Universal diffusion near the golden chaos border and
asymptotic statistics of Poincaré recurrences

The investigation of local diffusion rate near the critical golden curve in the Chirikov standard map was done in [77]. In agreement with the heuristic Chirikov's prediction it is found that the diffusion rate scales as $D \sim \vert r_n - r_g\vert^{5/2}$ where $r_n$ is the local rotation number converging to its limiting golden value $r_g$ via the principal rational approximants. The results demonstrate the universal self-similar structure of the local diffusion rate near the critical golden curve. The diffusion rate drops rapidly when an orbit approaches to the critical invariant curve. This should give the exponent $p=3$ for the decay of the Poincaré recurrences $P(\tau)$. However, this value of $p$ is in the contradiction with the values obtained from the numerical simulations of $P(\tau)$ (see section 3.2.1).

This contradiction is resolved in the recent paper [101]. There, the new numerical method was developed which allowed to study $P(\tau)$ up to very large $\tau \approx 10^9$. This allowed to establish that asymptotically in time $P(\tau) \sim 1/\tau^p$ with $p=3$. However, this behavior starts only after extrimely large times $\tau > \tau_g \sim 10^7$. For $\tau < \tau_g$ the exponent is different $p \sim 1.5$. The physical reason of this late asymptotic decay is explained on the basis of slow diffusion rate in the chaotic separatrix layer. The results obtained in [101] establish the universal law for Poincaré recurrences decay in hamiltonian systems with divided phase space. As the result, at $\tau \rightarrow \infty$ the correlation functions decay as $C(\tau) \sim 1/\tau^2$ and the diffusion rate remains finite ( $D \sim \int C d\tau < \infty$).