Table of contents for this issue | Previous article | Next article
Jacques-Élie Furter; Angela Maria Sitta
Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems
(Formulation des chemins pour des problèmes de bifurcation multi-paramétriques $\mathbb{D}_3$-équivariants)
Annales de l'institut Fourier, 60 no. 4 (2010), p. 1363-1400, doi: 10.5802/aif.2558
Article PDF | Reviews MR 2722245 | Zbl 1204.37054
Class. Math.: 37G40, 58K70, 58K40, 34F10, 34F15
Keywords: Equivariant bifurcation, degenerate bifurcation, path formulation, singularity theory, 1:1-resonance, reversible systems, subharmonic bifurcation
[1] V I Arnold, “Wavefront evolution and equivariant Morse lemma”, Comm. Pure. App. Math. 29 (1976), p. 557-582 Article | MR 436200 | Zbl 0343.58003
[2] J M Ball & D G Schaeffer, “Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions”, Math. Proc. Camb. Phil. Soc. 94 (1983), p. 315-339 Article | MR 715037 | Zbl 0568.73057
[3] Thomas J. Bridges & Jacques E. Furter, Singularity theory and equivariant symplectic maps, Lecture Notes in Mathematics 1558, Springer-Verlag, Berlin, 1993 MR 1290781 | Zbl 0799.58009
[4] J. W. Bruce, “Functions on discriminants”, J. London Math. Soc. (2) 30 (1984) no. 3, p. 551-567 Article | MR 810963 | Zbl 0605.58011
[5] J. W. Bruce, A. A. du Plessis & C. T. C. Wall, “Determinacy and unipotency”, Invent. Math. 88 (1987) no. 3, p. 521-554 Article | MR 884799 | Zbl 0596.58005
[6] Ernesto Buzano, Giuseppe Geymonat & Tim Poston, “Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross-section invariant under the dihedral group $D_n$”, Arch. Rational Mech. Anal. 89 (1985) no. 4, p. 307-388 Article | MR 792535 | Zbl 0568.73048
[7] Maria-Cristina Ciocci, “Generalized Lyapunov-Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms”, J. Difference Equ. Appl. 10 (2004) no. 7, p. 621-649 Article | MR 2064813 | Zbl 1055.37059
[8] Maria-Cristina Ciocci, “Subharmonic branching at a reversible $1:1$ resonance”, J. Difference Equ. Appl. 11 (2005) no. 13, p. 1119-1135 Article | MR 2183010 | Zbl 1085.37045
[9] João Carlos Ferreira Costa & Angela Maria Sitta, Path formulation for $Z_2\oplus Z_2$-equivariant bifurcation problems, Real and complex singularities, Trends Math., Birkhäuser, 2007, p. 127–141 MR 2280136 | Zbl 1128.58020
[10] James Damon, “The unfolding and determinacy theorems for subgroups of $\mathcal{A}$ and $\mathcal{K}$”, Mem. Amer. Math. Soc. 50 (1984) no. 306 MR 748971 | Zbl 0545.58010
[11] James Damon, “Deformations of sections of singularities and Gorenstein surface singularities”, Amer. J. Math. 109 (1987) no. 4, p. 695-721 Article | MR 900036 | Zbl 0628.14003
[12] James Damon, “On the legacy of free divisors: discriminants and Morse-type singularities”, Amer. J. Math. 120 (1998) no. 3, p. 453-492 Article | MR 1623404 | Zbl 0910.32038
[13] Ana Paula S. Dias & Ana Rodrigues, “Secondary bifurcations in systems with all-to-all coupling. II”, Dyn. Syst. 21 (2006) no. 4, p. 439-463 MR 2273688 | Zbl 1118.34033
