  This work presents a review of previous articles dealing with an original
turbulence theory proposed by the author, and provides new theoretical insights
into some related issues. The new theoretical procedures and methodological
approaches confirm and corroborate the previous results. These articles study
the regime of homogeneous isotropic turbulence for incompressible fluids and
propose theoretical approaches based on a specific Lyapunov theory for
determining the closures of the von K\'arm\'an-Howarth and Corrsin equations,
and the statistics of velocity and temperature difference. Furthermore, novel
theoretical issues are here presented among which we can mention the following
ones. The bifurcation rate of the velocity gradient, calculated along fluid
particles trajectories, is shown to be much larger than the corresponding
maximal Lyapunov exponent. On that basis, an interpretation of the energy
cascade phenomenon is given and the statistics of finite time Lyapunov exponent
of the velocity gradient is shown to be represented by normal distribution
functions. Next, the self--similarity produced by the proposed closures is
analyzed, and a proper bifurcation analysis of the closed von
K\'arm\'an--Howarth equation is performed. This latter investigates the route
from developed turbulence toward the non--chaotic regimes, leading to an
estimate of the critical Taylor scale Reynolds number. A proper statistical
decomposition based on extended distribution functions and on the
Navier--Stokes equations is presented, which leads to the statistics of
velocity and temperature difference.
