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This work presents a review of previous articles dealing with an originalturbulence theory proposed by the author, and provides new theoretical insightsinto some related issues. The new theoretical procedures and methodologicalapproaches confirm and corroborate the previous results. These articles studythe regime of homogeneous isotropic turbulence for incompressible fluids andpropose theoretical approaches based on a specific Lyapunov theory fordetermining the closures of the von K\'arm\'an-Howarth and Corrsin equations,and the statistics of velocity and temperature difference. Furthermore, noveltheoretical issues are here presented among which we can mention the followingones. The bifurcation rate of the velocity gradient, calculated along fluidparticles trajectories, is shown to be much larger than the correspondingmaximal Lyapunov exponent. On that basis, an interpretation of the energycascade phenomenon is given and the statistics of finite time Lyapunov exponentof the velocity gradient is shown to be represented by normal distributionfunctions. Next, the self--similarity produced by the proposed closures isanalyzed, and a proper bifurcation analysis of the closed vonK\'arm\'an--Howarth equation is performed. This latter investigates the routefrom developed turbulence toward the non--chaotic regimes, leading to anestimate of the critical Taylor scale Reynolds number. A proper statisticaldecomposition based on extended distribution functions and on theNavier--Stokes equations is presented, which leads to the statistics ofvelocity and temperature difference.

Statistical Lyapunov theory based on bifurcation analysis of energy cascade in isotropic homogeneous turbulence: a physical -- mathematical review