[14] J. E. Furter, “Geometric path formulation for bifurcation problems”, J. Natur. Geom. 12 (1997) no. 1 MR 1456087 | Zbl 0908.34030
[15] J-E. Furter, A. M. Sitta & I. Stewart, “Algebraic path formulation for equivariant bifurcation problems”, Mathematical Proceedings of the Cambridge Philosophical Society 124 (1998) no. 2, p. 275-304, Cited By (since 1996): 2 Article | MR 1631115 | Zbl 0920.58018
[16] Terence Gaffney, New methods in the classification theory of bifurcation problems, Multiparameter bifurcation theory (Arcata, Calif., 1985), Contemp. Math. 56, Amer. Math. Soc., 1986, p. 97–116 MR 855086 | Zbl 0625.58016
[17] Karin Gatermann, Computer algebra methods for equivariant dynamical systems, Lecture Notes in Mathematics 1728, Springer-Verlag, Berlin, 2000 MR 1755001 | Zbl 0944.65131
[18] Karin Gatermann & Reiner Lauterbach, “Automatic classification of normal forms”, Nonlinear Anal. 34 (1998) no. 2, p. 157-190 Article | MR 1635741 | Zbl 0947.34023
[19] Jean-Jacques Gervais, “Bifurcations of subharmonic solutions in reversible systems”, J. Differential Equations 75 (1988) no. 1, p. 28-42 Article | MR 957006 | Zbl 0664.34051
[20] M. Golubitsky & D. Schaeffer, “A theory for imperfect bifurcation via singularity theory”, Comm. Pure Appl. Math. 32 (1979) no. 1, p. 21-98 Article | MR 508917 | Zbl 0409.58007
[21] Martin Golubitsky & David Chillingworth, Bifurcation and planar pattern formation for a liquid crystal, Bifurcation, symmetry and patterns (Porto, 2000), Trends Math., Birkhäuser, 2003, p. 55–66 MR 2014355 | Zbl 1187.82132
[22] Martin Golubitsky & Mark Roberts, “A classification of degenerate Hopf bifurcations with ${\rm O}(2)$ symmetry”, J. Differential Equations 69 (1987) no. 2, p. 216-264 Article | MR 899161 | Zbl 0635.34036
[23] Martin Golubitsky & David Schaeffer, “Bifurcations with ${\rm O}(3)$ symmetry including applications to the Bénard problem”, Comm. Pure Appl. Math. 35 (1982) no. 1, p. 81-111 Article | MR 637496 | Zbl 0492.58012
[24] Martin Golubitsky, Ian Stewart & David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences 69, Springer-Verlag, New York, 1988 MR 950168 | Zbl 0691.58003
[25] Ali Lari-Lavassani & Yung-Chen Lu, “Equivariant multiparameter bifurcation via singularity theory”, J. Dynam. Differential Equations 5 (1993) no. 2, p. 189-218 Article | MR 1223447 | Zbl 0778.58014
[26] E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series 77, Cambridge University Press, Cambridge, 1984 MR 747303 | Zbl 0552.14002
[27] Jean Martinet, Déploiements versels des applications différentiables et classification des applications stables, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975), Springer, 1976, p. 1–44. Lecture Notes in Math., Vol. 535 MR 649264 | Zbl 0362.58004
[28] John N. Mather, “Stability of $C^{\infty }$ mappings. III. Finitely determined mapgerms”, Inst. Hautes Études Sci. Publ. Math. (1968) no. 35, p. 279-308 Numdam | MR 275459 | Zbl 0159.25001
[29] David Mond, Ciclos Evanescentes para las applicaciones Analíticas, Technical report, ICMSC-USP, São-Carlos, 1990
[30] David Mond & James Montaldi, Deformations of maps on complete intersections, Damon’s $\mathcal{K}_V$-equivalence and bifurcations, Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser. 201, Cambridge Univ. Press, 1994, p. 263–284 MR 1295079 | Zbl 0847.58007
[31] James Montaldi, The path formulation of bifurcation theory, Dynamics, bifurcation and symmetry (Cargèse, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 437, Kluwer Acad. Publ., 1994, p. 259–278 MR 1305382 | Zbl 0811.58015
[32] J. H. Rieger, “Versal topological stratification and the bifurcation geometry of map-germs of the plane”, Math. Proc. Cambridge Philos. Soc. 107 (1990) no. 1, p. 127-147 Article | MR 1021879 | Zbl 0696.58009
[33] J. H. Rieger, “$\mathcal{A}$-unimodal map-germs into the plane”, Hokkaido Math. J. 33 (2004) no. 1, p. 47-64 MR 2034807 | Zbl 1152.58314
[34] J. H. Rieger & M. A. S. Ruas, “Classification of $\mathcal{A}$-simple germs from $k^n$ to $k^2$”, Compositio Math. 79 (1991) no. 1, p. 99-108 Numdam | MR 1112281 | Zbl 0724.58008
[35] Mark Roberts, “Characterisations of finitely determined equivariant map germs”, Math. Ann. 275 (1986) no. 4, p. 583-597 Article | MR 859332 | Zbl 0582.58003
[36] Mark Roberts, “A note on coherent $G$-sheaves”, Math. Ann. 275 (1986) no. 4, p. 573-582 Article | MR 859331 | Zbl 0579.32013
[37] Kyoji Saito, “Theory of logarithmic differential forms and logarithmic vector fields”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) no. 2, p. 265-291 MR 586450 | Zbl 0496.32007
[38] Bernard Tessier, The hunting of invariants in the geometry of the discriminant, in P Holm, ed., Real and Complex Singularities, Oslo 1976, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, p. 565-677 MR 568901 | Zbl 0388.32010
[39] A. Vanderbauwhede, “Bifurcation of subharmonic solutions in time-reversible systems”, Z. Angew. Math. Phys. 37 (1986) no. 4, p. 455-478 Article | MR 854464 | Zbl 0603.58013
[40] A. Vanderbauwhede, “Subharmonic branching in reversible systems”, SIAM J.Math.Anal. 21 (1990), p. 954-979 Article | MR 1052881 | Zbl 0707.34038
Jacques-Élie Furter; Angela Maria Sitta
Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems
(Formulation des chemins pour des problèmes de bifurcation multi-paramétriques $\mathbb{D}_3$-équivariants)
Annales de l'institut Fourier, 60 no. 4 (2010), p. 1363-1400, doi: 10.5802/aif.2558
Article PDF | Reviews MR 2722245 | Zbl 1204.37054
Class. Math.: 37G40, 58K70, 58K40, 34F10, 34F15
Keywords: Equivariant bifurcation, degenerate bifurcation, path formulation, singularity theory, 1:1-resonance, reversible systems, subharmonic bifurcation
Résumé - Abstract
We implement a singularity theory approach, the path formulation, to classify $\mathbb{D}_3$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a $\mathbb{D}_3$-miniversal unfolding $F_{\!0}$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of $F_{\!0}$ onto its unfolding parameter space. We apply our results to degenerate bifurcation of period-$3$ subharmonics in reversible systems, in particular in the 1:1-resonance.
