Universal unfolding of symmetric resonances. (English) Zbl 1298.37042
Summary: We present the analysis of the bifurcation sequences of a family of resonant 2-DOF Hamiltonian systems invariant under spatial mirror symmetry and time reversion. The phase-space structure is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff-Gustavson normal form. Thresholds for the bifurcations of periodic orbits in generic position are computed as asymptotic series in terms of physical parameters of the original system.
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 70K45 | Normal forms for nonlinear problems in mechanics |
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
Keywords:
finite-dimensional Hamiltonian systems; perturbation theory; normal forms; bifurcation sequencesReferences:
| [1] | Belmonte, C., Boccaletti, D., Pucacco, G.: Stability of axial orbits in galactic potentials. Celest. Mech. Dyn. Astron. 95, 101-116 (2006) · Zbl 1219.70032 · doi:10.1007/s10569-006-9015-z |
| [2] | Belmonte, C., Boccaletti, D., Pucacco, G.: On the orbit structure of the logarithmic potential. Astrophys. J. 669, 202-217 (2007) · doi:10.1086/521423 |
| [3] | Broer, H.W., Vegter, G.: Bifurcational aspects of parametric resonance. Dyn. Report. New Ser. 1, 1-51 (1992) · Zbl 0893.58043 |
| [4] | Broer, H.W., Chow, S.N., Kim, Y., Vegter, G.: A normally elliptic Hamiltonian bifurcation. Z. Angew. Math. Phys. 44, 389-432 (1993) · Zbl 0805.58047 · doi:10.1007/BF00953660 |
| [5] | Broer, H.W., Lunter, G.A., Vegter, G.: Equivariant singularity theory with distinguished parameters: two case studies of resonant Hamiltonian systems. Phys. D 112, 64-80 (1998a) · Zbl 1194.34068 · doi:10.1016/S0167-2789(97)00202-9 |
| [6] | Broer, H.W., Hoveijn, I., Lunter, G.A., Vegter, G.: Resonances in a spring pendulum: algorithms for equivariant singularity theory. Nonlinearity 11, 1569-1605 (1998b) · Zbl 0932.37041 · doi:10.1088/0951-7715/11/6/009 |
| [7] | Broer, H.W., Hoveijn, I., Lunter, G.A., Vegter, G.: Bifurcations in Hamiltonian Systems: Computing Singularites by Gröbner Bases, Lecture Notes in Mathematics, vol. 1806. Springer, Berlin (2003) · Zbl 1029.37033 · doi:10.1007/b10414 |
| [8] | Celletti, A., Pucacco, G., Stella, D.: Lissajous and Halo orbits in the restricted three-body problem. J. Nonlinear Dyn. Syst. (2014) · Zbl 1344.70022 |
| [9] | Cicogna, G., Gaeta, G.: Symmetry and Perturbation Theory in Nonlinear Dynamics. Springer, Berlin (1999) · Zbl 1059.37044 |
| [10] | Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer, Berlin (2004) · Zbl 1041.85001 |
| [11] | Deprit, A., Elipe, A.: The Lissajous transformation. II. Normalization. Celest. Mech. Dyn. Astron. 51, 227-250 (1991) · Zbl 0756.70015 · doi:10.1007/BF00051692 |
| [12] | Ferrer, F., Hanßmann, H., Palacián, J., Yanguas, P.: On perturbed oscillators in 1:1:1 resonance: the case of axially symmetric cubic potentials. J. Geom. Phys. 40, 320-369 (2002) · Zbl 1037.34031 · doi:10.1016/S0393-0440(01)00041-9 |
| [13] | Giorgilli, A.: Notes on Exponential Stability of Hamiltonian Systems. Centro di Ricerca Matematica E. De Giorgi, Pisa (2002) |
| [14] | Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, 1, Applied Mathematical Sciences, vol. 51. Springer, Berlin (1985) · Zbl 0607.35004 · doi:10.1007/978-1-4612-5034-0 |
