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Lyapunov Exponents and Chaos
for Infinite Dimensional Random Dynamical Systems
.
12th Workshop on Markov Processes and Related Topics July 13-17, 2016 徐州师范大学
Kening Lu, BYU, 四川大学 Collaborators: Zeng Lian, 四川大学 Wen Huang,四川大学, 科大
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Content
Random Dynamical Systems
Basic Questions
Lyapunov Exponents
Brief History
Main Results
Applications
Lyapunov Exponets and Entropy
Entropy and Horseshoe
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1. Random Dynamical Systems
ndom Dynamical Systems
Example 1. Quasiperiodic ODE
。 =μ?
土 = f(() ,x) ,
where ()ε r_rrmμεIRrr飞 zε IRn , f is nonlinear.
Let ÇL = r_rrm , ()tω=μ+ω , JtD be the Haar measure on ÇL , cþ(t , ω)(xo) be the solution of
x' ( t) = f ( ()tω , x). Then
• (r2, β , JIÞ) a probability space , ()t preserves JIÞ
。(0 , ω) == Id • cþ(t + s , ω)== 功。, Ðsω) 0 cþ(s , ω).
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1. Random Dynamical Systems
1. Random Dynamical Systems
Example 2. Stochastic Differential Equations
dXt = fo仙词+艺 fk(Xt)dW
where
• x 巳 :ræn , fk are smooth ,
• Wt == (Wt1 , … ,Wt
k ) is a Brownian motion
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1. Random Dynamical Systems
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1. Random Dynamical Systems 1. Random Dynamical Systems
Example 3. Random PDEs
Ut == A(()tω)U + f(认 ()tω)
where u εX, a Banach space and ()t is a mea
surable dyna门1ical syste门lS over a probabilty
space (Q, F , JP).
Random dyna 门1ical system: solution operators
价(t , ω , uo) == u(t , 队 uo)
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1. Random Dynamical Systems
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1. Random Dynamical Systems
1. Random Dynamical Systems
A map <þ:ffi. xQxX • X , ( t , ω , x) ←→ <þ (t , w , x)
is called a random dynamical system over ()t if
(i) <þ is measurable;
(ii) cþ(t , ω) := cþ(t , 叭.) form a cocycle over ()t
<þ (O , ω) = Id , <þ(t + s , ω)= 功。, () sω)ocþ(s , w).
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{ω} x X {8sω} x X
cþ(s , ω)
二C
功。 + s , ω)
w 。sω
{ßt+sω} x X
cþ (t , 8sω) cþ( S , ω)x
= cþ(t + s , ω)x
。t8sω = 8t十sω
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1. Random Dynamical Systems
1. Random Dynamical Systems
• Time one map: ψ(灿 x) := </>(1 , ω , x)
• Random map: ψ(ω , x) generates RDS:
。(η? 叭.) = ψ(eη-1ω , .) 0 … oψ(ωγ) , η>0
1 , η=0
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2. Basic Problems
天气预报模型
当前气象数据
气象数据测量
气温, 气压,风力,风向,雨量等
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2. Basic Problems
Mathematical Model
丝丝 =f(x)dt
Let x(t , Xo) be the state at time t.
Computational Model
2=fω+物)Let y(t , YO) be the observed data.
2. Basic Problems
Question 2. Does y(t , υ0) stay close to x(t , xo)?
Can we trust what we see?
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2. Basic Problems
2. Basic Problems
1. Stability
2. Sensitive dependence of initial data.
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3. Lyapunov Exponents
yapunov Exponents
品 Deterministic Dynamical Systems
<> Stationary solution.
生 =f(x)dt
Eigenvalues '-
-
E
』
.. ..
• A = f'(O).
Eigenvectors
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3. Lyapunov Exponents
3. Lyapunov Exponents
品 Deterministic Dynamical Systems
。 Periodic orbits.
品 Floquet Theory:
x' = A(t)x = f'(p(t))X
Floquet exponents x=p(t).
Floquet spaces
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3. Lyapunov Exponents
品 Sta bi I ity:
• Orbits:
3. Lyapunov Exponents
cþ(η? 叫♂)=ψ(()n-l叭.) 0 … oψ(ω , x)
cþ(凡叭 y) = ψ(()n-lω , .) 0 … oψ(叭 ω
Question: Sensitivity in initial data?
功(η7ω , y) - cþ(η?ω , x) →?
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3. Lyapunov Exponents
• Linearized Systems:
DxCÞ(凡叭 x) = Dxψ(()n-lω , .) 0 … o Dxc.p(ω , x)
Example: ψ(队 x) = Ax
Eigenvalues and eigenvectors
Lyapunov Exponents
门leasure the average rate of separation of orbits starting from nearby initial points.
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3. Lyapunov Exponents
3. Lyapunov Exponents
• The Linear random dynamical system generated by
φ(凡 ω)==S(。η-1ω) … S(ω) , n > 0
1, η== o.
• Basic Problem: Find a 川 Lyapunov exponents
ι且川
飞八一
一ι扎
υ
ω
η
φ
qd o --n
mb H• n
Multiplicative Erogdic Theorem
Existence of Lyapunov Exponents and the associated invariant subspaces
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4. Brief History
Finite Dimensional Dynamical Systems
• Lyapunov,1900s, stability of orbits
• Furstenburg & Kesten,1960,limit properties of product of
random matrices
• 廖山涛, 1963, Lyapunov exponents, smooth vector fields
• V. Oseledets, 1968 (31 pages) , Multiplicative Ergodic Theorem
Existence of Lyapunov exponents,Invarant subspaces,.
Different Proofs:
Millionshchikov; Palmer, Johnson, & Sell; Margulis; Kingman; Raghunathan;
Ruelle; Mane; Crauel; Ledrappier; Cohen, Kesten, & Newman; Others.
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4. Brief History
Applications:
Deterministic Dynamical Systems
Pesin Theory, 1974, 1976, 1977
Nonuniform hyperbolicity
Entropy formula, chaotic dynamics
Random Dynamical Systems
Ruelle inequality, chaotic dynamics
Entropy Formula, Dimension Formula
Ruelle, Ladrappia, L-S. Young, …
Smooth conjugacy
W. Li and K. Lu
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4. Brief History
4. Brief History
4t Infinite Dimensional RDS
<> Ruelle , 1982 (Annals of Math)
Random Dynamical Systems in a Separable Hilbert Space.
• X is a Hilbert space ,
• S(ω) : X • X is a compact linear operator
• Q is a probabilty space ,
• () : Q • Q is a measurable metric DS.
Multiplicative Ergodic Theorem
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4. Brief History
4. Brief History
品 Basic Problem:
Establish Multiplicative Ergodic Theorem for RDS
• X is a Ba nach space ,
• Q is a proba bi Ity space ,
• () : Ç2• Q is a measurable metric DS.
Banach space such as
LP , C O, W 1 ,p.
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4. Brief History
~ Infinite Dimensional RDS
<> Mane, 1983
• X is a Ba nach space ,
4. Brief History
• S(ω) : X • X is a compact linear operator
• Q is a compact topological space ,
• () : Q •Ç2 is a homeomporphism.
Multiplicative Ergodic Theorem
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4. Brief History
4t Infinite Dimensional RDS
<> Thieullen , 1987
• X is a Ba nach space ,
4. Brief History
• S(ω) : X • X is a bou 门ded linear operator
• Q is a separable topological space ,
• () : Q • Q is a homeomporphism.
Multiplicative Ergodic Theorem
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4. Brief History
4. Brief History
4t Infinite Dimensional RDS
<) Flandoli and Schaumlffel , 1991
• X is a Hilbert space ,
• S(ω) : X • X is invertible (isomorphism)
门laps closed subspaces to closed subspaces
• Q is a probability space
• () : Q • Q is a measurable metric DS .
Multiplicative Ergodic Theorem
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4. Brief History
4t Infinite Dimensional RDS
<) Schaumlffel , 1991
• X is a Banach space ,门 convex"
• S(ω) : X • X is invertible (isomorphism)
4. Brief History
门laps closed subspaces to closed subspaces
• Q is a probability space
• () : Q • Q is a measurable metric DS .
