메뉴 건너뛰기




Volumn 40, Issue 3-4, 2002, Pages 320-369

On perturbed oscillators in 1–1–1 resonance: the case of axially symmetric cubic potentials

Author keywords

Axial symmetry; Genuine resonance; Invariant tori; Invariants; Normal forms; Periodic orbits; Reconstruction of the flow; Reductions; Relative equilibria

Indexed keywords


EID: 0005134856     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(01)00041-9     Document Type: Article
Times cited : (31)

References (62)
  • 1
    • 85018258484 scopus 로고
    • Foundations of Mechanics, 2nd Edition, Benjamin, New York
    • [1] R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd Edition, Benjamin, New York, 1978.
    • (1978)
    • Abraham, R.1    Marsden, J.E.2
  • 2
    • 0001167764 scopus 로고
    • On the stability of isolating integrals. I. Effect of the perturbation in the potential function
    • [2] Aizawa, Y., Saitô, N., On the stability of isolating integrals. I. Effect of the perturbation in the potential function. J. Phys. Soc. Jpn. 32 (1972), 1636–1640.
    • (1972) J. Phys. Soc. Jpn. , vol.32 , pp. 1636-1640
    • Aizawa, Y.1    Saitô, N.2
  • 3
    • 0002356196 scopus 로고
    • Small denominators and problems of stability of motion in classical and celestial mechanics
    • [3] Arnol'd, V.I., Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surv. 18:6 (1963), 85–191.
    • (1963) Russ. Math. Surv. , vol.18 , Issue.6 , pp. 85-191
    • Arnol'd, V.I.1
  • 4
    • 85018244031 scopus 로고
    • Mathematical Methods of Classical Mechanics, Springer, Berlin
    • [4] V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978.
    • (1978)
    • Arnol'd, V.I.1
  • 5
    • 0002321444 scopus 로고
    • A proof of the isoenergetic KAM-theorem from the ordinary one
    • [5] Broer, H.W., Huitema, G.B., A proof of the isoenergetic KAM-theorem from the ordinary one. J. Diff. Eq. 90:1 (1991), 52–60.
    • (1991) J. Diff. Eq. , vol.90 , Issue.1 , pp. 52-60
    • Broer, H.W.1    Huitema, G.B.2
  • 6
    • 85018241966 scopus 로고    scopus 로고
    • Quasi-periodic Motions in Families of Dynamical Systems Order Amidst Chaos, Lecture Notes in Mathematics, Vol. 1645, Springer, Berlin
    • [6] H.W. Broer, G.B. Huitema, M.B. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems Order Amidst Chaos, Lecture Notes in Mathematics, Vol. 1645, Springer, Berlin, 1996.
    • (1996)
    • Broer, H.W.1    Huitema, G.B.2    Sevryuk, M.B.3
  • 7
    • 0002208766 scopus 로고
    • Bifurcational aspects of parametric resonance
    • [7] Broer, H.W., Vegter, G., Bifurcational aspects of parametric resonance. Dyn. Rep. New Ser. 1 (1992), 1–53.
    • (1992) Dyn. Rep. New Ser. , vol.1 , pp. 1-53
    • Broer, H.W.1    Vegter, G.2
  • 8
    • 85018320317 scopus 로고
    • The 1:1 semisimple resonance, Ph.D. Thesis, University of California, Santa Cruz
    • [8] C. Cotter, The 1:1 semisimple resonance, Ph.D. Thesis, University of California, Santa Cruz, 1986.
    • (1986)
    • Cotter, C.1
  • 9
    • 85018261643 scopus 로고
    • Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, Berlin
    • [9] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, Berlin, 1992.
    • (1992)
    • Cox, D.1    Little, J.2    O'Shea, D.3
  • 10
    • 0010162361 scopus 로고
    • Geometry of the bifurcations of the normalized reduced Hénon–Heiles family
    • [10] Cushman, R., Geometry of the bifurcations of the normalized reduced Hénon–Heiles family. Proc. R. Soc. London A 382 (1982), 361–371.