| [15] | Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, 2, Applied Mathematical Sciences, vol. 69. Springer, Berlin (1988) · Zbl 0691.58003 · doi:10.1007/978-1-4612-4574-2 |
| [16] | Hanßmann, H.: Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems, Lecture Notes in Mathematics 1893. Springer, Berlin (2007) · Zbl 1120.37036 |
| [17] | Hanßmann, H., Sommer, B.: A degenerate bifurcation in the Hénon-Heiles family. Celest. Mech. Dyn. Astron. 81, 249-261 (2001) · Zbl 0999.70012 · doi:10.1023/A:1013252302027 |
| [18] | Henrard, J.: Periodic orbits emanating from a resonant equilibrium. Celest. Mech. 1, 437 (1969) · Zbl 0193.25302 · doi:10.1007/BF01231143 |
| [19] | Kas, A., Schlessinger, M.: On the versal deformation of a complex space with an isolated singularity. Math. Ann. 196, 23-29 (1972) · Zbl 0242.32014 |
| [20] | Kummer, M.: On resonant non linearly coupled oscillators with two equal frequencies. Commun. Math. Phys. 48, 53-79 (1976) · Zbl 0321.58017 · doi:10.1007/BF01609411 |
| [21] | Marchesiello, A., Pucacco, G.: Relevance of the 1:1 resonance in galactic dynamics. Eur. Phys. J. Plus 126, 104 (2011) · doi:10.1140/epjp/i2011-11104-y |
| [22] | Marchesiello, A., Pucacco, G.: The symmetric 1:2 resonance. Eur. Phys. J. Plus 128, 21 (2013) · doi:10.1140/epjp/i2013-13021-5 |
| [23] | Marchesiello, A., Pucacco, G.: Resonances and bifurcations in systems with elliptical equipotentials. MNRAS 428, 2029-2038 (2013) · doi:10.1093/mnras/sts174 |
| [24] | Marchesiello, A., Pucacco, G.: Equivariant singularity analysis of the 2:2 resonance. Nonlinearity 27, 43-66 (2014) · Zbl 1418.37091 · doi:10.1088/0951-7715/27/1/43 |
| [25] | Martinet, J.: Singularities of Smooth Functions and Maps, LMS Lecture Note Series, vol. 58. Cambridge University Press, Cambridge (1982) · Zbl 0522.58006 |
| [26] | Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the N Body Problem, Applied Mathematica Sciences, vol. 90. Springer, Berlin (2009) · Zbl 1179.70002 |
| [27] | Montaldi, J., Roberts, M., Stewart, I.: Existence of nonlinear normal modes of symmetric Hamiltonian systems. Nonlinearity 3, 695-730 (1990) · Zbl 0711.58011 · doi:10.1088/0951-7715/3/3/009 |
| [28] | Pucacco, G., Boccaletti, D., Belmonte, C.: Quantitative predictions with detuned normal forms. Celest. Mech. Dyn. Astron. 102, 163-176 (2008) · Zbl 1154.70328 · doi:10.1007/s10569-008-9141-x |
| [29] | Pucacco, G.: Resonances and bifurcations in axisymmetric scale-free potentials. MNRAS 399, 340-348 (2009) · doi:10.1111/j.1365-2966.2009.15284.x |
| [30] | Pucacco, G., Marchesiello, A.: An energy-momentum map for the time-reversal symmetric 1:1 resonance with \[{\mathbb{Z}}_2\times{\mathbb{Z}}_2\] Z2×Z2 symmetry. Phys. D 271, 10-18 (2014) · Zbl 1356.37066 · doi:10.1016/j.physd.2013.12.009 |
| [31] | Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer, Berlin (2007) · Zbl 1128.34001 |
| [32] | Schmidt, D.S.: Periodic solutions near a resonant equilibrium of a Hamiltonian system. Celest. Mech. 9, 81-103 (1974) · Zbl 0313.70020 · doi:10.1007/BF01236166 |
| [33] | Tuwankotta, J.M., Verhulst, F.: Symmetry and resonance in Hamiltonian systems. SIAM J. Appl. Math. 61, 1369-1385 (2000) · Zbl 1004.70015 |
| [34] | Verhulst, F.: Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies. Philos. Trans. R. Soc. Lond. Ser. A 290, 435 (1979) · Zbl 0394.34039 · doi:10.1098/rsta.1979.0006 |
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