Bibliography
[2] J M Ball & D G Schaeffer, “Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions”, Math. Proc. Camb. Phil. Soc. 94 (1983), p. 315-339 Article | MR 715037 | Zbl 0568.73057
[3] Thomas J. Bridges & Jacques E. Furter, Singularity theory and equivariant symplectic maps, Lecture Notes in Mathematics 1558, Springer-Verlag, Berlin, 1993 MR 1290781 | Zbl 0799.58009
[4] J. W. Bruce, “Functions on discriminants”, J. London Math. Soc. (2) 30 (1984) no. 3, p. 551-567 Article | MR 810963 | Zbl 0605.58011
[5] J. W. Bruce, A. A. du Plessis & C. T. C. Wall, “Determinacy and unipotency”, Invent. Math. 88 (1987) no. 3, p. 521-554 Article | MR 884799 | Zbl 0596.58005
[6] Ernesto Buzano, Giuseppe Geymonat & Tim Poston, “Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross-section invariant under the dihedral group $D_n$”, Arch. Rational Mech. Anal. 89 (1985) no. 4, p. 307-388 Article | MR 792535 | Zbl 0568.73048
[7] Maria-Cristina Ciocci, “Generalized Lyapunov-Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms”, J. Difference Equ. Appl. 10 (2004) no. 7, p. 621-649 Article | MR 2064813 | Zbl 1055.37059
[8] Maria-Cristina Ciocci, “Subharmonic branching at a reversible $1:1$ resonance”, J. Difference Equ. Appl. 11 (2005) no. 13, p. 1119-1135 Article | MR 2183010 | Zbl 1085.37045
[9] João Carlos Ferreira Costa & Angela Maria Sitta, Path formulation for $Z_2\oplus Z_2$-equivariant bifurcation problems, Real and complex singularities, Trends Math., Birkhäuser, 2007, p. 127–141 MR 2280136 | Zbl 1128.58020
[10] James Damon, “The unfolding and determinacy theorems for subgroups of $\mathcal{A}$ and $\mathcal{K}$”, Mem. Amer. Math. Soc. 50 (1984) no. 306 MR 748971 | Zbl 0545.58010
[11] James Damon, “Deformations of sections of singularities and Gorenstein surface singularities”, Amer. J. Math. 109 (1987) no. 4, p. 695-721 Article | MR 900036 | Zbl 0628.14003
[12] James Damon, “On the legacy of free divisors: discriminants and Morse-type singularities”, Amer. J. Math. 120 (1998) no. 3, p. 453-492 Article | MR 1623404 | Zbl 0910.32038
[13] Ana Paula S. Dias & Ana Rodrigues, “Secondary bifurcations in systems with all-to-all coupling. II”, Dyn. Syst. 21 (2006) no. 4, p. 439-463 MR 2273688 | Zbl 1118.34033
[14] J. E. Furter, “Geometric path formulation for bifurcation problems”, J. Natur. Geom. 12 (1997) no. 1 MR 1456087 | Zbl 0908.34030
[15] J-E. Furter, A. M. Sitta & I. Stewart, “Algebraic path formulation for equivariant bifurcation problems”, Mathematical Proceedings of the Cambridge Philosophical Society 124 (1998) no. 2, p. 275-304, Cited By (since 1996): 2 Article | MR 1631115 | Zbl 0920.58018
[16] Terence Gaffney, New methods in the classification theory of bifurcation problems, Multiparameter bifurcation theory (Arcata, Calif., 1985), Contemp. Math. 56, Amer. Math. Soc., 1986, p. 97–116 MR 855086 | Zbl 0625.58016
[17] Karin Gatermann, Computer algebra methods for equivariant dynamical systems, Lecture Notes in Mathematics 1728, Springer-Verlag, Berlin, 2000 MR 1755001 | Zbl 0944.65131
[18] Karin Gatermann & Reiner Lauterbach, “Automatic classification of normal forms”, Nonlinear Anal. 34 (1998) no. 2, p. 157-190 Article | MR 1635741 | Zbl 0947.34023
[19] Jean-Jacques Gervais, “Bifurcations of subharmonic solutions in reversible systems”, J. Differential Equations 75 (1988) no. 1, p. 28-42 Article | MR 957006 | Zbl 0664.34051
[20] M. Golubitsky & D. Schaeffer, “A theory for imperfect bifurcation via singularity theory”, Comm. Pure Appl. Math. 32 (1979) no. 1, p. 21-98 Article | MR 508917 | Zbl 0409.58007
[21] Martin Golubitsky & David Chillingworth, Bifurcation and planar pattern formation for a liquid crystal, Bifurcation, symmetry and patterns (Porto, 2000), Trends Math., Birkhäuser, 2003, p. 55–66 MR 2014355 | Zbl 1187.82132
