There is an intimate link between shadowing and two-point boundary value problems (2PBVPs) for ODEs. In contrast to the initial value problem (IVP)

which specifies all conditions at one point, a 2PBVP specifies conditions at two points, generally endpoints:

There are also multi-point BVPs, which place requirements at multiple points along the interval. These can be transformed into 2PBVPs by treating each sub-interval as an independent 2PBVP and then adding continuity conditions between sub-intervals.

Let be an approximate trajectory and let
be a true trajectory. Let *U*(*t*) and *S*(*t*) be the unstable and stable
subspaces along , respectively.
The boundary conditions that we apply for to be a shadow of
(cf. eqns. (14) and (15) and ensuing discussion) are

where ` ' denotes projection into a subspace. In practice, we
could relax these boundary conditions to allow and
into the ``forbidden'' subspaces by an amount less than the
shadowing distance. So, there exists a shadow *iff*
(17) has a solution.

The concept of hyperbolicity used in shadowing is identical to
the concept of *dichotomy* in BVPs [60, 16]:

*Definition*: For positive constants , the system

is said to be -hyperbolic [16] or to have
*exponential dichotomy* [5, p.116,]
on the interval [*a*,*b*] if for a given there exist
linear subspaces *S*(*s*) and *U*(*s*) such that, if ,

for and , and, if ,

for and . Here, is a solution to
the variational equation of the problem, *i.e., * a resolvent
(cf. footnote here ).

Dichotomy is closely related to well-conditioning of 2PBVPs, in the
sense that if there is a dichotomy, then it is easy to put bounds
on the conditioning constants. It is related to Green's functions,
in the sense
that if there are bounds, exponential in time, on the Green's functions,
then there is a dichotomy [5, section 3.4.3,].
The splitting of the solution components into the stable and unstable
subspaces is called *decoupling* in the BVP literature, and
standard algorithms exist to handle decoupling.

The GHYS/QT refinement algorithm, it turns out, is an implementation of
a form of *multiple shooting*, which is simply a method of solving
2PBVPs by splitting the interval into many sub-intervals, computing
an approximate solution on each sub-interval, and then requiring that
the solutions match at the boundaries. Many topics of potential
interest to the study of shadowing, including multiple shooting, are
discussed in Ascher *et al. * [5].
Enright [24] suggests ways to improve the performance of
multiple shooting, based on clever ways of choosing the sub-interval
endpoints. Krogh, Keener, and Enright [51]
show how it is possible to reduce the size of the resolvent matrix
since there are really only *k* unknowns at one end and *N*-*k* at
the other, instead of integrating the entire variational equation.
So, in the GHYS/QT algorithm, we could reduce the size of the
resolvents by half.
Chow and Van Vleck [16] show explicitly how shadowing
is related to multiple shooting.

The field of 2PBVPs is comparatively mature in comparison to the study
of shadowing. For example, methods for *a posteriori*, rigorous error
bounds on 2PBVPs existed in 1981 [50], six years before GHYS.
Clearly, the study
of shadowing may benefit from interaction with the study of 2PBVPs.

Fri Dec 27 17:41:39 EST 1996

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