Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and B. Eckmann
6811
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Proceedings of a Workshop
held ni Bremen, November 12-15, 1984
Edited by .L Arnold and .V Wihstutz
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Editors
Ludwig Arnold
Volker Wihstutz
Forschungsschwerpunkt Dynamische Systeme, Universit~t Bremen
BibliothekstraBe, 2800 Bremen 33, Federal Republic of Germany
Mathematics Subject Classification (1980): 34F, 60H, 93E; 28D, 58F, ?0L, 82A
ISBN 3-540-16458-8 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-38?-16458-8 Springer-Verlag New York Heidelberg Berlin Tokyo
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PREFACE
Lyapunov exponents have recently emerged as an important concept in
various fields such as
)i( products of random matrices and random maps,
(ii) linear stochastic systems, stability theory,
(iii) random Schr~dinger operators, wave propagation in random media,
(iv) nonlinear stochastic systems, stochastic flows on manifolds,
)v( chaos and phase transitions.
This volume contains, besides an introductory survey, 22 invited
papers of a workshop held at the University of Bremen in November 1984.
Several of those papers are survey articles. The workshop brought
together, for the first time, many of the leading specialists in the
above areas under the leitmotif of Lyaponov exponents.
We gratefully acknowledge the generous financial support of the
workshop by the UniverSity of Bremen.
August 1985 Ludwig Arnold, Volker Wihstutz
Forschungsschwerpunkt Dynamische Systeme
University of Bremen
TABLE OF CONTENTS
Preface
L. Arnold, V. Wihstutz: Lyapunov exponents: a survey
Part I: Products of random matrices and random maps
P. Bougerol: Oscillation de produits de matrices al~atoires
dont l'exposant de Lyapunov est nul 27
P. Walters: Unique ergodicity and random matrix products 37
F. Ledrappier: Positivity of the exponent for stationary
sequences of matrices 56
Y. Kifer: Characteristic exponents for random homeomorphisms
of metric spaces 74
Part II: Linear stochastic systems. Stability theory
L. Arnold, W. Kliemann, E. Oeljeklaus: Lyapunov exponents
of linear stochastic systems 85
ADDENDUM by L. San Martin: A note on transitivity of
coupled control systems 126
L. Arnold, E. Oeljeklaus, E. Pardoux: Almost sure and moment
stability for linear It~ equations 129
K. A. Loparo, G. L. Blankenship: Almost sure instability of
a class of linear stochastic systems with jump process
coefficients 160
O. B~hme: On a generalization of the Lyapunov exponent of a
second order linear stochastic differential equation 191
V. Wihstutz: Parameter dependence of the Lyapunov exponent
for linear stochastic systems. A survey 200
R. A. Johnson: An example concerning the geometric signi-
ficance of the rotation number. Integrated density of
states 216
VJ
Part III: Random Schr6din~er operators. Wave propa@ation in
random media
W. Kirsch: Lyapunov exponents for Schr~dinger operators
with random, but deterministic potentials 227
H. Englisch: Lyapunov exponents and one-dimensional alloys 242
F. Bentosela: Resonance states in disordered systems 246
W. Craig: The Lyapunov index, the density of states and
their regularity for general stochastic potentials 252
J. Lacroix: Computation of the sum of positive Lyapunov
exponents for the Lloyd model in a strip 258
.R Burridge, G. Papanicolaou, B. White:
Statistics of reflected pulses 265
Part IV: Nonlinear stochastic systems. Stochastic flows on
manifolds
H. Crauel: Lyapunov exponents and invariant measures of
stochastic systems on manifolds 271
A. Carverhill: Lyapunov exponents for stochastic flows on
manifolds 292
M. J. Chappell: Bounds for average Lyapunov exponents of
gradient stochastic systems 308
P. H. Baxendale: The Lyapunov spectrum of a stochastic flow
of diffeomorphisms 322
Part V: Chaos and phase transitions
F. Argoul, A. Arneodo: Lyapunov exponents and phase
transitions in dynamical systems 338
H. Daido: Coupling sensitivity and cliff in chaotic dynamics 361
Complete list of papers presented at the workshop 369
Subject Index 371
LYAPUNOV EXPONENTS: A SURVEY
L. Arnold and V. Wihstutz
Forschungsschwerpunkt Dynamische Systeme, Universit~t,
D-28OO Bremen 33, West Germany
Contents
Chapter :I History and classical results
.I The work of Lyapunov and its continuation
.2 The general notion of Lyapunov exponents
.3 Oseledec's multiplicative ergodic theorem
.4 Furstenberg's work on products of random matrices
.5 Has'minski~'s work on the stability of linear stochastic systems
Chapter II: Directions of further development
.I Products of random matrices and random maps
2. Linear stochastic systems. Stability theory
.3 Random Schr~dinger operators. Wave propagation in random media
.4 Nonlinear stochastic systems. StoGhastic flows on manifolds
.5 Smooth ergodic theory, iteration of maps, chaos and phase
transitions
References
Chapter I: History and classical results
In this chapter we review the pioneering work of Lyapunov (1892),
Furstenberg (1963), Has~inski~ (1966) and Oseledec (1968), which we
consider classical.
