 0 with f : R d → R d , f “well-behaved” (Existence and uniqueness of solution) Assumptions on deterministic dynamics ⊲ Attractors A 1 , A 2 , . . . ⊲ Domains of attraction B 1 , B 2 , . . . 2
Small random perturbations of autonomous systems x 0 = x det ∈ R d d x t = f ( x t ) d t + σ d W t , 0 ⊲ f : R d → R d “well-behaved” d -dim. (standard-) Brownian motion ⊲ { W t } t ≥ 0 ⊲ σ > 0 small Noise enables transitions between domains of attraction Questions Transition times? Transition probabilities? Where do typical transitions occur? 3
Diffusion exit from a domain: Exit problem Bounded domain D ∋ x 0 (with smooth boundary) τ = τ D = inf { t > 0: x t �∈ D} ⊲ first-exit time ⊲ first-exit location x τ ∈ ∂ D Distribution of τ and x τ ? Interesting case D positively invariant under deterministic flow Approaches ⊲ Mean first-exit times and locations via PDEs ⊲ Exponential asymptotics via Wentzell–Freidlin theory 4
Exponential asymptotics: Large deviations Large-deviation rate function � t I [0 ,t ] ( ϕ ) = 1 ϕ s − f ( ϕ s ) � 2 d s 0 � ˙ for ϕ ∈ H 1 2 I [0 ,t ] ( ϕ ) = + ∞ otherwise Large-deviation principle Probability ∼ exp {− I ( ϕ ) /σ 2 } to observe sample paths close to ϕ Assumptions (for the next two slides) ⊲ D positively invariant ⊲ unique, asymptotically stable equilibrium point at 0 ∈ D ⊲ ∂ D ⊂ basin of attraction of 0 (non-characteristic boundary) 5
Wentzell–Freidlin theory I Quasipotential ⊲ Quasipotential with respect to 0: Cost to go against the flow from 0 to z t> 0 inf { I [0 ,t ] ( ϕ ): ϕ ∈ C ([0 , t ] , R d ) , ϕ 0 = 0 , ϕ t = z } V (0 , z ) = inf ⊲ Minimum of quasipotential on boundary ∂ D : V := min z ∈ ∂ D V (0 , z ) Gradient case (reversible diffusion) Drift coefficient deriving from potential: f = −∇ V ⊲ Cost for leaving potential well: V = 2 H H ⊲ Attained for paths against the flow: ϕ t = − f ( ϕ t ) ˙ 6
Wentzell–Freidlin theory II Theorem [Wentzell & Freidlin � ’70,’84] (under above assumptions) For arbitrary initial condition x 0 ∈ D ⊲ Mean first-exit time E x 0 { τ } ∼ e V /σ 2 as σ → 0 ⊲ Concentration of first-exit times e ( V − δ ) /σ 2 � τ � e ( V + δ ) /σ 2 � � P x 0 → 1 as σ → 0 ( δ > 0 ) ⊲ Concentration of exit locations near minima of quasipotential P x 0 � � x τ − z ⋆ � < δ � → 1 as σ → 0 ( δ > 0 ) ( z ⋆ unique minimum of z �→ V (0 , z ) on ∂ D ) 7
Refined results in the gradient case Simplest case: V double-well potential First-hitting time τ hit of deeper well ⊲ E x 1 { τ hit } = c ( σ ) e 2 [ V ( z ) − V ( x 1 )] / σ 2 � � | det ∇ 2 V ( z ) | � 2 π � σ → 0 c ( σ ) = lim exists ! ⊲ det ∇ 2 V ( x 1 ) | λ 1 ( z ) | λ 1 ( z ) unique negative e.v. of ∇ 2 V ( z ) ([Eyring ’35], [Kramers ’40]; [Bovier, Gayrard, Eckhoff, Klein ’02–’05], [Helffer, Klein, Nier ’04]) ⊲ Subexponential asymptotics known; related to geometry at well and saddle / small eigenvalues of the generator τ hit > t E τ hit � ⊲ τ hit ≈ � = e − t exp. distributed: lim σ → 0 P ([Day ’82], [Bovier et al ’02]) 8
Slowly time-dependent systems Overdamped motion of a Brownian particle d x s = − ∂ ∂xV ( x s , εs ) d s + σ d W s in a periodically modulated potential V ( x, εs ) = − 1 2 x 2 + 1 4 x 4 + ( λ c − a 0 ) cos(2 πεs ) x ↓ a 3 / 2 ↑ 0 ← − − → √ a 0 V ( x, 0) V ( x, 1 / 4) = V ( x, 3 / 4) V ( x, 1 / 2) 9
Sample paths Amplitude of modulation A = λ c − a 0 Speed of modulation ε Noise intensity σ A = 0 . 00, σ = 0 . 30, ε = 0 . 001 A = 0 . 10, σ = 0 . 27, ε = 0 . 001 A = 0 . 24, σ = 0 . 20, ε = 0 . 001 A = 0 . 35, σ = 0 . 20, ε = 0 . 001 10
