The projection operator formalism yields a time evolution equation for the time correlation function $U_n\left(t\right)$ of the chaotic modes of interest in terms of the memory function ${\Gamma }_n\left(t\right)$. On the assumption of similarity between $U_n\left(t\right)$ and ${\Gamma }_n\left(t\right)$, this equation leads to a closed equation for $U_n\left(t\right)$, which yields the asymptotic behavior of the time correlation function $U_n\left(t\right)$ and the corresponding power spectrum $I_n\left(\omega \right)$ analytically. Thus it turns out that the time correlation function takes the algebraic form $1/\left(1+t^2\right)$ for $t\to 0$ as predicted previously, and can be classified into three decay forms for $t\to \infty$ according to the wave number $k_n$: the exponential decay $e^{-t}$, the oscillatory exponential decay $e^{-t}\text{cos}t$, and the oscillatory power-law decay $t^{-3/2}\text{cos}t$. All the corresponding power spectra form a dual structure which is Lorentzian as $\omega \to 0$ and decays exponentially as $\omega \to \infty$. In the entire domain $0\le t<\infty$, solutions to the closed equation are quite consistent with the numerical results for small $k_n$, while they are consistent with those for large $k_n$, except for the phase. In the case that the integral time scale of $U_n\left(t\right)$ is equal to that of ${\Gamma }_n\left(t\right)$, the closed equation is identical to the direct interaction approximation equation for fluid turbulence in the limit $k_n\to \infty$.

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