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

computing the homoclinic orbit, even for very small values ofǫ [56]. Carry
out all the following computations for (p, s)=(p0, s0).
2. Compute the slow manifolds Sǫ,l and Sǫ,r using the SMST algorithm.
3. Compute the unstable manifold of the equilibrium W u (q) by forward inte-
gration.
4. Define a sectionΣ={x1= c} where the constant c is chosen between x1,−
and x1,+ e.g. c=(x1,−+x1,+)/2. Compute the transversal intersection of
W s (S l,ǫ) and W u (S r,ǫ) onΣ, call the intersection point xsu= (c, x2,su, ysu) (see
Figure 4.2). Integrate forward and backward starting at xsu to obtain tra-
jectoriesγ f w andγbw.
5. The homoclinic orbit is approximated by a concatenation of the trajectory
segments on W u (q), S r,ǫ, W u (S r,ǫ)∩W s (S l,ǫ) and S l,ǫ computed in steps 1.-4.
The endpoints of these trajectory segments are exponentially close to one
another and therefore indistinguishable numerically.
All our figures for the fast wave case have been computed forǫ= 10 −3 , p0= 0
and s0≈1.2463. Jones et al. [69] proved the existence of homoclinic orbits in
this region for smallǫ. In Figure 4.3(a) we show the result from the SMST al-
gorithm and the unstable manifold of the equilibrium W u (q), i.e. the output of
steps 2 and 3. Due to the exponential separation along S r,ǫ the trajectory W u (q)
obtained from numerical integration cannot track the slow manifold for an O(1)
distance and escapes after following the slow manifold for a very short time.
This happens despite the fact that we have computed parameter values (p0, s0)
with maximal accuracy in double precision arithmetic at which we expect W u (q)
to follow S r,ǫ almost up to the fold point x1,+. This observation is relevant to
Figure 4.3(b) where the result of step 5 is shown. All the fast segments (red)
108