We study slow diffusion processes of orbits through thin chaotic layers of 2m (m ≥ 2)-dimensional symplectic maps (often referred to as Arnol'd diffusion) on a 4-dimensional mapping model of accelerator dynamics. Using a method proposed by Chirikov, we compute diffusion rates of vertical displacements, |yn|, near the flat beam case, when the horizontal motion occurs within a thin chaotic layer of the corresponding 2-dimensional xn, xn+1 map. Our computation distinguishes between regions of diffusive and quasiperiodic motion, in which the respective rates are found to differ by 10-12 orders of magnitude for N = 108 iterations. For diffusion, |yn| grows on the average with n (n ≤ 108, with a rate D(R) ∝ exp(aR) (a > 0, R2 = y20 + y21), which gives accurate estimates of escape times Nesc ∝ D-1. Modelling synchrotron oscillations by a periodic modulation on the "betratron" frequencies, we find that diffusion rates increase significantly and rapid escape occurs above certain thresholds in the ε, Ω parameter values, ε and Ω being the amplitude and frequency of the modulation respectively.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics