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Two-point Boundary Value Problems for ODEs

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.

Next: Discussion and future directions Up: Contents Previous: Shadowing lemmas for ODE

Wayne Hayes
Fri Dec 27 17:41:39 EST 1996

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