High-precision measurement of separatrix splitting in a nonlinear resonance

We present the results of numerical modeling and a theoretical analysis of the splitting of a nonlinear-resonance separatrix in the intermediate asymptotic region for the standard-map model. Direct measurements of the splitting angle α(K), where K is the small parameter of the system, have been carried out over a huge range, 0.1≳α≳10−208 (1⩾K⩾0.0004), with a relative accuracy greater than one part in 10−25 and an average accuracy of roughly one part in 10−30. This made it possible to compare in detail our results with those of the existing asymptotic theory and to detect a number of new effects. We find a relatively simple empirical expression for the α vs. K dependence in the intermediate asymptotic region, and this region proves to be surprisingly broad: K≲10−2. We also study the effect of noise, in particular, errors in measuring the angle, which proved to be much more significant and complicated than expected. Finally, we point out unresolved questions and possible directions of research involving this problem.