Multiplicative Ergodic Theorem
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4. Brief History
Difficulties: Random Dynamical Systems No topological structure of the base space Banach Space No inner product structure
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5. Main Results
:
ain Results
4t Settings and Assumptions
<> X - Separable Banach Space
。 Measurable metric dynamical system
(Q ,:F, P, e)
<> S(ω) : X • X is strongly measurable.
J S( ()n-lω) … S(ω) , n > 0 φ(饥?ω) = );
η== o.
log+ IIS(.) 川 ε L 1 (Q , F , P).
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5. Main Results
5. Main Results
命 Principal Lyapunov Exponent
κ(φ)(ω)=|iml|09||φ(凡 ω) 11
n一→寸-CX) n
~ Exponent of Noncompactness
α
ω
φ
Od o --n
∞
m+ n 一
一ω
α
lα(ω) κ(φ)(ω)
When LRDS is compact
lα(ω)= 一。。
T1== 巳lα
γ2= eκ(φ)
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5. Main Results
Theorem A (Lian and L 2010)
Assume that 叫φ)(ω) 共 lα(ω)
Then , (:3 Ð-invariant subset of full measure)
:3 countably 门lany Lyapunov exponents
入lCω) >…>入k(ω)Cω) >… >lα(ω)
and invariant splitting
X = El(ω) E9…⑦ Ek(忡ì) (ω) E9 F(ω)
5. Main Results
where Ei(ω) is finite dimensional and F(ω) is
infinite dimensiona l.
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5. Main Results
such that
(1) Invaria nce: 入i (()ω)=λ(ω)
S(ω)Ej(ω) 二 Ej(()ω)
S(ω)F(ω) C F(()ω)
(2) Lyapunov exponents:
for a 川
Hmf|09||φ(凡 ω)vll =λj(ω) 一-于 1二 cx二 n
uε Ej(ω) , 1 三 j 三 k
5. Main Results
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5. Main Results
(3) Exponential decay rate in F(ω) :
|imsup:|09||φ(川)1削 11 三 lα(ω)n一→-卜CX) I (J
5. Main Results
and ifvε F(ω) and (φ(η , ()-nω))-lV exists
then
|im Mfl|09||φ(一η?ω)川|三 -lα(ω)1ft一→十∞ η
(4) Measurabilities
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6. Application
Ut == ~u+b(8tw , x)u+uW, 0 < x < 1 , t > 0
with
the Dirichlet
or Neumann boundary conditions.
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6. Application 6. Application
Theorem. There exists a 8t -invariant set n c n of full P measure such that for each ωεQ
(i) Existence of infinitely many Lyapunov ex
ponents;
(ii) The associated Oseledets space E认ω) is
one-dimensional given by a stationary pro
cess:
。ii) L 2 (O , 1) ==⑦ρlEη(ω) .
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• Positive Lyapunov Exponents
Local Instability
(Sensitive dependence of initial data.)
Question?
• Positive Lyapunov Exponents
Chaotic Behavior
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7. Lyapunov Exponent and Entropy
apunov Exponent and Entropy
。 Measure-theoretic (metric) entropy.
Kolmogorov (1950'吟, Sinai
It measures the rate of increase in dynamical complexity as the system evolves with time.
。 Topological entropy.
Adler, Konheim and McAndrew (1965)
Bowen (1971) , Dinaburg (1970).
It measures the exponential growth rate of the number of distinguishable orbits as tir丁le advances.
。 Variational Principle.
htop = sup{hμ:μξ Pf(X)}
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7. Lyapunov Exponent and Entropy
7. Lyapunov Exponent and Entropy
。 Pesin Entropy Formula:
hμ(f) == I 汇入4 dimE4 dμυ 入i> O
Entropy = Sum of Positive Lyapunov Exponents
The Pesin formula holds if and only ifμis a
SRB 门leasu re.
Ledrappier and Young , Li and Shu
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8. Entropy and Horseshoe
ntropy and Horseshoe
d属 Problem:
What is the implication of positive entropy of a dyna 门1ical syste门17
• Sinai , 1964
An ergodic measure-preserving map T on a probability space (X; 只 μ).
