Global invariant manifolds in the transition to preturbulence in the Lorenz system
The authors study the first homoclinic bifurcation of the Lorenz system, where two primary periodic orbits of saddle type bifurcate from a symmetric pair of homoclinic loops. The transition through this homoclinic explosion point results in a dramatic change of the topological structure of the basins of the two attracting equilibria. A chaotic saddle is created in a tubular neighborhood of the two homoclinic loops and this invariant hyperbolic set gives rise to preturbulence. By means of numerical methods, the authors show how the two-dimensional stable manifolds of the origin and of the primary periodic orbits intersect a suitable sphere in the phase space. The computations reveal the bifurcating Cantor structure of invariant manifolds and basins, which explain where in phase space long transients -- the characterizing feature of preturbulence -- can be found.
scientific article; zbMATH DE number 5988385
scientific article