    • (1982) Proc. R. Soc. London A , vol.382 , pp. 361-371
    • Cushman, R.1
  • 11
    • 85018365235 scopus 로고    scopus 로고
    • Global Aspects of Classical Integrable Systems, Birkhauser, Basel
    • [11] R. Cushman, L.M. Bates, Global Aspects of Classical Integrable Systems, Birkhauser, Basel, 1997.
    • (1997)
    • Cushman, R.1    Bates, L.M.2
  • 12
    • 0033099574 scopus 로고    scopus 로고
    • Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator
    • [12] Cushman, R., Ferrer, S., Hanßmann, H., Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator. Nonlinearity 12 (1999), 389–410.
    • (1999) Nonlinearity , vol.12 , pp. 389-410
    • Cushman, R.1    Ferrer, S.2    Hanßmann, H.3
  • 13
    • 0000717287 scopus 로고
    • Canonical transformations depending on a small parameter
    • [13] Deprit, A., Canonical transformations depending on a small parameter. Celes. Mech. 1 (1969), 12–30.
    • (1969) Celes. Mech. , vol.1 , pp. 12-30
    • Deprit, A.1
  • 14
    • 0001289181 scopus 로고
    • The elimination of the parallax in satellite theory
    • [14] Deprit, A., The elimination of the parallax in satellite theory. Celes. Mech. 24 (1981), 111–153.
    • (1981) Celes. Mech. , vol.24 , pp. 111-153
    • Deprit, A.1
  • 15
    • 34249920395 scopus 로고
    • The Lissajous transformation — I. Basics
    • [15] Deprit, A., The Lissajous transformation — I. Basics. Celes. Mech. Dyn. Astron. 51 (1991), 201–225.
    • (1991) Celes. Mech. Dyn. Astron. , vol.51 , pp. 201-225
    • Deprit, A.1
  • 16
    • 0001364235 scopus 로고
    • The Lissajous transformation — II. Normalization
    • [16] Deprit, A., Elipe, A., The Lissajous transformation — II. Normalization. Celes. Mech. Dyn. Astron. 51 (1991), 227–250.
    • (1991) Celes. Mech. Dyn. Astron. , vol.51 , pp. 227-250
    • Deprit, A.1    Elipe, A.2
  • 17
    • 0002018756 scopus 로고
    • Non-integrability of the 1:1:2-resonance
    • [17] Duistermaat, J.J., Non-integrability of the 1:1:2-resonance. Ergod. Theory Dyn. Syst. 4 (1984), 553–568.
    • (1984) Ergod. Theory Dyn. Syst. , vol.4 , pp. 553-568
    • Duistermaat, J.J.1
  • 18
    • 0032340303 scopus 로고    scopus 로고
    • The monodromy of the Hamiltonian Hopf bifurcation
    • [18] Duistermaat, J.J., The monodromy of the Hamiltonian Hopf bifurcation. Z. Angew. Math. Phys. 49 (1998), 156–161.
    • (1998) Z. Angew. Math. Phys. , vol.49 , pp. 156-161
    • Duistermaat, J.J.1
  • 22
    • 0032094211 scopus 로고    scopus 로고
    • The Hénon and Heiles problem in three dimensions. II. Relative equilibria and bifurcations in the reduced system
    • [22] Ferrer, S., Lara, M., Palacián, J., San Juan, J.F., Viartola, A., Yanguas, P., The Hénon and Heiles problem in three dimensions. II. Relative equilibria and bifurcations in the reduced system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 8 (1998), 1215–1229.