.1 The work of Lyapunov and its continuation
In 1892, A. M. Lyapunov published his famous monograph "Probl~me
g~n~ral de la stabilit~ du mouvement" [ 25 ] in which he founded the
theory of characteristic exponents that today bear his name.
The original intention of Lyapunov was to determine criteria for the
stability (of the origin x ~ o) of
= A(t) x, x(O;Xo) O = 6 x R d, t 6 R +, )I(
A: + ~ R R (d'd) (= dxd matrices) continuous and bounded, and to conclude
from the stability of )I( the stability (of x ~ o) for the nonlinear
system
= A(t)x + f(t,x), )2(
f £ C °'I (R+,R+xR d) and If(t,x) i ~ Cixl I+~ for some C,~ > 0 .
Everybody knows that for constant A the eigenvalues of A determine
the stability behavior of (I), and if )I( is exponentially stable then
so is (2).
For periodic A(t) Floquet theory tells us that the results for
constant A remain true if the real parts of eigenvalues are replaced
by the characteristic exponents of A(t) (cf. Coddington and Levinson
[ 5 ,] chapters 3 and 13).
The Lyapunov exponent of a solution x(t;x o) of (I) is defined by
I
l(Xo) := lim ~ loglx(t;Xo) L
=~-t
Lyapunov proved that for every solution with O % x 0 l(x O) is finite.
Moreover, the set of all possible numbers which are Lyapunov exponents
of some nonzero solution of (I) is finite, with cardinality p ,
1~p~d,
Ip < ... < 11
Further, l(cx o) = l(x O) for c • 0 , and
~(ClX1+c2X2 ) ~ max(~(x]),l(x2)) ,
with equality if l(Xl) # l(x2)
Look at the subspaces
i : L {Xo6Rd: A(x o) ~ Ai }
They form a filtration of d R ,
O = Lp+ I c Lp c ... c I d = L R
with dim i = i L k , kp+ I = O < kp < ... < I = k d , and
l(Xo) = i I iff O x 6 i L ~ Li+ 1 , i = 1,...,p .
A fundamental system of solutions of (I) (identified with its initial
values Xl,...,Xd) is said to be normal if for all Cl,...,c d 6 R
d
X( I c x.] = max l(xi)
\i=I 1 i/ i:c.%o
1
Lyapunov proved that a normal basis always exists. Denote by i d the
multiplicity of Ai among the numbers l(x i) for a normal basis. Then
p
i d is the same for any normal basis, and i d = i k - ki+1, Z d : d .
I i
The numbers i X together with their multiplicities i d are called the
Lyapun__ov spectrum of (I). Of course, (I) is exponentially stable iff
I X < 0 .
It is in general not true that I X < O implies the stability of (2).
For this one needs a property of (I) which Lyapunov called regular. (I)
is said to be forward regular if
P I I t
d,l. = lira [ log det ¢(t) = lira ~ S trace A(s)ds , (3)
i=I i ± t_~o t-~o o
where @(t) is the fundamental matrix of (I) with 0(O) = id. For a
forward regular system, all lim are actually limits, and 11 < 0
implies exponential stability of (2).
