1
Large
Large Fluctuations
Fluctuations in
in
Chaotic
Chaotic Systems
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
2
Large
Large Fluctuations
Fluctuations in
in Chaotic
Chaotic Systems
Systems
Outline
● Large Fluctuational Approach and Model Reduction
● Escape in quasi-hyperbolic systems
● Escape in non-hyperbolic systems
● Conclusions
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Deterministic
Deterministic Chaos
Chaos and
and Noise:
Noise: Environment
Environment
Environment induces
Dissipation and Fluctuations
H = H S + H B + H SB
HS
System Hamiltonian
HB
Bath (Environmental) Hamiltonian
H SB
Hamiltonian of interaction
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
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Deterministic
Deterministic Chaos
Chaos and
and Noise:
Noise: Environment
Environment
Archetypical Example:
Example Environment as a Collection of
Linear Oscillators
H = H S + H B + H SB
p2
HS =
+ V ( q; t )
2m
The system is
a model of a particle in potential
 pn2 mn 2 2  The collection of
HB = ∑
+
ωn xn 
2
n =1  2 mn
 harmonic oscillators
N
2
c
n
= −q ∑ cn xn + q 2 ∑
2
n =1
n =1 2mnωn
N
H SB
N
Elimination leads to
mq&& + 2γ mq& +
Damping (dissipation)
Bath
Environment
Linear coupling between system and
bath
Dissipation and Fluctuations
have the same origin
∂V
= ξ (t )
∂q
Fluctuations
noise
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Chaos
Chaos and
and Noise:
Noise: Large
Large Fluctuations
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
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Chaos
Chaos and
and Noise:
Noise: Large
Large Fluctuation
Fluctuation Approach
Approach
x& = K ( x, t ) + Qξ (t ),
The system described by
Langevin equations:
ξα = 0, ξ α (t )ξ β ( s ) = Q δ (t − s )
Transition probability via fluctuations paths
ρ (x f , t f | x i , ti ) = ∑ ρ[x(t ) j ] ≈ρ[x(t )opt ]
j
The selection of the most probable (optimal) path
Path
xf
Path
Path
x1
x2
xi
xj
Path x opt
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Deterministic
Deterministic chaos
chaos and
and noise:
noise: optimal
optimal path
path approach
approach
deterministic
deterministic pattern
pattern of
of fluctuations
fluctuations
x
f
x& = K (x, t ) + ξ (t ),
xi
ξα = 0, ξα (t )ξ β ( s) = DQδ (t − s)
ρ[x(t ) j ]
The probability of fluctuational path
the probability
ρ[ξ (t ) j ]
is related to
of random force to have a realization
ξ (t) j
 1 tf

 1 
2

For Gaussian noise: ρ [ξ (t ) j ] = C exp − ∫ ξ (t ) j dt  = C exp − S 
 2t

 2 
i


Since the exponential form, the most probable path has a minimal S=Smin
Changing to dynamical variables:
Action S = S [ξ (t )]
x& = K (x, t ) + ξ (t )
ξ (t ) = x& − K (x, t )
S = [x(t )]
[
 S x(t ) opt
In the limit D → 0, ρ (x f ; x i ) = ρ (x(t ) opt ) = Const × exp −
D

[
]
S min = S xopt (t ) = min ∫ dt (x& − K (x, t ) )
2
]
Deterministic
minimization problem


