  The present work studies the isotropic and homogeneous turbulence for
incompressible fluids through a specific Lyapunov analysis, assuming that the
turbulence is due to the bifurcations associated to the velocity field. The
analysis consists in the calculation of the velocity fluctuation through the
Lyapunov analysis of the local deformation and the Navier-Stokes equations and
in the study of the mechanism of the energy cascade from large to small scales
through the finite scale Lyapunov analysis of the relative motion between two
particles. The analysis provides an explanation for the mechanism of the energy
cascade, leads to the closure of the von Karman-Howarth equation, and describes
the statistics of the velocity difference. Several tests and numerical results
are presented.
