Global invariant manifolds in the transition to preturbulence in the Lorenz system

We consider the homoclinic bifurcation of the Lorenz system, where two primary periodic orbits of saddle type bifurcate from a symmetric pair of homoclinic loops. The two secondary equilibria of the Lorenz system remain the only attractors before and after this bifurcation, but a chaotic saddle is created in a tubular neighbourhood of the two homoclinic loops. This invariant hyperbolic set gives rise to preturbulence, which is characterised by the presence of arbitrarily long transients.In this paper, we show how and where preturbulence arises in the three-dimensional phase space. To this end, we consider how the relevant two-dimensional invariant manifolds — the stable manifolds of the origin and of the primary periodic orbits — organise the phase space of the Lorenz system. More specifically, by means of recently developed and very robust numerical methods, we study how these manifolds intersect a suitable sphere in phase space. In this way, we show how the basins of attraction of the two attracting equilibria change topologically in the homoclinic bifurcation. More specifically, we characterise preturbulence in terms of the accessible boundary between the two basins, which accumulate on each other in a Cantor structure.