Memory Function Approach to Chaos and Turbulence and the Continued Fraction Expansion

The chaotic orbits of dynamical systems are deterministic and predictable on short timescales τ γ , but they become stochastic and random on long timescales T M (» τ γ ) due to the orbital instability of chaos. This randomization of chaotic orbits has been formulated recently by deriving a non-Markovian stochastic equation for macrovariables in terms of a fluctuating force and a memory function. In order to develop this memory function approach to chaos and turbulence, we explore the following problems by studying the Duffing oscillator and the Navier-Stokes equation for an incompressible fluid: 1) the physical meaning of the projection of macrovariables A(t) onto A(0); 2) the method of calculating the short-lived motion with short timescale τ γ , which determines the memory functions and the macroscopic transport coefficients due to chaos and turbulence; 3) the continued fraction expansion of the memory function, and the order estimation of short timescales τ γ and long timescales τM; 4) the relation between the memory function and the time correlation function of a nonlinear force, which gives computable theoretical expressions for the macroscopic transport coefficients.