Large Fluctuations in in Chaotic Systems Igor Khovanov Physics Department, Lancaster University V.S. Anishchenko, Saratov State University N.A. Khovanova, D.G. Luchinsky, P.V.E. McClintock, Lancaster University R. Mannella, Pisa University 1
Large Fluctuations in in Chaotic Systems Outline Large Fluctuational Approach and Model Reduction Escape in quasi-hyperbolic systems Escape in non-hyperbolic systems Conclusions 2
Deterministic Chaos and Noise: Environment Environment induces Dissipation and Fluctuations H = H + H + H H H H S B S B SB System Hamiltonian Bath (Environmental) Hamiltonian Hamiltonian of interaction SB Elimination of the environmental degrees of freedom leads to Dissipation and Bath Environment (external or internal degrees of freedom) Fluctuations Note: Elimination is,as a rule, a challenge task and it is often phenomenological 3
Deterministic Chaos and Noise: Environment Archetypical Example: Environment as a Collection of Linear Oscillators H = H + H + H H H S 2 p = + 2m Elimination leads to S B SB V( q; The system is a model of a particle in potential p m N 2 n n 2 2 B = + ωnxn n= 1 2mn 2 N N 2 2 n SB = n n + 2 n= 1 n= 1 2mnωn H q c x q c The collection of harmonic oscillators V mq&& + 2 γ mq& + = ξ ( q Damping (dissipation) Bath Environment Linear coupling between system and bath Dissipation and Fluctuations have the same origin Fluctuations noise 4
Chaos and Noise: Large Fluctuations The simplification of dynamics: considering dynamics related to Large Fluctuations Different manifestations of fluctuations: Diffusion in a vicinity of attractor Large fluctuations (deviations) from attractors t relax << t activ 5
Chaos and Noise: Large Fluctuation Approach The system described by Langevin equations: x& = K( x, + Qξ(, ξ α = 0, ξ ( ξ ( s) α β = Qδ ( t s) Transition probability via fluctuations paths ρ( x f, t f xi, ti ) = j ρ[ x( j ] ρ[ x( opt ] The selection of the most probable (optimal) path Path x 1 x f Path x 2 Path x j x i Path x opt 6
Deterministic chaos chaos and and noise: noise: optimal path path approach deterministic pattern of of fluctuations x f x i x& = K( x, + ξ(, ξ α = 0, ξ ( ξ ( s) α β = DQδ ( t s) The probability of fluctuational path ρ[ ξ( j ] ρ[ ξ( j ρ[ x( j ] is related to the probability of random force to have a realization For Gaussian noise: t f 1 2 1 ] = C exp = ξ( j dt C exp S 2 t 2 i Since the exponential form, the most probable path has a minimal S=S min ξ( j Changing to dynamical variables: Action S In the limit = S [ ξ( ] D 0, ρ( x x & = K( x, + ξ( ξ( = x& K( x, f ; x i ) = ρ [ x ( ] = min dt( x& K(, t ) Smin = S opt x ) ( x( ) 2 opt S = [ x( ] S = Const exp [ x( ] D opt Deterministic minimization problem 7
Large Large fluctuations and and Model reduction The initial model: the Hamiltonian for the system, the bath and coupling In general case the dimension is infinite. H = HS + HB + HSB Langevin model reduction: finite dimensional system with noise terms, The dimension is infinite x& = K( x, + ξ(, ξ α = 0, ξ ( ξ ( s) α β = DQδ ( t s) Large fluctuations reduction leads to a specific object: the optimal path as a solution of boundary value problem of the finite dimensional Hamilton system x f x i [ x ( ] = min dt( x& K(, t ) Smin = S opt x ) 2 Initial state : q ( ) x p( ) t =, t = 0, t ; i i i i Final state : ( ) ( ) q t = x, p t = 0, t. f f f f Formally the deterministic minimization problem can be formulated in the Hamiltonian form: 1 H = pqp + pk( q, ; 2 H H q& =, p& =, p q 8
Optimal path path approach deterministic pattern of of fluctuations Optimal paths are experimentally observable (Dykman 92) The prehistory probability of transition between states of bistable oscillator (electronic experimen ξ() t Optimal paths are essentially deterministic trajectories 9
Chaos and Noise: Quasi-hyperbolic Attractor Lorenz system q& 1 = σ ( q2 q1) q& = rq q qq 2 1 2 1 3 q& 3 = qq 1 2 bq3+ 2 D ξ ( Consider noise-induced escape from the chaotic attractor to the stable point in the limit D 0 The task is to determine the most probable (optimal) escape path σ = 10, b= 8/ 3, r = 24.08 Chaotic Attractor Stable points Saddle cycles 10
Chaos: Quasi-hyperbolic Attractor r 13.92 Lorenz Attractor Homoclinic loop -> Horseshoe L 2 L 1 The saddle point and its separatrices belong to chaotic attractor and form bad set or nonhyperbolic part of the attractor Γ 2 Γ 1 Separatrices Saddle point r 24.06 Loops between separatrices Γ 1 and Γ 2 and stable manifolds of cycles L 1 and L 2 generate The Lorenz attractor quasi-hyperbolic attractor 11
