Lie-point symmetries in bifurcation problems
Cicogna, G. ; Gaeta, G.
Annales de l'I.H.P. Physique théorique, Tome 56 (1992), p. 375-414 / Harvested from Numdam
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Publié le : 1992-01-01
@article{AIHPA_1992__56_4_375_0,
     author = {Cicogna, G. and Gaeta, Giuseppe},
     title = {Lie-point symmetries in bifurcation problems},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {56},
     year = {1992},
     pages = {375-414},
     mrnumber = {1175476},
     zbl = {0812.34032},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1992__56_4_375_0}
}

Cicogna, G.; Gaeta, G. Lie-point symmetries in bifurcation problems. Annales de l'I.H.P. Physique théorique, Tome 56 (1992) pp. 375-414. http://gdmltest.u-ga.fr/item/AIHPA_1992__56_4_375_0/

[Arn] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, Berlin, 1983. | MR 695786

[BC] G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations, Springer, New York, 1974. | MR 460846 | Zbl 0292.35001

[BK] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989. | MR 1006433 | Zbl 0698.35001

[CH] S.N. Chow and J. Hale, Methods of Bifurcation Theory, Springer, New York, 1982. | MR 660633 | Zbl 0487.47039

[CHW] B. Champagne, W. Hereman and P. Winternitz, The Computer Calculation of Lie Point Symmetries of Large Systems of Differential Equations, Preprint CRM-1689, Montreal, 1990. | MR 1125408

[Cic] G. Cicogna, A Nonlinear Version of the Equivariant Bifurcation Lemma, J. Phys. A, Vol. 23, 1990, p. L1339. | MR 1090003 | Zbl 0737.34024

[CG86] G. Cicogna and G. Gaeta, Spontaneous Linearization and Periodic Solutions in Hopf and Symmetric Bifurcations, Phys. Lett. A, Vol. 116, 1986, p. 303. | MR 847857

[CG90] G. Cicogna and G. Gaeta, Lie-Point Symmetries and Poincare' Normal Forms for Dynamical Systems, J. Phys. A., Vol. 23, 1990, p. L799; see also: Lie-Point Symmetries for Autonomous Symmetries and Resonance, Preprint 1991, to appear in J. Phys. A. | MR 1069467 | Zbl 0711.34058

[CK] J.D. Crawford and E. Knobloch, Symmetry and Symmetry-Breaking Bifurcations in Fluid Dynamics, Ann. Rev. Fluid Mech, Vol. 23, 1991, p. 341. | MR 1090332 | Zbl 0717.76007

[Cra] J.D. Crawford, M. Golubitsky, M.G.M. Gomes, E. Knobloch and I. Stewart, Boundary Conditions as Symmetry Constraints, in Singularity Theory and its Applications, Warwick, 1989, R. M. ROBERTS and I. STEWART Eds., Lecture Notes Math., No. 1463, Springer, Berlin, 1991. | MR 1129048 | Zbl 0735.35007

[Ehr] C. Ehresmann, Les prolongements d'une variété différentiable, I-V, C. R. Acad. Sci. Paris: I. Calcul des jets, prolongement principal, 233, 1951, p.598; II. L'espace des jets d'ordre r de Vn dans Vm, 233, 1951, p. 777; III. Transitivité des prolongements, 233, 1951, p. 1081; IV. Éléments de contact et éléments d'enveloppe, 234, 1952, p. 1028; V. Covariants différentiels et prolongements d'une structure infinitésimale, 234, 1952, p. 1424; Structures locales et structures infinitésimales, 234, 1952, p.587; Extension du calcul des jets aux jets non holonomes, 239, 1954, p. 1762; Applications de la notion de jet non holonome, 240, 1955, p. 397; Les prolongements d'un espace fibré différentiable, 240, 1955, p. 1755. Les prolongements d'une variété différentiable, Atti del IV congresso dell'Unione Matematica Italiana, 1951. | MR 66734 | Zbl 0043.17401

[Elp] C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet and G. Iooss, A Simple Global Characterization for Normal Forms of Singular Vector Fields, Physica D, Vol. 29, 1987, p. 95. | MR 923885 | Zbl 0633.58020

[Gae] G. Gaeta, Bifurcation Theory and Nonlinear Symmetry, Nonlinear Analysis, Vol. 17, 1991, p. 825. | Zbl 0776.58026

[Gae90a] G. Gaeta, Bifurcation and Symmetry Breaking, Phys. Rep., Vol. 189, 1990, p. 1. | MR 1057580

[Gae90b] G. Gaeta, Lie Point Symmetries and Periodic Solutions for Autonomous ODE, Preprint, C. P. Th., Palaiseau, 1990; Autonomous Dynamical Systems, LPTI Symmetries, Topology of Trajectories, and Periodic Solutions; Preprint, C. P. Th., Palaiseau, 1991.