[22] Martin Golubitsky & Mark Roberts, “A classification of degenerate Hopf bifurcations with ${\rm O}(2)$ symmetry”, J. Differential Equations 69 (1987) no. 2, p. 216-264 Article | MR 899161 | Zbl 0635.34036
[23] Martin Golubitsky & David Schaeffer, “Bifurcations with ${\rm O}(3)$ symmetry including applications to the Bénard problem”, Comm. Pure Appl. Math. 35 (1982) no. 1, p. 81-111 Article | MR 637496 | Zbl 0492.58012
[24] Martin Golubitsky, Ian Stewart & David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences 69, Springer-Verlag, New York, 1988 MR 950168 | Zbl 0691.58003
[25] Ali Lari-Lavassani & Yung-Chen Lu, “Equivariant multiparameter bifurcation via singularity theory”, J. Dynam. Differential Equations 5 (1993) no. 2, p. 189-218 Article | MR 1223447 | Zbl 0778.58014
[26] E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series 77, Cambridge University Press, Cambridge, 1984 MR 747303 | Zbl 0552.14002
[27] Jean Martinet, Déploiements versels des applications différentiables et classification des applications stables, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975), Springer, 1976, p. 1–44. Lecture Notes in Math., Vol. 535 MR 649264 | Zbl 0362.58004
[28] John N. Mather, “Stability of $C^{\infty }$ mappings. III. Finitely determined mapgerms”, Inst. Hautes Études Sci. Publ. Math. (1968) no. 35, p. 279-308 Numdam | MR 275459 | Zbl 0159.25001
[29] David Mond, Ciclos Evanescentes para las applicaciones Analíticas, Technical report, ICMSC-USP, São-Carlos, 1990
[30] David Mond & James Montaldi, Deformations of maps on complete intersections, Damon’s $\mathcal{K}_V$-equivalence and bifurcations, Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser. 201, Cambridge Univ. Press, 1994, p. 263–284 MR 1295079 | Zbl 0847.58007
[31] James Montaldi, The path formulation of bifurcation theory, Dynamics, bifurcation and symmetry (Cargèse, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 437, Kluwer Acad. Publ., 1994, p. 259–278 MR 1305382 | Zbl 0811.58015
[32] J. H. Rieger, “Versal topological stratification and the bifurcation geometry of map-germs of the plane”, Math. Proc. Cambridge Philos. Soc. 107 (1990) no. 1, p. 127-147 Article | MR 1021879 | Zbl 0696.58009
[33] J. H. Rieger, “$\mathcal{A}$-unimodal map-germs into the plane”, Hokkaido Math. J. 33 (2004) no. 1, p. 47-64 MR 2034807 | Zbl 1152.58314
[34] J. H. Rieger & M. A. S. Ruas, “Classification of $\mathcal{A}$-simple germs from $k^n$ to $k^2$”, Compositio Math. 79 (1991) no. 1, p. 99-108 Numdam | MR 1112281 | Zbl 0724.58008
[35] Mark Roberts, “Characterisations of finitely determined equivariant map germs”, Math. Ann. 275 (1986) no. 4, p. 583-597 Article | MR 859332 | Zbl 0582.58003
[36] Mark Roberts, “A note on coherent $G$-sheaves”, Math. Ann. 275 (1986) no. 4, p. 573-582 Article | MR 859331 | Zbl 0579.32013
[37] Kyoji Saito, “Theory of logarithmic differential forms and logarithmic vector fields”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) no. 2, p. 265-291 MR 586450 | Zbl 0496.32007
[38] Bernard Tessier, The hunting of invariants in the geometry of the discriminant, in P Holm, ed., Real and Complex Singularities, Oslo 1976, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, p. 565-677 MR 568901 | Zbl 0388.32010
[39] A. Vanderbauwhede, “Bifurcation of subharmonic solutions in time-reversible systems”, Z. Angew. Math. Phys. 37 (1986) no. 4, p. 455-478 Article | MR 854464 | Zbl 0603.58013
[40] A. Vanderbauwhede, “Subharmonic branching in reversible systems”, SIAM J.Math.Anal. 21 (1990), p. 954-979 Article | MR 1052881 | Zbl 0707.34038
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310