Construction of analytic KAM surfaces and effective stability bounds

A class of analytic (possibly) time-dependent Hamiltonian systems withd degrees of freedom and the “corresponding” class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct “explicitly” KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the HamiltonianH≡y 2/2+ε(cosx+cos(x−t)) and to the map (y,x)→(y+ε sinx,x+y+ε sinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of ∼10−5) of the golden-mean KAM surfaces for complex values of ε with |ε| less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between break-down thresholds and singularities in the complex ε-plane is pointed out.