    • (1998) Int. J. Bifurc. Chaos Appl. Sci. Eng. , vol.8 , pp. 1215-1229
    • Ferrer, S.1    Lara, M.2    Palacián, J.3    San Juan, J.F.4    Viartola, A.5    Yanguas, P.6
  • 24
    • 0005210494 scopus 로고    scopus 로고
    • Hamiltonian oscillators in 1–1–1 resonance: normalization and integrability
    • [24] Ferrer, S., Palacián, J., Yanguas, P., Hamiltonian oscillators in 1–1–1 resonance: normalization and integrability. J. Nonlinear Sci. 10 (2000), 145–174.
    • (2000) J. Nonlinear Sci. , vol.10 , pp. 145-174
    • Ferrer, S.1    Palacián, J.2    Yanguas, P.3
  • 26
    • 85018254446 scopus 로고
    • Die Lie–Reihen und ihre Anwendungen, Verlag der Wissenschaften, Berlin
    • [26] W. Gröbner, Die Lie–Reihen und ihre Anwendungen, Verlag der Wissenschaften, Berlin, 1967.
    • (1967)
    • Gröbner, W.1
  • 27
    • 85018334209 scopus 로고
    • Chaos in Classical and Quantum Mechanics: Interdisciplinary Applied Mathematics, Vol. 1, Springer, New York
    • [27] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics: Interdisciplinary Applied Mathematics, Vol. 1, Springer, New York, 1990.
    • (1990)
    • Gutzwiller, M.C.1
  • 28
    • 0000318075 scopus 로고    scopus 로고
    • Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems
    • [28] Haller, G., Wiggins, S., Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems. Physica D 90 (1996), 319–365.
    • (1996) Physica D , vol.90 , pp. 319-365
    • Haller, G.1    Wiggins, S.2
  • 29
    • 85018264239 scopus 로고
    • Quasi-periodic motions of a rigid body — a case study on perturbations of superintegrable systems, Ph.D. Thesis, Rijksuniversiteit Groningen
    • [29] H. Hanßmann, Quasi-periodic motions of a rigid body — a case study on perturbations of superintegrable systems, Ph.D. Thesis, Rijksuniversiteit Groningen, 1995.
    • (1995)
    • Hanßmann, H.1
  • 30
    • 85018274863 scopus 로고    scopus 로고
    • Equivariant perturbations of the Euler top
    • H.W. Broer, et al. (Eds.), Nonlinear Dynamical Systems and Chaos, Groningen, 1995, Progress in Nonlinear Differential Equations and Their Applications, Vol. 19, Birkhauser, Basel
    • [30] H. Hanßmann, Equivariant perturbations of the Euler top, in: H.W. Broer, et al. (Eds.), Nonlinear Dynamical Systems and Chaos, Groningen, 1995, Progress in Nonlinear Differential Equations and Their Applications, Vol. 19, Birkhauser, Basel, 1996, pp. 227–252.
    • (1996) , pp. 227-252
    • Hanßmann, H.1
  • 31
    • 85018357628 scopus 로고    scopus 로고
    • On the Hamiltonian Hopf bifurcations in the 3D Hénon–Heiles family
    • in preparation.
    • [31] H. Hanßmann, J.-C. van der Meer, On the Hamiltonian Hopf bifurcations in the 3D Hénon–Heiles family, in preparation.
    • Hanßmann, H.1    van der Meer, J.-C.2
  • 32
    • 85018249417 scopus 로고    scopus 로고
    • A degenerate bifurcation in the Hénon–Heiles family, Inst. Reine and Angew. Math., RWTH Aachen, Preprint
    • [32] H. Hanßmann, B. Sommer, A degenerate bifurcation in the Hénon–Heiles family, Inst. Reine and Angew. Math., RWTH Aachen, Preprint, 2000.
    • (2000)
    • Hanßmann, H.1    Sommer, B.2
  • 33
    • 0002519122 scopus 로고
    • The applicability of the third integral of motion: some numerical experiments
    • [33] Hénon, M., Heiles, C., The applicability of the third integral of motion: some numerical experiments. Astron. J. 69 (1964), 73–79.