[Gae91] G. Gaeta, Michel Theorem and Critical Sections of Gauge Functionals, Preprint, C. P. Th., Palaiseau, 1991.

[GMS] M. Golubitsky, J. Marsden and D. Schaeffer, Bifurcation Problems with Hidden Symmetry, in Partial Differential Equations and Dynamical Systems, W. E. FITZGIBBON Ed., Res. Notes Math., Vol. 101, Pitman, Boston, 1984. | Zbl 0567.58004

[GG] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, New York, 1973. | MR 341518 | Zbl 0294.58004

[GS] M. Golubitsky and I. Stewart, Hopf Bifurcation in the Presence of Symmetry, Arch. Rat. Mech. Anal., Vol. 87, 1985, p. 107. | MR 765596 | Zbl 0588.34030

[GSS] M. Golubitsky, I. Stewart and D. Schaeffer, Singularity and Groups in Bifurcation Theory, Vol. II, Springer, New York, 1988. | Zbl 0691.58003

[GH] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer, New York, 1983. | MR 709768 | Zbl 0515.34001

[HPS] M. Hirsch, D. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, No. 583, Springer, New York, 1975. | MR 650811 | Zbl 0355.58009

[IJ] G. Ioos and D.D. Joseph, Elementary Stability and Bifurcation Theory, Springer, Berlin, 1981; second edition, 1990. | MR 1026101 | Zbl 0691.34002

[JS] D.D. Joseph and D.H. Sattinger, Bifurcating Time Periodic Solutions and Their Stability, Arch. Rat. Mech. Anal., Vol. 45, 1972, p. 75. | MR 387844 | Zbl 0239.76057

[MM] J. E. MARSDEN and M. MCCRACKEN Eds., The Hopf Bifurcation and its Applications, Springer, New York, 1976. | MR 494309 | Zbl 0346.58007

[Olv] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986. | MR 836734 | Zbl 0588.22001

[Ovs] L.V. Ovsjannikov, Group Properties of Differential Equations (in russian), Novosibirsk, 1962, Group Analysis of Differential Equations, Academic Press, New York, 1982. | MR 142695

[Rue73] D. Ruelle, Bifurcations in the Presence of a Symmetry Group, Arch. Rat. Mech. Anal., Vol. 51, 1973, p. 136. | MR 348796 | Zbl 0259.58009

[Rue89] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, London, 1989. | MR 982930 | Zbl 0684.58001

[Sat73] D.H. Sattinger, Topics in Stability and Bifurcation Theory, Lect. Notes Math., No. 309, Springer, Berlin, 1973. | MR 463624 | Zbl 0248.35003

[Sat79] D.H. Sattinger, Group Theoretic Methods in Bifurcation Theory, Lect. Notes Math., No. 762, Springer, Berlin, 1979. | MR 551626 | Zbl 0414.58013

[Sat83] D.H. Sattinger, Branching in the Presence of Symmetry, S.I.A.M., Philadelphia, 1983. | MR 764932 | Zbl 0516.58016

[SW] D.H. Sattinger and O. Weaver, Lie Groups and Algebras, Springer, New York, 1986. | MR 835009 | Zbl 0595.22017

[Van] A. Vanderbauwhede, Local Bifurcation and Symmetry, Pitman, Boston, 1982. | MR 697724 | Zbl 0539.58022

[Vin84] A.M. Vinogradov, Local Symmetries and Conservation Laws, Acta Appl. Math., Vol. 2, 1984, p. 21. | MR 736872 | Zbl 0534.58005

[Vin89] A.M. Vinogradov, Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results, Acta Appl. Math., Vol. 15, 1989, p. 3. | MR 1007340 | Zbl 0692.35002

[Win] P. Winternitz, Group Theory and Exact Solutions of Partially Integrable Differential Systems, in: Partially Integrable Evolution Equations in Physics, R. CONTE and N. BOCCARA Eds., Kluwer, 1990. | MR 1073563 | Zbl 0726.35127

[Wul] C.E. Wulfman, Limit Cycles as Invariant Functions of Lie Groups, J. Phys. A, Vol. 12, 1979, p. L73. | MR 523806 | Zbl 0398.34026