Ebook: Semi-hyperbolicity and bi-shadowing

Preface

Hyperbolicity is a very important concept in dynamical systems theory. It
has been extensively investigated over the past half century along with its
associated concepts of robustness of dynamical behavior [11, 13, 18, 19, 24{26,
37, 61, 62, 87, 92, 94, 102]. Many deep results have been obtained and there are
now numerous monographs and textbooks on the subject [1, 14, 20, 109, 114].

Most of the work on hyperbolicity, however, concerns abstract differentiable
dynamical systems and it is often very difficult to show that the results
apply to systems generated by specic differential equations such as the Lorenz
equations. In addition, in recent years, there have been many interesting and
useful generalizations and extensions of dynamical systems ideas to what were
previously `non-mainstream' applications arising, for example, in nonsmooth
systems, numerically generated systems and systems with hysteresis, to name
just a few. The existing theory of hyperbolicity does not apply directly to these
situations, but many of the associated results on robustness and shadowing
are nevertheless very important and useful for them too.

Since we were not able to not nd any suitable `working' tools in the
literature to handle such real, numerical or physical applications, during the
1990s, we developed a practical approach, first for noninvertible differentiable
mappings and later for Lipschitz mappings, to investigate `robustness' issues
for `real-world' systems. We called this concept semi-hyperbolicity. It arose
indirectly in the context of our research on the eects of spatial discretization
on the behavior of a dynamical system, in particular by using finite machine
arithmetic in computer representations of dynamical systems.

Subsequently, this idea rapidly broadened into a series of papers in which
differing aspects and applications of the concept were explored. These and
some recent papers form the basis of this monograph, the aim of which is
to present a more complete and systematic development of the concept of
semi-hyperbolicity, as well as to illustrate its usefulness.

The concept of bi-shadowing was also developed in the above papers. Shadowing
is a well known consequence of hyperbolicity, and also holds in weaker
situations of semi-hyperbolicity. Essentially it says that there is always a true
solution near a pseudo-solution, which could be arbitrary or, more interestingly,
a solution of some approximating system. Bi-shadowing includes the
converse eect and is denitely nontrivial when the pseudo-solutions that are
near a given true solution come from a specied class or approximating solution.
We would like to stress that semi-hyperbolicity has, essentially, very little
in common with `classical' hyperbolicity | no invariant splitting, no `real'
smoothness of a system, no invertibility, and so on. Of course, when all of
these properties are present in a system, then semi-hyperbolicity is often a
sufficient condition for hyperbolicity of the system. For this reason, in this
monograph we cite only very general facts about hyperbolic systems that we
need here, rather than going into very deep problems of hyperbolicity that are
usually considered in monographs on the subject. We reiterate that while the
connection with the theory of hyperbolic systems is important and cannot be
ignored, much of our motivation comes from our interest and background in
applications of dynamical systems and this has naturally in
fluenced the types
of questions asked and investigated here.

The theory of semi-hyperbolicity and bi-shadowing is developed systematically
in this monograph in nine chapters. There are also two appendices,
one by Marcin Mazur, Jacek Tabor and Piotr Koscielniak on the relationship
between hyperbolicity and semi-hyperbolicity in linear systems and one
by Janosch Rieger on semi-hyperbolicity and bi-shadowing in set-valued systems.

Brisbane-Frankfurt-Moscow-Cork
1992-2012

Phil Diamond
Peter Kloeden
Victor Kozyakin
Alexei Pokrovskii