    • (1964) Astron. J. , vol.69 , pp. 73-79
    • Hénon, M.1    Heiles, C.2
  • 34
    • 0001230718 scopus 로고
    • A generalisation of the Runge–Lenz constant of classical motion in a central potential
    • [34] Holas, A., March, N.H., A generalisation of the Runge–Lenz constant of classical motion in a central potential. J. Phys. A 23 (1990), 735–749.
    • (1990) J. Phys. A , vol.23 , pp. 735-749
    • Holas, A.1    March, N.H.2
  • 35
    • 85018345513 scopus 로고
    • Aspects of resonance in dynamical systems, Ph.D. Thesis, Universiteit Utrecht
    • [35] I. Hoveijn, Aspects of resonance in dynamical systems, Ph.D. Thesis, Universiteit Utrecht, 1993.
    • (1993)
    • Hoveijn, I.1
  • 36
    • 0042371863 scopus 로고
    • On the problem of degeneracy in quantum mechanics
    • [36] Jauch, J.M., Hill, E.L., On the problem of degeneracy in quantum mechanics. Phys. Rev. 57 (1940), 641–645.
    • (1940) Phys. Rev. , vol.57 , pp. 641-645
    • Jauch, J.M.1    Hill, E.L.2
  • 37
    • 38249019449 scopus 로고
    • On resonant classical Hamiltonians with n frequencies
    • [37] Kummer, M., On resonant classical Hamiltonians with n frequencies. J. Diff. Eq. 83 (1990), 220–243.
    • (1990) J. Diff. Eq. , vol.83 , pp. 220-243
    • Kummer, M.1
  • 38
    • 0040331855 scopus 로고
    • Painlevé analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type
    • [38] Lakshmanan, M., Sahadevan, R., Painlevé analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type. Phys. Rep. 224 (1993), 1–93.
    • (1993) Phys. Rep. , vol.224 , pp. 1-93
    • Lakshmanan, M.1    Sahadevan, R.2
  • 39
    • 85018366244 scopus 로고    scopus 로고
    • Bifurcations of Hamiltonian systems — computing singularities by Gröbner bases, Ph.D. Thesis, Rijksuniversiteit Groningen
    • [39] G.A. Lunter, Bifurcations of Hamiltonian systems — computing singularities by Gröbner bases, Ph.D. Thesis, Rijksuniversiteit Groningen, 1999.
    • (1999)
    • Lunter, G.A.1
  • 40
    • 85018267473 scopus 로고
    • Dynamical Systems — A Differential Geometric Approach to Symmetry and Reduction, Wiley, New York
    • [40] G. Marmo, E.J. Saletan, A. Simoni, B. Vitale, Dynamical Systems — A Differential Geometric Approach to Symmetry and Reduction, Wiley, New York, 1985.
    • (1985)
    • Marmo, G.1    Saletan, E.J.2    Simoni, A.3    Vitale, B.4
  • 41
    • 0031522248 scopus 로고    scopus 로고
    • KAM theorem for generic analytic perturbations of the Euler system
    • [41] Mazzocco, M., KAM theorem for generic analytic perturbations of the Euler system. Z. Angew. Math. Phys. 48 (1997), 193–219.
    • (1997) Z. Angew. Math. Phys. , vol.48 , pp. 193-219
    • Mazzocco, M.1
  • 42
    • 0004111695 scopus 로고
    • The Hamiltonian Hopf bifurcation
    • Vol. 1160, Springer, Berlin
    • [42] J.-C. van der Meer, The Hamiltonian Hopf bifurcation, Lecture Notes in Mathematics, Vol. 1160, Springer, Berlin, 1985.
    • (1985) Lecture Notes in Mathematics
    • van der Meer, J.-C.1
  • 43
    • 84966219641 scopus 로고
    • Generic bifurcation of periodic points
    • [43] Meyer, K.R., Generic bifurcation of periodic points. Trans. AMS 149 (1970), 95–107.
