2012 NCTS Workshop on Dynamical Systems National Center for - PowerPoint PPT Presentation

Slowly driven systems Stochastic resonance Saddle–node MMOs 2012 NCTS Workshop on Dynamical Systems National Center for Theoretical Sciences, National Tsing-Hua University Hsinchu, Taiwan, 16–19 May 2012 The Effect of Gaussian White Noise on Dynamical Systems: Bifurcations in Slow–Fast Systems Barbara Gentz University of Bielefeld, Germany Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ ˜ gentz

Slowly driven systems Stochastic resonance Saddle–node MMOs Slowly driven systems in dimension n = 1 Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 1 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Slowly driven systems Recall from yesterday’s lecture Parameter dependent ODE, perturbed by Gaussian white noise d x s = ˜ ( x s ∈ R 1 ) f ( x s , λ ) d s + σ d W s Assume parameter varies slowly in time: λ = λ ( ε s ) d x s = ˜ f ( x s , λ ( ε s )) d s + σ d W s Rewrite in slow time t = ε s d x t = 1 ε f ( x t , t ) d t + σ √ ε d W t Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 2 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Assumptions yesterday Existence of a uniformly asymptotically stable equlibrium branch x ⋆ ( t ) ∃ ! x ⋆ : I → R s.t. f ( x ⋆ ( t ) , t ) = 0 and a ⋆ ( t ) = ∂ x f ( x ⋆ ( t ) , t ) � − a 0 < 0 Then there exists an adiabatic solution ¯ x ( t , ε ) x ( t , ε ) = x ⋆ ( t ) + O ( ε ) ¯ and ¯ x ( t , ε ) attracts nearby solutions exp. fast x ⋆ ( t ) x det ( t ) Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 3 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Defining the strip describing the typical spreading ⊲ Let v ( t ) be the variance of the solution z ( t ) of the linearized SDE for the deviation x t − ¯ x ( t , ε ) ⊲ v ( t ) /σ 2 is solution of a deterministic slowly driven system admitting a uniformly asymptotically stable equilibrium branch ⊲ Let ζ ( t ) be the adiabatic solution of this system ⊲ ζ ( t ) ≈ 1 / | a ( t ) | , where a ( t ) = ∂ x f (¯ x ( t , ε ) , t ) ≤ − a 0 / 2 < 0 � Define a strip B ( h ) around ¯ x ( t , ε ) of width ≃ h ζ ( t ) and the first-exit time τ B ( h ) � B ( h ) = { ( x , t ): | x − ¯ x ( t , ε ) | < h ζ ( t ) } τ B ( h ) = inf { t > 0: ( x t , t ) �∈ B ( h ) } Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 4 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Concentration of sample paths x ⋆ ( t ) x t ¯ x ( t , ε ) B ( h ) Theorem [Berglund & G ’02, ’05] � t ≤ const 1 � h � � σ e − h 2 [1 −O ( ε ) −O ( h )] / 2 σ 2 � � τ B ( h ) < t a ( s ) d s P � � ε � 0 Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 5 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Avoided bifurcation: Stochastic Resonance Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 6 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Overdamped motion of a Brownian particle in a periodically modulated potential d x t = − 1 ∂ x V ( x t , t ) d s + σ ∂ √ ε d W t ε V ( x , t ) = − 1 2 x 2 + 1 4 x 4 + ( λ c − a 0 ) cos(2 π t ) x Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 7 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs 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 Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 8 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs 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 Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 9 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Synchronisation regime II Characterised by 3 small parameters: 0 < σ ≪ 1 , 0 < ε ≪ 1 , 0 < a 0 ≪ 1 Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 10 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Effective barrier heights and scaling of small parameters Theorem [Berglund & G ’02 ] (informal version; exact formulation uses first-exit times) ∃ threshold value σ c = ( a 0 ∨ ε ) 3 / 4 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 | ⊲ Transitions occur near instants of minimal barrier height; window ≍ σ 2 / 3 Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 11 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Deterministic dynamics ⊲ For t ≤ − const : x det reaches ε -nbhd of x ⋆ + ( t ) t in time ≍ ε | log ε | (Tihonov ’52) x ⋆ + ( t ) ⊲ For − const ≤ t ≤ − ( a 0 ∨ ε ) 1 / 2 : x det − x ⋆ + ( t ) ≍ ε/ | t | x det t t ⊲ For | t | ≤ ( a 0 ∨ ε ) 1 / 2 : 0 ( t ) ≍ ( a 0 ∨ ε ) 1 / 2 ≥ √ ε x det − x ⋆ x ⋆ 0 ( t ) t (effective barrier height) ⊲ For ( a 0 ∨ ε ) 1 / 2 ≤ t ≤ + const : x ⋆ − ( t ) x det − x ⋆ + ( t ) ≍ − ε/ | t | t ⊲ For t ≥ + const : | x det − x ⋆ + ( t ) | ≍ ε t Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 12 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Below threshold: σ ≤ σ c = ( a 0 ∨ ε ) 3 / 4 σ 2 σ 2 v ( t ) ∼ curvature ∼ ( | t | 2 ∨ a 0 ∨ ε ) 1 / 2 Approach for stable case can still be used 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 with κ + = 1 − O ( ε ) − O ( h ), κ − = 1 + O ( ε ) + O ( h ) + O (e − c 2 t /ε ) Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 13 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Above threshold: σ ≫ σ c = ( a 0 ∨ ε ) 3 / 4 ⊲ Typical paths stay below x det � + h ζ ( t ) t ⊲ For t ≪ − σ 2 / 3 : Transitions unlikely; as below threshold ⊲ At time t = − σ 2 / 3 : Typical spreading is σ 2 / 3 ≫ x det − x ⋆ 0 ( t ) t Transitions become likely ⊲ Near saddle: Diffusion dominated dynamics ⊲ δ 1 > δ 0 with f ≍ − 1 ; δ 0 in domain of attraction of x ⋆ − ( t ) Drift dominated dynamics ⊲ Below δ 0 : behaviour as for small σ Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 14 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Above threshold: σ ≫ σ c = ( a 0 ∨ ε ) 3 / 4 Idea of the proof With probability ≥ δ > 0, in time ≍ ε | log σ | /σ 2 / 3 , the path reaches ⊲ x det if above t ⊲ then the saddle ⊲ finally the level δ 1 σ 4 / 3 In time σ 2 / 3 there are ε | log σ | attempts possible During a subsequent timespan of length ε , level δ 0 is reached (with probability ≥ δ ) Finally, the path reaches the new well Result ≤ e − const σ 4 / 3 /ε | log σ | ∀ s ∈ [ − σ 2 / 3 , t ] � � ( t ≥ − γσ 2 / 3 , γ small) x s > δ 0 P Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 15 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Space–time sets for stochastic resonance Below threshold Above threshold Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 16 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Saddle–node bifurcation Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 17 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Saddle–node bifurcation (e.g. f ( x , t ) = − t − x 2 ) σ ≪ σ c = ε 1 / 2 σ ≫ σ c = ε 1 / 2 x t σ = 0: Solutions stay at distance ε 1 / 3 above bif. point until time ε 2 / 3 after bif. Theorem ⊲ If σ ≪ σ c : Paths likely to stay in B ( h ) until time ε 2 / 3 after bifurcation; maximal spreading σ/ε 1 / 6 . ⊲ If σ ≫ σ c : Transition typically for t ≍ − σ 4 / 3 ; transition probability � 1 − e − c σ 2 /ε | log σ | Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 18 / 35

Slowly driven systems Stochastic resonance Saddle–node MMOs Mixed-mode oscillations Bifurcations in Slow–Fast Systems Barbara Gentz NCTS, 18 May 2012 19 / 35