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

eturns directly to q. Figure 4.5(b) shows a homoclinic orbit that makes one
additional large excursion around S m,ǫ after it was close to S r,ǫ and then returns
to q; hence we refer to the orbit in 4.5(b) as a double-pulse homoclinic orbit.
The same double-pulse phenomenon exists for fast waves as well. In this case
the double-pulse orbit has no additional interaction with the middle branch S m
and therefore it is difficult to distinguish between different pulse types for fast
waves numerically and graphically as the second loop follows the first one very
closely.
4.4.3 A Model of Reciprocal Inhibition
This section on a 4D fast-slow model of coupled neurons appeared in the origi-
nal paper [55] but has been omitted here for brevity.
4.5 Conclusion
We have illustrated how slow manifolds of saddle type appear in the bifurca-
tion analysis of slow-fast systems. From the perspective of simulation via initial
value solvers, these manifolds are ephemeral objects. Different methods are
needed to compute them accurately. Heretofore, collocation and continuation
methods incorporated into the program AUTO [34] have been used to compute
periodic and homoclinic orbits in multiple time scale systems, but this approach
becomes increasingly difficult as one approaches the singular limit. Our expe-
rience [53] in using AUTO has been that increasingly fine meshes are required
to analyze stiff systems as the ratio of time scales becomes more extreme, es-
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