    • (1970) Trans. AMS , vol.149 , pp. 95-107
    • Meyer, K.R.1
  • 44
    • 0000457036 scopus 로고
    • Normal forms for the general equilibrium
    • [44] Meyer, K.R., Normal forms for the general equilibrium. Funkcial. Ekva. 27 (1984), 261–271.
    • (1984) Funkcial. Ekva. , vol.27 , pp. 261-271
    • Meyer, K.R.1
  • 45
    • 0011280146 scopus 로고
    • The Lissajous transformation — III. Parametric bifurcations
    • [45] Miller, B.R., The Lissajous transformation — III. Parametric bifurcations. Celes. Mech. Dyn. Astron. 51 (1991), 251–270.
    • (1991) Celes. Mech. Dyn. Astron. , vol.51 , pp. 251-270
    • Miller, B.R.1
  • 46
    • 84980148530 scopus 로고
    • Regularization of Kepler's problem and the averaging method on a manifold
    • [46] Moser, J., Regularization of Kepler's problem and the averaging method on a manifold. Commun. Pure Appl. Math. 23 (1970), 609–636.
    • (1970) Commun. Pure Appl. Math. , vol.23 , pp. 609-636
    • Moser, J.1
  • 47
    • 0000234399 scopus 로고
    • Action–angle variables and their generalizations
    • [47] Nehorošev, N.N., Action–angle variables and their generalizations. Trans. Mosc. Math. Soc. 26 (1972), 180–198.
    • (1972) Trans. Mosc. Math. Soc. , vol.26 , pp. 180-198
    • Nehorošev, N.N.1
  • 48
    • 9344225119 scopus 로고    scopus 로고
    • A note on focus–focus singularities
    • [48] Nguyen, T.Z., A note on focus–focus singularities. Diff. Geom. Appl. 7:2 (1997), 123–130.
    • (1997) Diff. Geom. Appl. , vol.7 , Issue.2 , pp. 123-130
    • Nguyen, T.Z.1
  • 49
    • 0030209313 scopus 로고    scopus 로고
    • Visualizing relative equilibria and bifurcations by painting Hamiltonians on personal computers
    • [49] Pascua, P., Rubio, J.L., Viartola, A., Ferrer, S., Visualizing relative equilibria and bifurcations by painting Hamiltonians on personal computers. Int. J. Bifurc. Chaos Appl. Sci. Eng. 6 (1996), 1411–1424.
    • (1996) Int. J. Bifurc. Chaos Appl. Sci. Eng. , vol.6 , pp. 1411-1424
    • Pascua, P.1    Rubio, J.L.2    Viartola, A.3    Ferrer, S.4
  • 50
    • 45349111974 scopus 로고
    • The Painlevé property and singularity analysis of integrable and non-integrable systems
    • [50] Ramani, A., Grammaticos, B., Bountis, T., The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys. Rep. 180 (1989), 159–245.
    • (1989) Phys. Rep. , vol.180 , pp. 159-245
    • Ramani, A.1    Grammaticos, B.2    Bountis, T.3
  • 51
    • 0001369721 scopus 로고
    • A formal third integral of motion in a nearly spherical stellar system
    • [51] Saaf, A.F., A formal third integral of motion in a nearly spherical stellar system. Astrophys. J. 154 (1968), 483–498.
    • (1968) Astrophys. J. , vol.154 , pp. 483-498
    • Saaf, A.F.1
  • 52
    • 85018346244 scopus 로고
    • Averaging Methods in Nonlinear Dynamical Systems, Springer, Berlin
    • [52] J.A. Sanders, F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer, Berlin, 1985.
    • (1985)
    • Sanders, J.A.1    Verhulst, F.2
  • 53
    • 0007018954 scopus 로고
    • Singularities of functions on boundaries, corners, etc.
