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Analytic integrals of the semistandard map, and splitting of separatrices. (English. Russian original) Zbl 0725.58028
Leningr. Math. J. 1, No. 2, 427-445 (1990); translation from Algebra Anal. 1, No. 2, 116-131 (1989).
The behaviour of stochastic layers of a perturbed integrable Hamiltonian system is investigated. If the perturbation is measured by a small parameter \(\epsilon\), and is analytic, then the thickness of the stochastic layer is of order \(\exp (-const/\sqrt{\epsilon}).\) An important characteristic is the angle of the transversal intersection of separatrices of the arising hyperbolic periodic trajectories.
In his previous investigations (1984-1988), the author presented a formula for the asymptotics (\(\epsilon\to 0)\) of the angle mentioned above, based on the hypothesis of the existence of an analytic integral in the stochastic layer of the semistandard mapping \[ u_ 1=u+v_ 1(mod 2\pi i),\quad v_ 1=v+\exp (u). \] In this paper, the hypothesis is proved.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70K30 Nonlinear resonances for nonlinear problems in mechanics
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