Quantitative predictions with detuned normal forms. (English) Zbl 1154.70328
Summary: The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.
MSC:
| 70H05 | Hamilton’s equations |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 70F15 | Celestial mechanics |
| 85A05 | Galactic and stellar dynamics |
References:
| [1] | Belmonte C., Boccaletti D., Pucacco G. (2006) Stability of axial orbits in galactic potentials Celest. Mech. Dyn. Astr. 95: 101–116 · Zbl 1219.70032 · doi:10.1007/s10569-006-9015-z |
| [2] | Belmonte C., Boccaletti D., Pucacco G. (2007) On the orbit structure of the logarithmic potential. Astrophys. J. 669: 202–217 · doi:10.1086/521423 |
| [3] | Bender C.M., Orszag S.A. (1978) Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York · Zbl 0417.34001 |
| [4] | Birkhoff, G.D.: Dynamical Systems. Amer. Math. Soc. Coll. Publ. 9. New York, USA (1927) · JFM 53.0732.01 |
| [5] | Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press (1987) · Zbl 1130.85301 |
| [6] | Boccaletti D., Pucacco G. (1999) Theory of Orbits: Vol. 2. Springer-Verlag, Berlin · Zbl 0927.70002 |
| [7] | Broer H.W., Hoveijn I., Lunter G., Vegter G. (2003) Bifurcations in Hamiltonian Systems, vol. 1806 of Lecture Notes in Mathematics. Springer-Verlag, Berlin · Zbl 1029.37033 |
| [8] | Contopoulos G. (1978) Higher order resonances in dynamical systems. Celest. Mech. 18: 195–204 · Zbl 0396.58002 · doi:10.1007/BF01228716 |
| [9] | Contopoulos G. (2004) Order and Chaos in Dynamical Astronomy. Springer-Verlag, Berlin · Zbl 1041.85001 |
| [10] | Contopoulos G., Efthymiopoulos C., Giorgilli A. (2003) Nonconvergence of formal integrals of motion. J. Phys. A: Math. Gen. 36: 8639–8660 · Zbl 1078.70019 · doi:10.1088/0305-4470/36/32/306 |
| [11] | Cushman R.H., Bates L.M. (1997) Global Aspects of Classical Integrable Systems. Birkhäuser, Boston · Zbl 0882.58023 |
| [12] | de Zeeuw T., Merritt D. (1983) Stellar orbits in a triaxial galaxy. I. Orbits in the plane of rotation. Astrophys. J. 267: 571–595 · doi:10.1086/160894 |
| [13] | Efthymiopoulos C., Giorgilli A., Contopoulos G. (2004) Nonconvergence of formal integrals: II. Improved estimates for the optimal order of truncation. J. Phys. A: Math. Gen. 37: 10831–10858 · Zbl 1094.70012 · doi:10.1088/0305-4470/37/45/008 |
| [14] | Fridman T., Merritt D. (1997) Periodic orbits in triaxial galaxies with weak cusps. Astron. J. 114: 1479–1487 · doi:10.1086/118578 |
| [15] | Gerhard O., Saha P. (1991) Recovering galactic orbits by perturbation theory. MNRAS 251: 449–467 · Zbl 0736.70006 |
| [16] | Giorgilli A. (2002) Notes on Exponential Stability of Hamiltonian Systems Centro di Ricerca Matematica E. De Giorgi, Pisa |
| [17] | Giorgilli, A., Locatelli, U.: Canonical perturbation theory for nearly integrable systems. In: Steves, B.A., et al. (eds.) Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems, NATO Science Series II-vol. 227, pp.1–41. Springer, Dordrecht (2006) · Zbl 1142.70009 |
| [18] | Gustavson F. (1966) On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71: 670–686 · doi:10.1086/110172 |
| [19] | Khovanskii A.N. (1963) The Application of Continued Fractions and their Generalizations to Problems in Approximation Theory. Noordhoff, Groningen |
| [20] | Kummer M. (1977) On resonant Hamiltonians with two degrees of freedom near an equilibrium point. Lect. Notes Phys. 93: 57–75 · doi:10.1007/BFb0021738 |
| [21] | Miralda-Escudé J., Schwarzschild M. (1989) On the orbit structure of the logarithmic potential. Astrophys. J. 339: 752–762 · doi:10.1086/167333 |
| [22] | Moser J. (1968) Lectures on Hamiltonian systems. Mem. Am. Math. Soc. 81: 1–60 · Zbl 0172.11401 |
| [23] | Sanders J.A., Verhulst F. (1985) Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York · Zbl 0586.34040 |
| [24] | Scuflaire R. (1995) Stability of axial orbits in analytic galactic potentials. Celest. Mech. & Dyn. Astron. 61: 261–285 · Zbl 0827.70007 · doi:10.1007/BF00051897 |
| [25] | Tuwankotta J.M., Verhulst F. (2000) Symmetry and resonance in Hamiltonian systems. SIAM J. Appl. Math. 61: 1369–1385 · Zbl 1004.70015 |
| [26] | Verhulst F. (1996) Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin · Zbl 0854.34002 |
| [27] | Yanguas P. (2001) Perturbations of the isochrone model. Nonlinearity 14: 1–34 · Zbl 0992.34029 · doi:10.1088/0951-7715/14/1/301 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
