Microscopic Aspects of Nonlinearity in Condensed Matter by F. T. Arecchi (auth.), A. R. Bishop, V. L. Pokrovsky, V. PDF

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3D ,31 Let us begin with a simpler problem of an isolated critical KAM curve whose rotation number is some irrational 'T". According to the KAM theory most invariant curves are preserved under a sufficiently weak perturbation in the sense that they remain 28 Per) -, 10 -9 10 Fig. 3. Statistical properties of motion with chaos border: (a) Poincare's recurrences; (1)) correlation decay. 7)27 while circles are our data for ,\ = 3. Straight lines indicate power law with the exponent shown. Dashed curve is the effect of noise.

6) is the periodicity not only in X but also in y with the same period 27r. 3). 3) for the diffusion rate as K ~ KG. For K > KG the last (most robust) KAM curve is destroyed and transformed into a chaotic layer comprising all critical scales qn :<:, qE: where qE: '" c- 1 , and c = K - KG ~ 0 (see Eq. 19) and around). This chaotic layer is just the critical 'bottleneck' which controls the transition time between integer resonances r = m, and hence, global diffusion. The time scale in the layer is '" qE:, and the same is for the exit time (t) from the layer.

M. , Lecture Notes in Physics 38 (1975) 112; J. , 93 (1979) 232. 13. B. G. Konopelchenko, Nonlinear Integrable Equations, Lecture Notes in Physics 270 (1987). 14. B. V. Chirikov and V. V. , Academic Press, 1990, p. 219. 15. V. I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics, Benjamin, 1968. 16. B. V. Chirikov, Proc. Roy. Soc. Lond. A 413 (1987) 145. 17. A. , Phys. Rev. A 23 (1981) 2664. 18. B. V. Chirikov, D. L. Shepelyansky, Radiofizika 29 (1986) 1041. 19. F. Vivaldi, private communication.

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