Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems

[Formulation des chemins pour des problèmes de bifurcation multi-paramétriques $\mathbb{D}_3$-équivariants]

Furter, Jacques-Élie ; Sitta, Angela Maria

Annales de l'Institut Fourier, Tome 60 (2010), p. 1363-1400 / Harvested from Numdam

Résumé

Nous utilisons une approche de la théorie des singularités pour classifier des problèmes de bifurcation $\mathbb{D}_3$-équivariants de corang 2, avec un ou deux paramètres de bifurcation distingués, et leurs perturbations. Les diagrammes de bifurcation sont identifiés avec des sections sur des chemins dans l’espace des paramètres d’un déployement miniversel $\mathbb{D}_3$-équivariant $F_{\!0}$ de leur noyau. Les équivalences entre les chemins sont données par des difféomorphismes qui se relèvent le long de la projection de l’ensemble des zéros de $F_{\!0}$ dans l’espace de ses paramètres. Nos résultats sont appliqués aux bifurcations dégénérées de solutions sous-harmoniques de période 3 dans des systèmes dynamiques réversibles, en particulier dans la résonance 1 :1.

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.

Détail

Publié le : 2010-01-01

MR 2722245 | Zbl 1204.37054

DOI : https://doi.org/10.5802/aif.2558
Classification:  37G40,  58K70,  58K40,  34F10,  34F15
Mots clés: bifurcation équivariante, bifurcation dégénérée, formulation des chemins, théorie des singularités, résonance 1 :1, systèmes dynamiques réversibles, bifurcation sous-harmonique

Bibliographie

[1] Arnold, V I Wavefront evolution and equivariant Morse lemma, Comm. Pure. App. Math., Tome 29 (1976), pp. 557-582 | Article | MR 436200 | Zbl 0343.58003

[2] Ball, J M; Schaeffer, D G Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions, Math. Proc. Camb. Phil. Soc., Tome 94 (1983), pp. 315-339 | Article | MR 715037 | Zbl 0568.73057

[3] Bridges, Thomas J.; Furter, Jacques E. Singularity theory and equivariant symplectic maps, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1558 (1993) | MR 1290781 | Zbl 0799.58009

[4] Bruce, J. W. Functions on discriminants, J. London Math. Soc. (2), Tome 30 (1984) no. 3, pp. 551-567 | Article | MR 810963 | Zbl 0605.58011

[5] Bruce, J. W.; Du Plessis, A. A.; Wall, C. T. C. Determinacy and unipotency, Invent. Math., Tome 88 (1987) no. 3, pp. 521-554 | Article | MR 884799 | Zbl 0596.58005

[6] Buzano, Ernesto; Geymonat, Giuseppe; Poston, Tim Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross-section invariant under the dihedral group $D_n$, Arch. Rational Mech. Anal., Tome 89 (1985) no. 4, pp. 307-388 | Article | MR 792535 | Zbl 0568.73048

[7] Ciocci, Maria-Cristina 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., Tome 10 (2004) no. 7, pp. 621-649 | Article | MR 2064813 | Zbl 1055.37059

[8] Ciocci, Maria-Cristina Subharmonic branching at a reversible $1:1$ resonance, J. Difference Equ. Appl., Tome 11 (2005) no. 13, pp. 1119-1135 | Article | MR 2183010 | Zbl 1085.37045

[9] Costa, João Carlos Ferreira; Sitta, Angela Maria Path formulation for $Z_2\oplus Z_2$-equivariant bifurcation problems, Real and complex singularities, Birkhäuser, Basel (Trends Math.) (2007), pp. 127-141 | MR 2280136 | Zbl 1128.58020

[10] Damon, James The unfolding and determinacy theorems for subgroups of $\mathcal{A}$ and $\mathcal{K}$, Mem. Amer. Math. Soc., Tome 50 (1984) no. 306, pp. x+88 | MR 748971 | Zbl 0545.58010

[11] Damon, James Deformations of sections of singularities and Gorenstein surface singularities, Amer. J. Math., Tome 109 (1987) no. 4, pp. 695-721 | Article | MR 900036 | Zbl 0628.14003

[12] Damon, James On the legacy of free divisors: discriminants and Morse-type singularities, Amer. J. Math., Tome 120 (1998) no. 3, pp. 453-492 | Article | MR 1623404 | Zbl 0910.32038

[13] Dias, Ana Paula S.; Rodrigues, Ana Secondary bifurcations in systems with all-to-all coupling. II, Dyn. Syst., Tome 21 (2006) no. 4, pp. 439-463 | MR 2273688 | Zbl 1118.34033

[14] Furter, J. E. Geometric path formulation for bifurcation problems, J. Natur. Geom., Tome 12 (1997) no. 1, pp. xii+100 | MR 1456087 | Zbl 0908.34030

[15] Furter, J-E.; Sitta, A. M.; Stewart, I. Algebraic path formulation for equivariant bifurcation problems, Mathematical Proceedings of the Cambridge Philosophical Society, Tome 124 (1998) no. 2, pp. 275-304 www.scopus.com (Cited By (since 1996): 2) | Article | MR 1631115 | Zbl 0920.58018

[16] Gaffney, Terence New methods in the classification theory of bifurcation problems, Multiparameter bifurcation theory (Arcata, Calif., 1985), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 56 (1986), pp. 97-116 | MR 855086 | Zbl 0625.58016

