  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 study of the mechanism of the energy cascade
from large to small scales through the Lyapunov analysis of the relative motion
between two particles and in the calculation of the velocity fluctuation
through the Lyapunov analysis of the local deformation and the Navier-Stokes
equations.
  The analysis provides an explanation for the mechanism of the energy cascade,
leads to the closure of the von K\'arm\'an-Howarth equation, and describes the
statistics of the velocity difference.
  Several tests and numerical results are presented.
