,
,
,
,
, with
a smooth (or topological) dynamical system, typically
generated by a differential or difference equation
or 
,
to a random differential equation
or random difference equation
.
Roughly speaking, a random dynamical system is a combination
of a measure-preserving dynamical system in the sense of ergodic theory,
,
,
,
,
, with
a smooth (or topological) dynamical system, typically
generated by a differential or difference equation
or ,
to a random differential equation
or random difference equation
.
Both components have been very well investigated separately. However,
a symbiosis of them leads to a new research program which has only partly
been carried out. As we will see, it also leads to new problems
which do not emerge if one only looks at ergodic theory and smooth
or topological dynamics separately.
From a dynamical systems point of view this book just deals with those dynamical systems that have a measure-preserving dynamical system as a factor (or, the other way around, are extensions of such a factor). As there is an invariant measure on the factor, ergodic theory is always involved.
Our book is a ``continuation'' of that by Guckenheimer and Holmes
[6]on
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of
Vector Fields. In their own words (Preface, page xi), their
book ``should be seen as an attempt to extend the work
of Andronov et al. (i.e. the analysis of a single degree
of freedom nonlinear oscillator, L. A.) by one dimension
(i.e. by adding a small periodic forcing term, L. A.)''. Specifically,
they look at certain equations of the form
in
where
is periodic. We will go further and beyond the periodic ``noise'' to a
general measure-preserving dynamical system to which the ordinary
differential equation is coupled. In yet other words, we take
the step from autonomous systems
to
nonautonomous systems, but of the special kind
,
i.e. to those which are coupled to a dynamical ``bath''.
If the flow
in the equation
is a flow of homeomorphisms of a compact space we are in the realm
of skew-product flows in the sense of Sacker, Sell and Johnson
(see e.g. [9],
[16],
[17],
[18]).
We go beyond this again by stripping off all the topology from
,
and instead adding an invariant measure - shortly, by going
from ``almost periodic'' to ``random''.
We also extend and generalize Mañé's book
[14]
on Ergodic Theory and Differentiable Dynamics.
He has a measure
invariant with respect to the flow
of a deterministic vector field
on a manifold M. Here, we have a measure
on
with marginal
on
invariant with respect to the skew-product flow
,
where
is the solution flow generated by the random vector field
.
From a probabilistic point of view this book offers another look at the quite classical subject of random difference equations and of random and stochastic differential equations, i.e. ordinary differential equations driven by real or white noise.
During the last 20 to 30 years an impressive structure called ``stochastic analysis'' has been erected, part of which is a theory of differential equations with semimartingale (rather than only Gaussian white noise, or Wiener) driving processes, providing us with a unified theory of random and stochastic differential equations.
Around 1980 it was discovered by Elworthy, Baxendale, Bismut, Ikeda, Watanabe, Kunita and others (see e.g. [5], [2], [3], [8], [12]) that a stochastic differential equation generates ``for free'' a much richer structure than just a family of stochastic processes, each solving the stochastic differential equation for a given initial value. It gives us in fact a flow of random diffeomorphisms. We can now bridge the gap between stochastic analysis and dynamical systems by proving that a random or stochastic differential equation generates a random dynamical system.
This makes it possible to re-evaluate and improve all the classical results (which are based on one-point motions and Markov transition probabilities) on stochastic stability, existence of invariant measures, etc. by Kushner [13], Khasminskii [10], Bunke [4] and many others. In [1] I have described the extension of the horizon when going from Markov processes to stochastic flows and cocycles.
The present book also adds a new chapter to the volume by Horsthemke and Lefever [7] entitled Noise-Induced Transitions and re-interprets their findings: Their noise-induced transitions are nothing but bifurcations on the static level of the Fokker-Planck equation. We will also study bifurcation scenarios on the dynamic level.
The book closest to ours in spirit and content is the one by Kifer [11] on Ergodic Theory of Random Transformations. He, however, deals exclusively with the i.i.d. case, i.e. with the case of iterations of random mappings chosen independently with identical distribution. In this case the orbits in state space form a Markov chain. We go beyond that by allowing a stationary stochastic sequence of mappings to be iterated, keeping the i.i.d. case as an important particular case.
It is a characteristic feature of the theory of random dynamical systems that every problem involves some ergodic theory and ergodic theorems. The most crucial and most important ergodic theorem applies to the linearization of smooth random dynamical systems. It is traditionally called the Multiplicative Ergodic Theorem and was proved by Oseledets [15] in 1968. This theorem provides a random substitute of linear algebra and hence makes a local theory of smooth random dynamical systems possible. Without it the whole field (in particular this book) would not exist.
Structure of the Book
As this is the first monograph on random dynamical systems, my main intention is foundational. This forces me to adopt a systematic, maybe sometimes even pedantic, style, and put my emphasis on theory rather than applications. I hope nevertheless to present a useful, reliable, and rather complete source of reference which lays the foundations for future work and applications.
Part I (Random Dynamical Systems and Their Generators) introduces the subject matter, settles the subtle perfection question, develops the theory of invariant measures (Chap. 1) and gives a (hopefully ultimate) treatment of the problem of which random dynamical systems have infinitesimal generators (Chap. 2).
Part II (Multiplicative Ergodic Theory) is the heart of the book. I
first present and prove the classical Multiplicative Ergodic Theorem for
products of random matrices in 
(Chap. 3), then present its
various modifications and the concept of random norms which turns out to
be basic (Chap. 4). In Chap. 5 the multiplicative ergodic
theory of related linear random dynamical systems obtained by
taking the inverse, the adjoint, and exterior and tensor products is
studied. The same is done with the systems
induced by a linear random dynamical system on
the unit sphere, the projective space, and Grassmannian manifolds,
culminating in the Furstenberg-Khasminskii formulas for Lyapunov
exponents. Finally, a multiplicative ergodic theorem for rotation numbers
is proved (Chap. 6).
Part III (Smooth Random Dynamical Systems) addresses the three most fundamental problems regarding nonlinear systems. The first one is the construction of invariant manifolds (Chap. 7). We adopt the new method of Wanner [19] which provides a unified approach towards invariant manifolds and the Hartman-Grobman theorem. The second basic problem is the simplification of a random dynamical system by means of a smooth coordinate transformation (normal form problem) (Chap. 8). I finally present the state of the art of random bifurcation theory (Chap. 9) which is still in its infancy and is not much more than a collection of (numerical) examples.
Part IV (Appendices) collects some facts from measurable dynamics
(Appendix A) and smooth dynamics (Appendix B).