On Analytic Integrals along the Unstable Separatrix

One of the most recent solved problem of the theory of hamiltonian dynamical systems with two degrees of freedom is finishing a proof of the exponentially small transversality of the separatrices for the standard map ([1], [2], [3], [4]). The central point if the proof is a construction of an analytic integral along the unstable separatrix. Only the article [4] contains the complete evidence of the existing the analytic integral in a neighbourhood of the unstable separatrix. Now we will describe the main reason of introducing the analytic integral along the unstable separatrix for the standard map: )) sin( ), sin( ( ) , ( : x y x y x y x SM ε ε + + + a , where ) 2 / ( sinh 4 2 h = ε . For the unstable invariant manifold, separatrix ) , ( h t W u , there was introduced new coordinates, E(h)) (t(h), , such that 0 ) , ( = E t W u for all t (see fig. 1).