multiple time scale dynamics with two fast variables and one slow ...

(a) In the case when the slow flow is one-dimensional (y∈R) it seems to be
very helpful (possibly essential) to rescale the vector field:
x ′ = f (x, y,ǫ)
y ′ = ǫg(x, y,ǫ)
The slow flow will be normalized to unit speedǫtg(x, y,ǫ)↦→ ˜t so that:
dx
d˜t
dy
d˜t
= f (x, y,ǫ)
ǫg(x, y,ǫ)
= 1
Note that it remains a question for future work how to access the influence
of the re-scaling on the numerical method including the case of higher-
dimensional slow manifolds (y∈R n , n>1).
(b) The standard boundary value solvers in MatLab bvp4c and bvp5c [101]
do not seem to be able to solve the problem, although the same bound-
ary conditions and initial guess as for our scheme presented above were
used. Note that it is unclear whether this is a problem on the implemen-
tation, user or algorithmic side. In fact, for moderate values ofǫ= O(10 −2 )
we would expect that almost any standard 2-point boundary value solver
should be able to solve for a trajectory in the slow manifold manifold.
Furthermore a different strategy for the linear test problem (4.3) was imple-
mented in AUTO [34]. First we re-write the fast-slow system as z ′ = F(z) where
z=(x, y)∈R N and F= ( f,ǫg). Let T denote the final time of a trajectory and
re-scale t↦→ tT. Then we consider the following boundary value problem for
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