Different parameter regimes and stochastic resonance Synchronisation I ⊲ For matching time scales: 2 π/ε = T forcing = 2 T Kramers ≍ e 2 H/σ 2 ⊲ Quasistatic approach: Transitions twice per period likely (physics’ literature; [Freidlin ’00], [Imkeller et al , since ’02]) ⊲ Requires exponentially long forcing periods Synchronisation II ⊲ For intermediate forcing periods: T relax ≪ T forcing ≪ T Kramers and close-to-critical forcing amplitude: A ≈ A c ⊲ Transitions twice per period with high probability ⊲ Subtle dynamical effects: Effective barrier heights [Berglund & G ’02] SR outside synchronisation regimes ⊲ Only occasional transitions ⊲ But transition times localised within forcing periods ⊲ Cycling: Distribution of first-exit locations doesn’t converge ([Day ’92], [Maier & Stein ’96], [Berglund & G ’04], [Berglund & G ’05]) 11
Synchronisation regime II Characterised by 3 small parameters: 0 < σ ≪ 1 , 0 < ε ≪ 1 , 0 < a 0 ≪ 1 Recall: Motion of a Brownian particle d x s = − ∂ ∂xV ( x s , εs ) d s + σ d W s V ( x, εs ) = − 1 2 x 2 + 1 4 x 4 + ( λ c − a 0 ) cos(2 πεs ) x , 2 λ c = √ 3 3 Rewrite in slow time t = εs : d x t = 1 εf ( x t , t ) d t + σ √ ε d W t with drift term f ( x, t ) = − ∂ ∂xV ( x, t ) = x − x 3 − ( λ c − a 0 ) cos(2 πt ) 12
Small-barrier-height regime 13
Effective barrier heights and scaling of small parameters Theorem [ Berglund & G, Annals of Appl. Probab. ’02 ] (informal version; exact formulation uses first-exit times from space–time sets) σ c = ( a 0 ∨ ε ) 3 / 4 ∃ threshold value Below: σ ≤ σ c ⊲ Transitions unlikely ⊲ Sample paths concentrated in one well σ σ ⊲ Typical spreading ≍ � 1 / 4 ≍ � 1 / 2 � | t | 2 ∨ a 0 ∨ ε � curvature ≤ e − const σ 2 c /σ 2 ⊲ Probability to observe a transition Above: σ ≫ σ c ⊲ 2 transitions per period likely (back and forth) ⊲ with probabilty ≥ 1 − e − const σ 4 / 3 /ε | log σ | ⊲ Transtions occur near instants of minimal barrier height; ⊲ Transition window ≍ σ 2 / 3 14
Step 1: Deterministic dynamics ⊲ For t ≤ − const : x det reaches ε -nbhd of x ⋆ + ( t ) t x ⋆ + ( t ) in time ≍ ε | log ε | (Tihonov ’52) − const ≤ t ≤ − ( a 0 ∨ ε ) 1 / 2 : ⊲ For x det t x det − x ⋆ + ( t ) ≍ ε/ | t | t | t | ≤ ( a 0 ∨ ε ) 1 / 2 : ⊲ For x ⋆ 0 ( t ) 0 ( t ) ≍ ( a 0 ∨ ε ) 1 / 2 ≥ √ ε x det − x ⋆ t (effective barrier height) ( a 0 ∨ ε ) 1 / 2 ≤ t ≤ + const : ⊲ For x det − x ⋆ x ⋆ − ( t ) + ( t ) ≍ − ε/ | t | t ⊲ For t ≥ + const : | x det − x ⋆ + ( t ) | ≍ ε t 15
σ ≤ σ c = ( a 0 ∨ ε ) 3 / 4 Step 2: Below threshold Behaviour of y t = x t − x det ? t Linearizing the drift coefficent − → nonautonomous linear SDE t = 1 t d t + σ d y 0 εa ( t ) y 0 √ ε d W t , y 0 = 0 � t a ( t ) = ∂ x f ( x det α ( t, s ) := , t ) = curvature ; s a ( u ) du t � t t = σ 0 e α ( t,s ) /ε d W s y 0 Solution is a Gaussian process √ ε � t v ( t ) = σ 2 σ 2 0 e 2 α ( t,s ) /ε d s ∼ Variance ε curvature t | > δ } ≤ e − δ 2 / 2 v ( t ) P {| y 0 Concentration result for y 0 t : Aim: Analogous resultat for the whole sample path { y t } t ≥ 0 16
σ ≤ σ c = ( a 0 ∨ ε ) 3 / 4 Step 2: Below threshold σ 2 σ 2 v ( t ) ∼ curvature ∼ ( | t | 2 ∨ a 0 ∨ ε ) 1 / 2 ζ ( t ) := v ( t ) σ 2 � � ( x, t ): | x − x det � B ( h ) := ζ ( t ) | < h t τ B ( h ) = first-exit time of ( x t , t ) from B ( h ) 17
σ ≤ σ c = ( a 0 ∨ ε ) 3 / 4 Step 2: Below threshold Theorem ([Berglund & G ’02], [Berglund & G ’05]) ∃ h 0 , c 1 , c 2 , c 3 > 0 ∀ h ≤ h 0 C ( h/σ, t, ε ) e − κ − h 2 / 2 σ 2 ≤ P ≤ C ( h/σ, t, ε ) e − κ + h 2 / 2 σ 2 � � τ B ( h ) < t κ − = 1 + c 1 h + c 1 e − c 2 t/ε ; with κ + = 1 − c 1 h , � � � � � ε e − c 3 h 2 /σ 2 + e − c 1 t/ε + ε 2 | α ( t ) | h σ + t C ( h/σ, t, ε ) = 1 + O π ε σ h Basic idea local approximation of y t by y 0 t ; Gaussian process is a rescaled Brownian motion; results on the density of the first-passage time for BM through nonlinear curves 18
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