Nonconvergence of formal integrals: II. Improved estimates for the optimal order of truncation

Abstract

We investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 DOF Hamiltonian systems. In particular, we study the dependence of the optimal order of truncation N opt on the distance ρ from the elliptic equilibrium, by numerical and analytical means. The function N opt (ρ) determines the region of Nekhoroshev stability of the orbits and the time of practical stability. We find that the function N opt (ρ) decreases by abrupt steps. The decrease is roughly approximated with an average power law N opt = O(ρ −a), with a 1. We find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via Lie series and in the direct construction of first integrals. Precisely, we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity. Our results agree with those obtained by Servizi G et al (1983 Phys. Lett. A 95 11) for a conservative map of the plane. Moreover, our analytical considerations allow us to explain the results of our previous paper (Contopoulos G et al 2003 J. Phys. A: Math. Gen. 36 8639), including in particular the different behaviour observed for low-order and higher order resonances.

Ordered and chaotic Bohmian trajectories

“…These terms act in the series in a similar way as in the classical Birkhoff series, i.e., by producing repetitions of Fourier terms with the same small divisors at every second iteration step (see Efthymiopoulos et al 2004 for a description of the classical case). That is, if x n contains a Fourier term of the form f x,n = A n cos[(m 1 + m 2 /c 2 )t], where A n is a coefficient of order n in a, b, then, via Eq.…”

“…n , the factor n! being due to repeated actions of a Poisson bracket operator (see Efthymiopoulos et al 2004 for details). On the contrary, in the quantum series the factor n n ∼ n!…”

Special Features of Galactic Dynamics

“…the numerator of the Fourier terms of I (k) (see Efthymiopoulos et al 2004 for a more detailed analysis). Putting these remarks together, the size of Fourier terms (31) can be estimated as ||f (k) || ∼ k!…”

“…A particular application in the problem of stability of the Trojan asteroids (Giorgilli and Skokos 1997) showed that the optimal order of truncation of the integrals in this case is beyond n = 32 (in some cases we find n opt > 60, Efthymiopoulos and Sándor 2005). But a precise treatment of the problem was made only very recently (Contopoulos et al 2003, Efthymiopoulos et al 2004. In these works scaling formulae are given yielding the optimal order of truncation as a function of the distance from the elliptic equilibrium and of the number of degrees of freedom.…”

Periodic orbits in the logarithmic potential

“…On the other hand, the normal form provides us with an efficient way of constructing series with an asymptotic character: this implies that at some point we should achieve an "optimal" value for the expansion order N opt (hopefully > N min ) that provides the best possible result (Efthymiopoulos et al 2004). The optimal order depends on the size of the phase-space region in which we are interested.…”