One can see in this movie that for small epsilon, the solution appears perfectly regular. For large epsilon, the solution becomes singular.
If you continue with this tuning, you find an evolution which appears to "blow" off lots of energy, leaving behind an almost static solution. In fact, this static-looking solution looks remarkably like a solution found explicitly by assuming staticity. Here I show the results of one such tuning plotted along with the evolution of one of the static solutions. Shown is the field chi(r,t) (the tuned solution[magenta]; the explicitly static solution[light blue]) as well as the energy density as a function of r and t, rho(r,t) (the tuned solution[yellow]; the explicitly static solution[navy blue]).
One thing to note is that the static solution is remarkably static. That is, what you're seeing is the output from our code when given the static solution as initial data. In other words, this static solution looks static despite the inevitable "perturbations" provded by the numerics. This seems consistent with the fact that for "spread-out" static solutions, the instability is very slow-growing. More importantly, one sees the energy that is blown off, and that what remains appears to be the static solution.
Keep in mind here, that there is a continuous family of static solutions. As the tuning proceeds, it would seem that different static solutions are approached. Hence, it doesn't seem as if any of the static solutions represent any kind of attractor (e.g. an intermediate attractor).
Here's another recently made movie which
I'm hoping will help explain at least the difference
that we're seeing between super- and sub- critical initial data. Shown
is the energy density
for four different evolutions of the Turok-Spergel data:
For an ingoing Gaussian perturbation (r_0=0.5, del=0.2, amp=+/-0.01), I show the n=0, the n=1, and the n=2 solutions. Looking at just the n=0 and n=2 cases, one observes all three evolutions (amp=-0.01, amp=0, amp=+0.01) collapsing, suggesting that the solutions are not on threshold. However, the case of n=1 is unique in that the amp=+0.01 case collapses quite quickly while the amp=-0.01 disperses. These movies then suggest that only the n=1 solution sits on threshold, and presumably this would indicate the self-similar solution we are seeing in tuning of the Gaussian initial data is indeed the n=1 self-similar solution.