6 research outputs found
A test for a conjecture on the nature of attractors for smooth dynamical systems
Dynamics arising persistently in smooth dynamical systems ranges from regular
dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov,
uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The
latter include many classical examples such as Lorenz and H\'enon-like
attractors and enjoy strong statistical properties.
It is natural to conjecture (or at least hope) that most dynamical systems
fall into these two extreme situations. We describe a numerical test for such a
conjecture/hope and apply this to the logistic map where the conjecture holds
by a theorem of Lyubich, and to the Lorenz-96 system in 40 dimensions where
there is no rigorous theory. The numerical outcome is almost identical for both
(except for the amount of data required) and provides evidence for the validity
of the conjecture.Comment: Accepted version. Minor modifications from previous versio
Thermodynamic formalism for contracting Lorenz flows
We study the expansion properties of the contracting Lorenz flow introduced
by Rovella via thermodynamic formalism. Specifically, we prove the existence of
an equilibrium state for the natural potential for the contracting Lorenz flow and for in an interval
containing . We also analyse the Lyapunov spectrum of the flow in terms
of the pressure