Speaker
Description
It is known that the dynamics of a Lorenz-like attractor are described by a singular one-dimensional map (the quotient of the Poincare map over the strong-stable invariant foliation). For Lorenz attractors emerging out of a variety of homoclinic bifurcations, this map takes a universal form — it is a C^1-small perturbation of the map
X --> |1 - c X^b|
where the parameter c can be arbitrary, and 0 < b < 1, so the map has an infinite derivative at zero and positive Schwarzian. We give a complete description of the attractors for this family of maps and for all its low-regularity perturbations. In particular, we determine the region in the parameter plane (b,c) for which the Lyapunov exponent is positive for all orbits. The main difficulty is that for small values of b the map is contracting for long series of consecutive iterations, but we show that the expansion always prevails. This is a joint work with Klim Safonov.