[17] Gatermann, Karin Computer algebra methods for equivariant dynamical systems, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1728 (2000) | MR 1755001 | Zbl 0944.65131

[18] Gatermann, Karin; Lauterbach, Reiner Automatic classification of normal forms, Nonlinear Anal., Tome 34 (1998) no. 2, pp. 157-190 | Article | MR 1635741 | Zbl 0947.34023

[19] Gervais, Jean-Jacques Bifurcations of subharmonic solutions in reversible systems, J. Differential Equations, Tome 75 (1988) no. 1, pp. 28-42 | Article | MR 957006 | Zbl 0664.34051

[20] Golubitsky, M.; Schaeffer, D. A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., Tome 32 (1979) no. 1, pp. 21-98 | Article | MR 508917 | Zbl 0409.58007

[21] Golubitsky, Martin; Chillingworth, David Bifurcation and planar pattern formation for a liquid crystal, Bifurcation, symmetry and patterns (Porto, 2000), Birkhäuser, Basel (Trends Math.) (2003), pp. 55-66 | MR 2014355 | Zbl 1187.82132

[22] Golubitsky, Martin; Roberts, Mark A classification of degenerate Hopf bifurcations with ${\rm O}(2)$ symmetry, J. Differential Equations, Tome 69 (1987) no. 2, pp. 216-264 | Article | MR 899161 | Zbl 0635.34036

[23] Golubitsky, Martin; Schaeffer, David Bifurcations with ${\rm O}(3)$ symmetry including applications to the Bénard problem, Comm. Pure Appl. Math., Tome 35 (1982) no. 1, pp. 81-111 | Article | MR 637496 | Zbl 0492.58012

[24] Golubitsky, Martin; Stewart, Ian; Schaeffer, David G. Singularities and groups in bifurcation theory. Vol. II, Springer-Verlag, New York, Applied Mathematical Sciences, Tome 69 (1988) | MR 950168 | Zbl 0691.58003

[25] Lari-Lavassani, Ali; Lu, Yung-Chen Equivariant multiparameter bifurcation via singularity theory, J. Dynam. Differential Equations, Tome 5 (1993) no. 2, pp. 189-218 | Article | MR 1223447 | Zbl 0778.58014

[26] Looijenga, E. J. N. Isolated singular points on complete intersections, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 77 (1984) | MR 747303 | Zbl 0552.14002

[27] Martinet, Jean Déploiements versels des applications différentiables et classification des applications stables, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975), Springer, Berlin (1976), p. 1-44. Lecture Notes in Math., Vol. 535 | MR 649264 | Zbl 0362.58004

[28] Mather, John N. Stability of $C^{\infty }$ mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. (1968) no. 35, pp. 279-308 | Numdam | MR 275459 | Zbl 0159.25001

[29] Mond, David Ciclos Evanescentes para las applicaciones Analíticas, ICMSC-USP, São-Carlos (1990) (Lecture Notes)

[30] Mond, David; Montaldi, James Deformations of maps on complete intersections, Damon’s $\mathcal{K}_V$-equivalence and bifurcations, Singularities (Lille, 1991), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 201 (1994), pp. 263-284 | MR 1295079 | Zbl 0847.58007

[31] Montaldi, James The path formulation of bifurcation theory, Dynamics, bifurcation and symmetry (Cargèse, 1993), Kluwer Acad. Publ., Dordrecht (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.) Tome 437 (1994), pp. 259-278 | MR 1305382 | Zbl 0811.58015

[32] Rieger, J. H. Versal topological stratification and the bifurcation geometry of map-germs of the plane, Math. Proc. Cambridge Philos. Soc., Tome 107 (1990) no. 1, pp. 127-147 | Article | MR 1021879 | Zbl 0696.58009

[33] Rieger, J. H. $\mathcal{A}$-unimodal map-germs into the plane, Hokkaido Math. J., Tome 33 (2004) no. 1, pp. 47-64 | MR 2034807 | Zbl 1152.58314

[34] Rieger, J. H.; Ruas, M. A. S. Classification of $\mathcal{A}$-simple germs from $k^n$ to $k^2$, Compositio Math., Tome 79 (1991) no. 1, pp. 99-108 | Numdam | MR 1112281 | Zbl 0724.58008

[35] Roberts, Mark Characterisations of finitely determined equivariant map germs, Math. Ann., Tome 275 (1986) no. 4, pp. 583-597 | Article | MR 859332 | Zbl 0582.58003

[36] Roberts, Mark A note on coherent $G$-sheaves, Math. Ann., Tome 275 (1986) no. 4, pp. 573-582 | Article | MR 859331 | Zbl 0579.32013

[37] Saito, Kyoji Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 27 (1980) no. 2, pp. 265-291 | MR 586450 | Zbl 0496.32007

[38] Tessier, Bernard; Holm, P The hunting of invariants in the geometry of the discriminant, Real and Complex Singularities, Oslo 1976, Sijthoff and Noordhoff, Alphen aan den Rijn (1977), pp. 565-677 | MR 568901 | Zbl 0388.32010

[39] Vanderbauwhede, A. Bifurcation of subharmonic solutions in time-reversible systems, Z. Angew. Math. Phys., Tome 37 (1986) no. 4, pp. 455-478 | Article | MR 854464 | Zbl 0603.58013

[40] Vanderbauwhede, A. Subharmonic branching in reversible systems, SIAM J.Math.Anal., Tome 21 (1990), pp. 954-979 www.scopus.com | Article | MR 1052881 | Zbl 0707.34038