Lyapunov Exponents and Chaos for Infinite Dimensional Random …math0.bnu.edu.cn/probab/Workshop2016/Talks/LuKN.pdf · 2016-07-17 · for Infinite Dimensional Random Dynamical Systems

1

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

2

Content

Random Dynamical Systems

Basic Questions

Lyapunov Exponents

Brief History

Main Results

Applications

Lyapunov Exponets and Entropy

Entropy and Horseshoe

3

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).

4



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

5


6

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)

7


8



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).

9

{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

10



• 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

11

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

12

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?

13

2. Basic Problems

2. Basic Problems

1. Stability

2. Sensitive dependence of initial data.

14

3. Lyapunov Exponents

yapunov Exponents

品 Deterministic Dynamical Systems

<> Stationary solution.

生 =f(x)dt

Eigenvalues '-

-

E

』

.. ..

• A = f'(O).

Eigenvectors

15



品 Deterministic Dynamical Systems

。 Periodic orbits.

品 Floquet Theory:

x' = A(t)x = f'(p(t))X

Floquet exponents x=p(t).

Floquet spaces

16


品 Sta bi I ity:

• Orbits:


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) →?

17


• 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.

18



• 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

19

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.

20

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

21

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

22

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.

23

4. Brief History

~ Infinite Dimensional RDS

<> Mane, 1983


4. Brief History

• S(ω) : X • X is a compact linear operator

• Q is a compact topological space ,

• () : Q •Ç2 is a homeomporphism.


24

4. Brief History


<> Thieullen , 1987


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.


25

4. Brief History

4. Brief History


<) 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 .


26

4. Brief History


<) 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 .


27

4. Brief History

Difficulties: Random Dynamical Systems No topological structure of the base space Banach Space No inner product structure

28

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).

29

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)

30

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.

31

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

32

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

33

6. Application

Ut == ~u+b(8tw ， x)u+uW， 0 < x < 1 , t > 0

with

the Dirichlet

or Neumann boundary conditions.

34

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) .

35

• Positive Lyapunov Exponents

Local Instability

(Sensitive dependence of initial data.)

Question?

• Positive Lyapunov Exponents

Chaotic Behavior

36

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)}

37



。 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

38

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.

39



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)

40



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.

41



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.

42



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;

43



~ 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.

44



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.

45


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}

46



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.

47



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.

48



,. 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)

49

Lyapunov Exponents and Chaos for Infinite Dimensional Random …math0.bnu.edu.cn/probab/Workshop2016/Talks/LuKN.pdf · 2016-07-17 · for Infinite Dimensional Random Dynamical Systems

Documents