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

exit point of a trajectory starting at q∈ M∩{|b|=δ} that spends a time t=O(1/ǫ) in
B. Let V be a neighborhood of q in M. The image of V under the time t map is close in
the C 1 -norm to
{|a|=δ, yi− yi(0)=0, i>1}
where yi(0) denotes the y-coordinates of q. In particular M is C 1 -close to{|a|=δ, yi−
yi(0)=0, i>1} near ¯q.
Remark: The key feature of both Exchange Lemmas (Theorems 2.3.2 and
2.4.1) is that we have traded information about transversality and variation of
certain center directions (here: yi with yi> 1) near q with new information near
the exit point ¯q given by a C 1 -closeness result to a certain submanifold of the
unstable manifold W u (Sǫ). The idea used by Jones and Kopell [68] to achieve a
tracking of the tangent spaces to M is to consider (k+1)-differential forms that
are dual to (k+1) planes in k+l+n space and describe their evolution by a dif-
ferential equation. As usual we shall simply write (k+1)-forms with the implicit
assumption that all forms are sufficiently differentiable. A basic (k+1)-form is
given by
Pσ1···σk+1
:= dσ1∧...∧dσk+1
whereσ j∈{ai, bi, yi}. Furthermore we know that a basis of all forms is given by
restricting to increasing indices. Here we agree on the ordering
a1< a2