If its 门leasure theoretic entropy is positive, then
T contains a factor which is semi-conjugate to
a shift map.
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8. Entropy and Horseshoe
8. Entropy and Horseshoe
品 Finite Dimenional Dynamical Systems
Theorem. [Katok , Pub. Math. IHES, 1980]
Let M - 2D , compact , C∞ Riemannian manifold ,
f ε Diffl+α(M)
If htop(f) > 0 , then :3 k εN and a closed fk_
invariant set r such that fkl , has Horseshoe
of two symbols.
- k 0 k 2k 3k 4k
1 1 2 1 2 2
bi-infinite sequence of 2 symbols
f-k(X) Ul U2
fk(x)
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8. Entropy and Horseshoe
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Re阿lark.
• If f E三 Diffl(M) or M is high-dimension ,
then Katok's result is not true.
• If M is 2D and htop(f) > 0 , then f is hyperbolic.
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8. Entropy and Horseshoe
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Without assuming any hyperbolicity, Glasner,
Kolyada , and Maass showed that
Theorem. [BGKM , J. Reine Angew. Math. 2002]
Let X be a compact metric space and
T:X • X be a homeomorphism.
If htop(T) > 0 , then (X , T) is chaotic in the sense of Li-Yorke.
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8. Entropy and Horseshoe
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Definition of Chaos. (Li-Yorke , 1975)
3κ> 0 , E C X ,vhich is a union of countably nlan}T Cantor sets,
such that for every pair Xl , X2 of distinct points in E , we have
I i m .i n f d ( cþ (凡 ω)(Xl) , cþ(凡 ω)(X2)) == 0 , n一今→-00
lim sup d( cþ (η?ω)(X l) , cþ (η7ω)(X2)) 主 κn→+∞
More some results about positive entropy and chaos: See S.Numi ,ETDS 2003; W.Huang and X.Ye, ISR 2006; W.Huang , CMP 2008;
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8. Entropy and Horseshoe
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~ Dynamical Systems in Hilbert Spaces
Theorem . [Lian & Young , Ann. Henri Poincaré , 2011]
o f -C2 differentiable maps in a Hilbert space.
。 f has a nonuniformly hyperbolic compact invariant set supported by an invariant 仔1easure.
丁hen 曹 Katok's result holds.
Theorem. [Lian & Young , JAMS , 2012]
o ft - C 2 semiflow in a Hilbert space.
。 ft has a nonuniformly hyperbolic compact invariant set supported by an invariant measure.
丁hen, the positive entropy implies the existence of horseshoes.
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8. Entropy and Horseshoe
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4- Problem:
How to characterize the chaotic behavior of orbits topologically or geometrically (in terms of
horseshoe) in the presence of ONLY positive entropy?
without assuming any hyperbolic structures.
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8. Entropy and Horseshoe
d属 Setting:
。 Infinite Dimensional RDS
功(η7ω , x) , η>0
<) Random Invariant Set
A c Q x X , meαsurαble
。(凡 ω)A(ω) == (A(enω)) p 一 α. s. ,
ωhere A(ω) == {x εX(灿 x) ε A}
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8. Entropy and Horseshoe
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Theorem (H ua吨 and L)
Let A(ω) be a compact random invariant set
If the topological entropy is positive, i.e. ,
htop飞中、 λl) > 0 、
then,
(1) the dynamics of cþ restricted to A is chaotic;
(2) the dynamics of (队 A) has a weak horseshoe.
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8. Entropy and Horseshoe
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Corollary. (Hua吨 and L)
Let A be a global attractor of deterministic PDEs.
生 = Au + F(t ,u) dt
If the topological entropy is positive, i.e. j
h top ( u , A) > 0 ,
then, (认 A) has a full horseshoe.
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,. A Horseshoe with two symbols:
::1 an infinite subsequence of integers with positive
density in N: 0 < nl < γl也…,< nk < …
such that for any infinite seque丑ce of 2 symbols nl n2 n3 … nk
2 1 2 ... 1
X
Ul hMU
hMU
hMU
。 n2 nk
U2
nl t n3
fηl(X) fη3 (X)
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