Chaos: Quasi-hyperbolic Attractor Stable manifold W S Unstable manifold W U Cycle L 1 The loop between separatrices Γ 1 and Γ 2 and stable manifolds of cycles L 1 and L 2 persists by varying parameters Separatrix Γ 2 Saddle point 12
Large Fluctuations: the prehistory probability ξ α The prehistory approach x& = K( x, + ξ(, = 0, ξ ( ξ ( s) α β = DQδ ( t s) 1. Select the regime D 0 i.e. rare large fluctuations t << relax t activ 2. Record all trajectories x j ( arrived to the final state and build the prehistory probability density p h (x, The maximum of the density corresponds to the most probable (optimal) path 3. Simultaneously noise realizations ξ( are collected and give us the optimal fluctuational force x f 13
Chaos: Quasi-hyperbolic Attractor The Poincaré section q 3 = r -1 + Separatrix Stable manifold of the saddle point Chaotic attractor Stable point P 1 Stable point P 2 o Saddle cycle 14
Chaos and Noise: Quasi-hyperbolic Attractor Probability density p(q 1 ) for Poincaré section q3 = r 1 q 1 q 1 The difference in the tail (low probable part of the density) D=0 In the absence of noise D=0.001 In the presence of noise Noise does not change significantly the probability density Are noise-induced tail important? 15
Chaos and Noise: Quasi-hyperbolic Attractor q 3 Escape from quasi-hyperbolic attractor Escape trajectories W S is the stable manifold and Γ 1 and Γ 2 are separatrices of the saddle point O L 1 and L 2 are saddle cycles T 1 and T 2 are trajectories which are tangent to W S Noise-induced tail q 1 q 2 The escape process is connected with the non-hyperbolic structure of attractor: stable and unstable manifolds of the saddle point The distribution of escape trajectories (exit distribution) 16
Chaos and Noise: Quasi-hyperbolic Attractor The optimal path and fluctuational force from analysis of fluctuations prehistory Optimal force Optimal path Three parts: 1) Deterministic part, the force equals to zero; The point A is the initial state x i 2) Noise-assisted motion along stable and unstable manifolds of the saddle point 3) Slow diffusion to overcome the deterministic drift of unstable manifold of the saddle cycle and cross the cycle 17
Chaos and Noise: Non-hyperbolic Attractor Non-autonomous nonlinear oscillator Uqt (, ) q&& +Γ q& + = 2 Dξ( The potential U(q) q is monostable 2 ω0 2 β 3 γ 4 Uqt (, ) = q+ q+ q+ qhsinωt 2 3 4 Γ= 0.05 ω = 0.597 β = γ = 1 0 The motion is underdamped Ω=1.005 Co-existence of two cycles of period 1 Chaos region Co-existence of two cycles of period 2 18
Chaos and Noise: Non-hyperbolic Attractor The co-existence of chaotic attractor and the limit cycle h = 0.13 Ω= 0.95 :? Initial state ( t ) ( t ) q = x, p = 0, i i i ti ; Final state : ( tf ) f ( t f ) q = x, p = 0, t f q&. The stable cycle q& Stable cycle Saddle cycle Chaotic attractor 19
Chaos and Noise: Non-hyperbolic Attractor Probability density pqq& (, ) for Poincaré section Ω t = 0 D = 0 D = 510 6 q q q& q& A weak noise significantly changes the probability density 20
Chaos and Noise: Non-hyperbolic Attractor The prehistory probability density p( qqt,&, ) There is a miximum in the prehistory probability density h D = 510 4 q q& 21
Chaos and Noise: Non-hyperbolic Attractor Escape trajectories follow a narrow tube Q: Do any sets form the escape path? To answer we take initial conditions along the path and try to localize any sets q& Saddle cycle time T 2π = Ω 22
Chaos and Noise: Non-hyperbolic Attractor The prehistory probability density p ( q, Saddle cycles form the escape path S5 h Saddle cycle of period 5 Saddle cycle of period 3 S3 23
Chaos and Noise: Non-hyperbolic Attractor The escape is a sequence of jumps between saddle cycles. Escape trajectory is a heteroclinic trajectories connected saddle cycles of Hamilton system. 1 H = pqp + pk( q, ; 2 q& H H q& =, p& =, Saddle cycle of period 3 p q is outside the attractor Initial state : cycle S5 q( ti) = xi, p( ti) = 0, ti ; S3 Final state : cycle UC1 Escape trajectory q( tf ) = xf, p( t f ) = 0, tf. S5 Saddle cycle of period 5 belongs to chaotic attractor 24
Large Fluctuations Summary For a quasi-hyperbolic attractor, its non-hyperbolic part plays an essential role in the escape process. For a non-hyperbolic attractor, saddle cycles embedded in the attractor and basin of attraction are important. Escape from a non-hyperbolic attractor occurs in a sequence of jumps between saddle cycles. For both types of chaotic attractor we can select specific sets which are connected with Large Fluctuations and the most probable paths. Thus the description of large fluctuations is reduced to specification of a particular trajectory (the optimal path). 25