Contemporary Mathematics
Volume 469, 2008
Nonconvergence examples in averaging
Victor Bakhtin and Yuri Kifer
To Misha Brin on his 60th birthday
Systems which combine fast and slow motions lead to complicated
two scale equations and the averaging principle suggests to approximate the
slow motion by averaging in fast variables. When the fast motion does not
depend on the slow one this approximation usually works for all or almost all
initial conditions but when the slow and fast motions depend on each other
(fully coupled), as is usually the case, the averaging prescription cannot always
be applied, and when it is valid then only in the sense of convergence in
measure (or in average) with respect to initial conditions. A nonconvergence
example for fixed initial conditions with Hamiltonian fast motions appeared in
[16] and in this paper we construct corresponding nonconvergence examples in
the discrete time averaging setup with fast motions being expanding maps and
Markov chains. We provide also an example where the averaging principle fails
completely (even in the weaker sense mentioned above) if averaging measures
are not chosen appropriately.
Models of many real systems exhibit combinations of slow and fast motions
which lead to complicated double scale equations and the averaging principle, ap-
plied to celestial mechanics computations already in the 19th century, suggests to
simplify the problem approximating the slow motion by the averaged one obtained
by averaging parameters in fast variables. Traditionally, averaging methods were
employed in the study of two scale ordinary differential equations describing a con-
tinuous time motion. On the other hand, it is well known that the study of discrete
time dynamical systems and Markov processes enables us to deal with a wider class
of models and examples and to reveal new effects.
When the fast motion does not depend on the slow one (and on the correspond-
ing small parameter) the averaging principle was justified rigorously in the middle
of the 20th century by Bogolyubov (see
and references there) and it provides
an approximation of the slow motion moving with the speed of order
on time
2000 Mathematics Subject Classification. Primary: 34C29 Secondary: 34D20, 60Fl0, 60J25.
Key words and phrases. averaging principle, large deviations, expanding transformations,
Markov chains.
Part of the work was done during the visit of the 1st author to the Hebrew University.
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