With an introductory level of understanding of shadowing behind us, it may be instructive to devote some time to ``devil's advocate'' questions and answers.
:
Science is usually interested in generally applicable results.
Why should we care about finding an exact solution
to a specific initial condition?
: We don't. What shadowing tries to do, empirically for now,
is show that given any particular initial condition and cheaply
integrated solution, that a solution of a more expensive integration
exists nearby. Ultimately, using GHYS's containment algorithm, we
aim to show that any particular noisy solution has a true solution nearby.
If we cannot find such solutions, then we may have good reason to
suspect the reliability of our simulations.
:
OK, but if the system is chaotic, then it is supposed to
have sensitive dependence on initial conditions.
Shouldn't it be fine that there is noise in our orbits? i.e., isn't
our theory robust enough to withstand small perturbations? Isn't
computational noise roughly equivalent to chaos anyway?
: We (or at least I) do not know enough about chaotic systems to
say, so for now we should play it safe. In fact,
one way of looking at shadowing is that we are actually trying to
discover if the theory is robust enough!
:
There is a large body of theory behind numerical integration of ODE's.
In particular, the theory of defect-based backwards error analysis (DBBEA)
shows that, for any particular ODE y'=f(t,y), a numerical method
will give the exact solution to a slightly perturbed problem
, where 
is a function whose
magnitude is small, roughly the size of the tolerance requested of the
numerical ODE method. Furthermore, there exists a theorem of ODEs
that states equivalence between the problem
of how solutions change with changing initial conditions, and how they change
with changing parameters. Isn't this good enough when we realize that
can be considered a parameter with 
giving
the true solution?
: Again, we don't know enough about chaotic systems to say. If
I want to integrate the gravitational N-body problem, DBBEA tells me
that I will get the exact solution to the problem
But this is not the N-body problem. Generally in science, one tries to change as few things as possible between the real world and the model of the world. It seems to me that asking for an exact solution to the same ODE with sligthly different initial conditions is changing less than asking for an exact solution to a slightly different ODE. It is certainly not clear that the latter is good enough, especially considering that we are dealing specifically with problems that we know have sensitive dependence on small changes. Furthermore, Hamiltonian systems have many special properties that will be preserved if we change the initial conditions, but may not be preserved by equation 1.11.
What is the ultimate goal of shadowing?
: That's a tough question. For one, we are answering question 0
above. A possible ultimate goal of shadowing could be to identify,
learn about, recognize, and finally eliminate glitches from our
simulations, but it is unclear if all of these are feasible, especially
the last. I suspect the GHYS containment procedure may allow us to
eliminate glitches when we detect them, but this line of reasoning
requires more thought.