Assume now that (I) is defined on the whole time axis R ,
= A(t) x, t6R, A:R ~ R (d'd) bounded and continuous. (I)'
We can define backward exponents by
l-(Xo) =: ~ 1 loglx(t;Xo) I
t~-~ I~i
There will be p , I < p- < d , exponents
l-_< P ... <k 1
with multiplicities d7 for AT and a filtration
1 i t
0 : Lp_+1 c Lp-c ... c 1 L = d R
wi th
l-(x )o = ki iff ° x 6 L~ ~ Li+ ] , i : I ..... p
We call (1)' backward regular if
P I I t
ZI=i d~?l i = mil~-~t I~I log det ~(t) = mil~-~t I~I So trace A(s)ds. (3)'
In general, the forward and backward spectrum of (I)' are not related.
We introduce such a relation as follows: The system (I)' is called
regular, if
)i( it is forward and backward regular,
(ii) p = P' di = dp+1-i'li = -lp+l-i' i=1,...p,
(iii) Li+ I N Lp+I_ i = {O}, i=I ..... p-1.
For a regular system the subspaces i = i E N L Lp+1_ i, i = 1,...,p
have dim i i = E d and form a splitting of d R with
P P
L. = @ E. , i=1,...,p, d = R @ E ,
l j=i 3 j=1 3
and
I
lim ~ loglx(t;Xo) I = I. iff x 6 E.~{O}.
t~+~ l o 1
For example, (I)' is regular if A is constant or periodic. In the
latter case the i I are the characteristic exponents.
Although regularity might be hard to verify for a particular system,
we know that it happens with probability one in many cases involving a
flow with an invariant probability measure. This is the key statement
of Oseledec's famous multiplicative ergodic theorem (see section 1.3).
From the Russian school which has been carrying on Lyapunov's
tradition let us just mention the profound monograph by Bylov, Vinograd,
Grobman and Nemycki~ [ 2 ] and the work of Mllllons• cikov , VV (see e.g. [26 ]).
An offspring of the Russian school is the group in Hanoi (Vietnam)
around H. H. Duong which was represented at the Workshop by H. H. Duong,
N. T. Hoan and T. V. Nhung (see Duong [ 9 ,] Hoan [ 19]). Duong
works with the concept of a characteristic vector l(m) (x o) = (~o,...,~m)
of a solution x(t;Xo) of (I), m ~ o , defined by
" I
~o(Xo) := lim ~ log Ix(t;Xo) i ,
t-~=
-~ t
I~( (Xo) := lti-m~ ~ I log(Ix(t;Xo) le o ) ,
-~2 -~m-1
~m(Xo):= lim I_____ log(ix(t;Xo) le-~Ott-~1(log )t ...(log m_2 t) ),
t-~o l°gm t
where logot = t , logjt = log(logj_it) , j=1,...,m. Again, for any
m ~ O , system (I) has at most d different characteristic vectors,
which can be ordered lexicographically,
)m( < . < ~ )m(
p "" ~I "
and "tA (m) < O is sufficient for asymptotic stability of )I( . This concept
is particularly useful for concluding stability of (2) from that of )I(
in critical cases (where I A . )m( = (O,. .. ,O,c) with c < O)
2. The general notion of Lyapunov exponent s
The fundamental matrix ~(t,A) of (I)' is a Gl(d,R)-valued function
which satisfies the relation
~(t+s,A) = ~(t,FsA)@(s,A), all t,s 6 R . )4(
(4) says that ¢ is a (multiplicative) cocycle associated with the
dynamical system consisting of the group of shifts Ft, t 6 R , in the
space of bounded continuous functions A: R ~ R (d'd) defined by
FtA(s) = A(s+t), s,t 6 R .
Lyapunov exponents as defined in section I.I thus just describe the
asymptotic behavior of a certain cocycle at a point A. We will formalize
and generalize this.
Let (X,F) be a measurable space, and let
Ft, t 6 T , T = Z or R ,
be a measurable flow (i.e. a group of hi-measurable bijeetions of X to
itself such that for each measurable f: X ~ R f(Ftx) is measurable in
(t,x)). We now attach to this flow a co~cle C(t,x) which is a
