Bryuno Function and the Standard Map

Abstract

For the standard map the homotopically non-trivial invariant curves of rotation number ω satisfying the Bryuno condition are shown to be analytic in the perturbative parameter ε, provided |ε| is small enough. The radius of convergence ρ(ω) of the Lindstedt series -sometimes called critical function of the standard map -is studied and the relation with the Bryuno function B(ω) is derived: the quantity | log ρ(ω) + 2B(ω)| is proved to be bounded uniformily in ω. cleare, Sez. Tor Vergata

Aspects of Ergodic, Qualitative and Statistical Theory of Motion

“…Davie's result implies the upper bound Q 2 (ω) < C The results stated in proposition (9.2.1) were first conjectured in [BM94], supported by numerical results, and a proof was given for the resonances p/q = 0/1 and p/q = 1/2. Validity of Bryuno's interpolation formula |Q 2 (ω)| < C was recently proved in [BG01], where a bound Q 2 (ω) > C was found via a refinement of the techniques discussed in the present section. §9.4 Scaling laws for the standard map…”

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

“…for suitable universal constants C 1 and C 2 ; see [27,9] for a proof of the last statement. The proof of the lower bound in (12.3) relies on deeper cancellations than those discussed in Section 7; see [9] for details.…”

“…All the results of the previous sections can be extended to rotation vectors satisfying the Bryuno condition: see [39] for maximal and lower-dimensional tori, [9] for the standard map, and [40] for dissipative systems. For d = 2 one can write ω = (ω 1 , ω 2 ) = (1, α)ω 1 , where α = ω 2 /ω 1 is the rotation number.…”

“…The proof of the lower bound in (12.3) relies on deeper cancellations than those discussed in Section 7; see [9] for details. Another Diophantine condition considered in the literature is the so-called Rüssmann condition [75,76,73], which has a somewhat intricate definition if compared to (12.1).…”

“…Finally, the analysis developed so far for ordinary differential equations, can be extended to partial differential equations, such as the nonlinear wave equation 9) and the nonlinear Schrödinger equation…”