A fractal set of perpetually bouncing universe?

For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.