multiple time scale dynamics with two fast variables and one slow ...
multiple time scale dynamics with two fast variables and one slow ...
multiple time scale dynamics with two fast variables and one slow ...
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(0, 0, 0).<br />
2.3 The Exchange Lemma I<br />
The Exchange Lemma was initially proved by J<strong>one</strong>s <strong>and</strong> Kopell [68]. We shall<br />
mainly follow their original paper <strong>and</strong> an expository presentation in [74]. Recall<br />
from the problem of finding homoclinic orbits in the FitzHugh-Nagumo <strong>with</strong><br />
more than <strong>one</strong> jump discussed in the previous section that we have to track an<br />
invariant manifold in phase phase. For the general situation we start <strong>with</strong> a<br />
<strong>fast</strong>-<strong>slow</strong> system<br />
ǫ ˙x = f (x, y,ǫ)<br />
˙y = g(x, y,ǫ) (2.17)<br />
<strong>with</strong> x∈R m <strong>and</strong> y∈R n . We include in (2.17) possible parameters in the sys-<br />
tem in the vector y as <strong>slow</strong> <strong>variables</strong>. In the FitzHugh-Nagumo equation that<br />
would mean including the equation for the wave speed ˙s=0 <strong>and</strong> re-labeling s<br />
to a suitable indexed y-coordinate. Let S 0 denote some compact normally hy-<br />
perbolic submanifold of the critical manifold <strong>and</strong> let Sǫ be the corresponding<br />
<strong>slow</strong> manifold. Note that we shall assume that S 0 is in fact uniformly normally<br />
hyperbolic meaning that Dx f (p) has eigenvalues uniformly bounded away from<br />
zero for each p∈S 0.<br />
Recall from Theorem 1.1.3 that we can transform a <strong>fast</strong>-<strong>slow</strong> system near a<br />
30