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

whereµ∈R p are parameters. Usually it is assumed thatµis in some compact
region in R p . The extended system with the trivial equation ˙µ = 0 does not
satisfy the invariance assumptions but it is quite easy to modify the equation
˙µ=0 near the boundary of the manifolds
Ñ i :={(N i (µ),µ)} for i=1, 2
to make the manifold Ñ 1 overflowing invariant and Ñ 2 inflowing invariant. The
reason why the extended system (2.4) is important is the problem that for some
fixedµ0 the manifolds N 1 and N 2 might be tangential. Varyingµcan poten-
tially break this tangential intersection to make it transverse for Ñ 1 and Ñ 2 . This
naturally leads to the main problem in applying Theorem 2.1.1 for a concrete
example: the verification of the transversality hypothesis might be very diffi-
cult. This will be our next step. Suppose we are in the situation described in
Theorem 2.1.1. Since N 1 and N 2 intersect there exists a value y=y0 such that in
the fast subsystem
x ′ = f (x, y0, 0)
there is a heteroclinic orbit between two equilibrium points, say p1 =
(x1(y0), y0)∈W u (P 1 ) and p2= (x2(y0), y0)∈W s (P 2 ). Denote the heteroclinic orbit
by (x0(t), y0)∈R m+n . The intersection between N 1 and N 2 at a point p=(x0(t), y0)
for some t is transverse if and only if
T pN 1 ⊕ T pN 2 = R m+n
Direct dimension counting always gives:
If we define
dim(T pN 1 ⊕ T pN 2 )=dim(T pN 1 )+dim(T pN 2 )−dim(T pN 1 ∩ T pN 2 )
d=dim(T pN 1 )+dim(T pN 2 )−m−n (2.5)
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