    • [53] Siersma, D., Singularities of functions on boundaries, corners, etc. Quart. J. Math. Oxford 32:2 (1981), 119–127.
    • (1981) Quart. J. Math. Oxford , vol.32 , Issue.2 , pp. 119-127
    • Siersma, D.1
  • 54
    • 85018324651 scopus 로고    scopus 로고
    • Über eine Bifurkation der Hénon–Heiles-Familie, Diplomarbeit, RWTH Aachen
    • [54] B. Sommer, Über eine Bifurkation der Hénon–Heiles-Familie, Diplomarbeit, RWTH Aachen, 1997.
    • (1997)
    • Sommer, B.1
  • 55
    • 0000744221 scopus 로고
    • Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies
    • [55] Verhulst, F., Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies. Phil. Trans. R. Soc. London A 290 (1979), 435–465.
    • (1979) Phil. Trans. R. Soc. London A , vol.290 , pp. 435-465
    • Verhulst, F.1
  • 56
    • 85018267328 scopus 로고
    • A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge
    • [56] E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge, 1904.
    • (1904)
    • Whittaker, E.T.1
  • 57
    • 85018261670 scopus 로고    scopus 로고
    • The Mathematica Book, 3rd Edition, Wolfram Media/Cambridge University Press, Cambridge
    • [57] S. Wolfram, The Mathematica Book, 3rd Edition, Wolfram Media/Cambridge University Press, Cambridge, 1996.
    • (1996)
    • Wolfram, S.1
  • 58
    • 85018366126 scopus 로고    scopus 로고
    • Integrability, normalization and symmetries of Hamiltonian systems in 1–1–1 resonance, Ph.D. Thesis, Universidad Pública de Navarra
    • [58] P. Yanguas, Integrability, normalization and symmetries of Hamiltonian systems in 1–1–1 resonance, Ph.D. Thesis, Universidad Pública de Navarra, 1998.
    • (1998)
    • Yanguas, P.1
  • 59
    • 0000858395 scopus 로고
    • Motion in the core of a triaxial potential
    • [59] de Zeeuw, T., Motion in the core of a triaxial potential. Mon. Not. R. Astron. Soc. 215 (1985), 731–760.
    • (1985) Mon. Not. R. Astron. Soc. , vol.215 , pp. 731-760
    • de Zeeuw, T.1
  • 60
    • 85018264256 scopus 로고
    • Zeeuw, Dynamical models for axisymmetric and triaxial galaxies, in: T. de Zeeuw, (Ed.), Structure and Dynamics of Elliptical Galaxies, IAU Symp., Vol. 127, Reidel, Dordrecht
    • [60] T. de Zeeuw, Dynamical models for axisymmetric and triaxial galaxies, in: T. de Zeeuw, (Ed.), Structure and Dynamics of Elliptical Galaxies, IAU Symp., Vol. 127, Reidel, Dordrecht, 1987, pp. 271–290.
    • (1987) , pp. 271-290
    • de, T.1
  • 61
    • 12044257911 scopus 로고
    • Structure and dynamics of elliptical galaxies
    • [61] de Zeeuw, T., Franx, M., Structure and dynamics of elliptical galaxies. Annu. Rev. Astron. Astrophys. 29 (1991), 239–274.
    • (1991) Annu. Rev. Astron. Astrophys. , vol.29 , pp. 239-274
    • de Zeeuw, T.1    Franx, M.2
  • 62
    • 0007247629 scopus 로고
    • Stellar orbits in a triaxial galaxy. I. Orbits in the plane of rotation
    • [62] de Zeeuw, T., Merritt, D., Stellar orbits in a triaxial galaxy. I. Orbits in the plane of rotation. Astrophys. J. 267 (1983), 571–595.
    • (1983) Astrophys. J. , vol.267 , pp. 571-595
    • de Zeeuw, T.1    Merritt, D.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.