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

Taylor expansion agrees with that of exp(−δ/ǫ) through terms of degree 4, and
its value always lies in the interval (0, 1). Thus the solutions of the boundary
value equation converge geometrically toward the slow manifold along its sta-
ble manifold with increasing time. If the mesh intervals have lengthδ≤ǫ, then
the relative error of the decrease satisfies
0< ρ j( δ
ǫ )−exp(δ
ǫ )
exp( δ
ǫ )
< 0.0015
For large values ofδ/ǫ, the solution is no longer accurate near t=a if the bound-
ary conditions do not satisfy y0 = (x1)0+ǫ. A similar, but simpler argument
establishes that the solution of the discretized problem converges to the slow
manifold at an exponential rate with decreasing time from t = b. Thus, the
boundary value solver is stable and yields solutions that qualitatively resem-
ble the exact solution for all meshes when applied to this linear problem. In
particular, the solution of the discretized problem is exponentially close to the
slow manifold away from the ends of the time interval [a, b]. As the mesh size
decreases to zero, the algorithm has fourth-order convergence to the exact solu-
tion.
4.4 Numerical Examples
4.4.1 Bursting Neurons
This section on the Terman modification of the Morris-Lecar model appeared in
the original paper [55] but has been omitted here for brevity.
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