Improved accuracy of the Birkhoff-Gustavson normal form and its convergence properties

Kaluža, Matjaž; Robnik, Marko

doi:10.1088/0305-4470/25/20/013

Cited by 22 publications

(9 citation statements)

References 16 publications

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“…In particular, the question of the optimal exponents in Nekhoroshev-type estimates near elliptic equilibria has been investigated by Lochak (1992), Fassò et al (1998) and Niederman (1998). On the other hand, numerical results connected to our study are only sporadic in the literature (Servizi et al 1983, Kaluza and Robnik 1992, Morbidelli and Giorgilli 1997.…”

Section: Introductionmentioning

confidence: 93%

“…In particular, the behaviour of the functions R r (ρ) and N opt (ρ) allows us to prove a Nekhososhev (1977)-like result for the approximate constancy of formal integrals, namely that the variations of such integrals remain bounded over times exponentially long in the inverse of ρ. Furthermore, truncated expressions of the formal integrals are known to reproduce theoretically, with great accuracy, the invariant curves corresponding to invariant KAM tori on a Poincaré surface of section (Contopoulos and Moutsoulas 1965, Gustavson 1966, Kaluza and Robnik 1992, Contopoulos et al 2003. In this respect, the limit N opt → 0 provides interesting information on the dynamics, since it is connected with the phenomenon of breakdown of the invariant tori and with the introduction of a large degree of chaos in phase space.…”

Section: Introductionmentioning

confidence: 99%

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

Efthymiopoulos¹,

Giorgilli²,

Contopoulos³

2004

J. Phys. A: Math. Gen.

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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.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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

Efthymiopoulos¹,

Giorgilli²,

Contopoulos³

2004

J. Phys. A: Math. Gen.

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“…This may be compared to the results of Gustavson (1966, figure 10) obtained at the same energy: he obtained analytical sections in good qualitative agreement with the sections plotted from computed orbits. More recent works (see Kaluza & Robnik 1992;Robnik 1993;Contopoulos et al 2003) give higher orders of formal integrals and show a better agreement at energy E=1/8 where the chaotic orbits occupy a large volume of the phase space.…”

Section: Hénon and Heiles Potentialmentioning

confidence: 82%

Approximate integrals of motion

Bienaymé

Traven

2013

A&A

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“…Contrastingly, our method is free from such a spurious diverging behavior and works even for such highly-excited molecules. Some other possible methods to improve validity ranges are using different styles of normalization [88] and using Kolmogorov normal form [89,90]. Both of the methods can be used combining with our method.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

Teramoto¹,

Toda²,

Komatsuzaki³

2014

Gregory S. Ezra

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Validity ranges of Lie Canonical Perturbation Theory (LCPT) are investigated in terms of non blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non blow-up regions enclose the region inside the separatrix of the Hamiltonian but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non blow-up regions do not necessarily cover the whole regions inside the HCN well.We propose a new perturbation method to improve non blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in [1,2].

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Improved accuracy of the Birkhoff-Gustavson normal form and its convergence properties

Cited by 22 publications

References 16 publications

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

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

Approximate integrals of motion

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

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