TITLE:
Numerical study of discrete Lorenz-like attractors.

AUTHORS:
Kazakov A.^(1), Murillo A.^(2)$, Vieiro A.$^(2,3) and Zaichikov K.^(1)
(1)  National Research University Higher School of Economics,
     25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
(2)  Departament de Matem{\`a}tiques i Inform{\`a}tica, Universitat de Barcelona, 
      Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
(3) Centre de Recerca Matem{\`a}tica (CRM), 08193 Bellaterra, Spain 

ABSTRACT:
We consider a homotopic to the identity family of maps, obtained as a
discretization of the Lorenz system, such that the dynamics of the last is
recovered as a limit dynamics when the discretization parameter tends to zero.
We investigate the structure of the discrete Lorenz-like attractors that the
map shows for different values of parameters. In particular, we check the
pseudohyperbolicity of the observed discrete attractors and show how to use
interpolating vector fields to compute kneading diagrams for near-the-identity
maps. For larger discretization parameter values, the map exhibits what appears
to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic
pseudohyperbolic attractors with a negative second Lyapunov exponent. The
numerical methods used are general enough to be adapted for arbitrary
near-identity discrete systems with similar phase space structure.