Basin-filling Peano omega-branches and structural stability of a chaotic attractor

Abstract

Attractors demonstrating randomly transitional (or chaotic) phenomena usually exhibit the following three outstanding properties:
·Sensitive dependence on initial conditions.
·Fractal structure of alpha-branches of principal saddles.
·Transitive property, i.e., one trajectory or sequence of images can come arbitrarily close to every point in the attractor.
Among the three properties given above, the third property is always observed in simulation experiments. However, rigorous mathematical demonstration has not been provided yet.
It supports the proposition that the omega branches of the principal saddle in a chaotic attractor must be basin-filling Peano curves.