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Large
Large fluctuations
fluctuations and
and Model
Model reduction
reduction
The initial model: the Hamiltonian for
the system, the bath and coupling
In general case the dimension is infinite.
Langevin model reduction: finite
dimensional system with noise terms,
The dimension is infinite
H = H S + H B + H SB
x& = K (x, t ) + ξ (t ),
ξα = 0, ξα (t )ξ β ( s ) = 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
[
]
S min = S xopt (t ) = min ∫ dt (x& − K ( x, t ) )
f
2
xi
Initial state :
q ( ti ) = xi , p ( ti ) = 0, ti → −∞;
Final state :
q ( t f ) = x f , p ( t f ) = 0, t f → ∞.
Formally the deterministic minimization
problem can be formulated in the
Hamiltonian form:
1
H = pQp + pK (q, t );
2
∂H
∂H
q& =
, p& = −
,
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∂p
∂q
Optimal
Optimal path
path approach
approach
deterministic
deterministic pattern
pattern of
of fluctuations
fluctuations
Optimal paths are experimentally observable (Dykman’92)
The prehistory
probability of
transition
between states
of bistable
oscillator
(electronic
experiment)
ξ (t )
Optimal paths are essentially deterministic trajectories
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Quasi
Quasi-hyperbolic
Attractor
Lorenz system
σ = 10, b = 8 / 3, r = 24.08
q&1 = σ ( q2 − q1 )
Stable points
q&2 = rq1 − q2 − q1q3
q&3 = q1q2 − bq3 + 2 D ξ (t )
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
Chaotic
Attractor
Saddle cycles
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Chaos:
-hyperbolic Attractor
Chaos: Quasi
Quasi-hyperbolic
Attractor
Lorenz Attractor
r ≈ 13.92
Homoclinic
loop -> Horseshoe
L2
The saddle point and its
separatrices belong to
chaotic attractor and
form “bad set” or nonhyperbolic part of the
attractor
L1
Γ2
Γ1
Separatrices
Saddle point
r ≈ 24.06
Loops between separatrices Γ1 and Γ2 and stable
manifolds of cycles L1 and L2 generate
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The Lorenz attractor – quasi-hyperbolic attractor
Chaos:
-hyperbolic Attractor
Chaos: Quasi
Quasi-hyperbolic
Attractor
Stable
manifold WS
Unstable
manifold WU
Cycle L1
Separatrix Γ2
The loop between separatrices
Γ1 and Γ2 and stable manifolds
of cycles L1 and L2 persists by
varying parameters
Saddle point
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Large
Large Fluctuations:
Fluctuations: the
the prehistory
prehistory probability
probability
The prehistory approach
x& = K (x, t ) + ξ (t ),
ξα = 0, ξα (t )ξ β ( s ) = DQδ (t − s )
1. Select the regime D → 0
i.e. rare large fluctuations
t relax << t activ
2. Record all trajectories
xj(t) arrived to the final
state and build the
prehistory probability
density ph(x,t)
The maximum of the density
corresponds to the most
probable (optimal) path
3. Simultaneously noise
realizations ξ(t) are
collected and give us the
optimal fluctuational force
xf
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Chaos:
-hyperbolic Attractor
Chaos: Quasi
Quasi-hyperbolic
Attractor
The Poincaré section q3 = r - 1
“+” Separatrix
Stable manifold of
the saddle point
Chaotic
attractor
Stable point P1
Stable point P2
“o” Saddle cycle
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Quasi
Quasi-hyperbolic
Attractor
Probability density p(q1) for Poincaré section
q3 = r − 1
The difference
in the tail (low
probable part
of the density)
q1
q1
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?
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Quasi
Quasi-hyperbolic
Attractor
Escape from quasi-hyperbolic attractor
Escape trajectories
q3
WS is the stable manifold and
Γ1 and Γ2 are separatrices of the
saddle point O
L1 and L2 are saddle cycles
T1 and T2 are trajectories which
are tangent to WS
Noise-induced tail
q1
q2
The escape process is connected with
the non-hyperbolic structure of
attractor: stable and unstable
manifolds of the saddle point
The distribution of escape
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trajectories (exit distribution)
Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Quasi
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 xi
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
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
Non-hyperbolic
Attractor
Non-autonomous nonlinear oscillator
q&& + Γq& +
U ( q, t ) =
∂U (q, t )
= 2 Dξ (t )
∂q
ω02
2
q2 +
β
q3 +
γ
The potential U(q)
is monostable
q 4 + q h sin Ωt
3
4
Γ = 0.05 ω0 = 0.597 β = γ = 1
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
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
Non-hyperbolic
Attractor
The co-existence of chaotic attractor and the limit cycle
h = 0.13 Ω = 0.95
Initial state :
?
q ( ti ) = xi , p ( ti ) = 0,
q&
Stable cycle
Saddle cycle
ti → −∞;
Final state : The stable
cycle
q ( t f ) = x f , p ( t f ) = 0,
t f → ∞.
q&
Chaotic attractor
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
Non-hyperbolic
Attractor
Probability density p (q, q& ) for Poincaré section Ωt = 0
D = 5 ⋅10−6
D=0
q
q
q&
q&
A weak noise significantly changes the probability density
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
Non-hyperbolic
Attractor
The prehistory probability density ph (q, q& , t )
D = 5 ⋅10−4
There is a miximum in
the prehistory
probability density
q
q&
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
Non-hyperbolic
Attractor
Escape trajectories follow a narrow tube
Q: Do any
sets form the
escape path?
q&
Saddle cycle
To answer we
take initial
conditions along
the path and try to
localize any sets
time
T=
2π
Ω
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
Non-hyperbolic
Attractor
The prehistory probability density
Saddle cycles
form the
escape path
S5
ph (q, t )
Saddle cycle of period 5
Saddle cycle of period 3
S3
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Chaos
-hyperbolic Attractor
Chaos and
and Noise:
Noise: Non
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, t );
2
∂H
∂H
q& =
, p& = −
,
∂p
∂q
Initial state : cycle S5
q&
Saddle cycle of period 3
is outside the attractor
S3
q ( ti ) = xi , p ( ti ) = 0, ti → −∞;
Final state : cycle UC1
q ( t f ) = x f , p ( t f ) = 0, t f → ∞.
Escape
trajectory
S5
Saddle cycle of period 5
belongs to chaotic attractor
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Large
Large Fluctuations
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).
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