Attractors from Lorenz-like Maps
by
Youngna Choi
Northwestern University

The original Lorenz equation has been analyzed only numerically, yet the linear version, the geometrical model has revealed a lot of facts. Assuming the foliation along the stable manifold, we reduced the three dimensional equation to a map f:[-A, A] --> [-A, A] with f^'(x) > \surd2 off the discontinuity. It was proved that such a map is topologically transitive on the whole interval and any open set U of (-A, A) covers (-A, A) after some iteration, hence the title locally eventually onto. Recent result on a generalized condition, expanding maps with a single discontinuity is, there is a compact invariant subset of the original domain on which the restricted map is topologically transitive and the stable manifold of the invariant set is open and dense in the domain, which suggest the existence of an attractor. In this paper were shown the general structure of the above invariant set analogous to that of the original Lorenz map as well as an explicit proof of the existence of a trapping region for the invariant set. Moreover the condition which guarantees the trapping is provided with its stability in the sense that, when the invariant set is not an attractor due to the lack of a trapping region, it can be approximated by attractors nearby. The paper concentrates on a single discontinuity case, yet all the theories can be generalized for maps with finite discontinuities.

Date received: March 18, 1998