Path formulation for multiparameter 𝔻 3 -equivariant bifurcation problems

Jacques-Élie Furter[1]; Angela Maria Sitta[2]

  • [1] Brunel University Department of Mathematical Sciences Uxbridge UB8 3PH (United Kingdom)
  • [2] Universidade Estadual Paulista - UNESP Departamento de Matemática - IBILCE Campus de São José do Rio Preto - SP (Brazil)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 4, page 1363-1400
  • ISSN: 0373-0956

Abstract

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We implement a singularity theory approach, the path formulation, to classify 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 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.

How to cite

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Furter, Jacques-Élie, and Sitta, Angela Maria. "Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems." Annales de l’institut Fourier 60.4 (2010): 1363-1400. <http://eudml.org/doc/116307>.

@article{Furter2010,
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.},
affiliation = {Brunel University Department of Mathematical Sciences Uxbridge UB8 3PH (United Kingdom); Universidade Estadual Paulista - UNESP Departamento de Matemática - IBILCE Campus de São José do Rio Preto - SP (Brazil)},
author = {Furter, Jacques-Élie, Sitta, Angela Maria},
journal = {Annales de l’institut Fourier},
keywords = {Equivariant bifurcation; degenerate bifurcation; path formulation; singularity theory; 1:1-resonance; reversible systems; subharmonic bifurcation; equivariant bifurcation},
language = {eng},
number = {4},
pages = {1363-1400},
publisher = {Association des Annales de l’institut Fourier},
title = {Path formulation for multiparameter $\mathbb\{D\}_3$-equivariant bifurcation problems},
url = {http://eudml.org/doc/116307},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Furter, Jacques-Élie
AU - Sitta, Angela Maria
TI - Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 4
SP - 1363
EP - 1400
AB - 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.
LA - eng
KW - Equivariant bifurcation; degenerate bifurcation; path formulation; singularity theory; 1:1-resonance; reversible systems; subharmonic bifurcation; equivariant bifurcation
UR - http://eudml.org/doc/116307
ER -

References

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  1. V I Arnold, Wavefront evolution and equivariant Morse lemma, Comm. Pure. App. Math. 29 (1976), 557-582 Zbl0343.58003MR436200
  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), 315-339 Zbl0568.73057MR715037
  3. Thomas J. Bridges, Jacques E. Furter, Singularity theory and equivariant symplectic maps, 1558 (1993), Springer-Verlag, Berlin Zbl0799.58009MR1290781
  4. J. W. Bruce, Functions on discriminants, J. London Math. Soc. (2) 30 (1984), 551-567 Zbl0605.58011MR810963
  5. J. W. Bruce, A. A. du Plessis, C. T. C. Wall, Determinacy and unipotency, Invent. Math. 88 (1987), 521-554 Zbl0596.58005MR884799
  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), 307-388 Zbl0568.73048MR792535
  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), 621-649 Zbl1055.37059MR2064813
  8. Maria-Cristina Ciocci, Subharmonic branching at a reversible 1 : 1 resonance, J. Difference Equ. Appl. 11 (2005), 1119-1135 Zbl1085.37045MR2183010
  9. João Carlos Ferreira Costa, Angela Maria Sitta, Path formulation for Z 2 ⊕ Z 2 -equivariant bifurcation problems, Real and complex singularities (2007), 127-141, Birkhäuser, Basel Zbl1128.58020MR2280136
  10. James Damon, The unfolding and determinacy theorems for subgroups of 𝒜 and 𝒦 , Mem. Amer. Math. Soc. 50 (1984) Zbl0545.58010MR748971
  11. James Damon, Deformations of sections of singularities and Gorenstein surface singularities, Amer. J. Math. 109 (1987), 695-721 Zbl0628.14003MR900036
  12. James Damon, On the legacy of free divisors: discriminants and Morse-type singularities, Amer. J. Math. 120 (1998), 453-492 Zbl0910.32038MR1623404
  13. Ana Paula S. Dias, Ana Rodrigues, Secondary bifurcations in systems with all-to-all coupling. II, Dyn. Syst. 21 (2006), 439-463 Zbl1118.34033MR2273688
  14. J. E. Furter, Geometric path formulation for bifurcation problems, J. Natur. Geom. 12 (1997) Zbl0908.34030MR1456087
  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), 275-304 Zbl0920.58018MR1631115
  16. Terence Gaffney, New methods in the classification theory of bifurcation problems, Multiparameter bifurcation theory (Arcata, Calif., 1985) 56 (1986), 97-116, Amer. Math. Soc., Providence, RI Zbl0625.58016MR855086
  17. Karin Gatermann, Computer algebra methods for equivariant dynamical systems, 1728 (2000), Springer-Verlag, Berlin Zbl0944.65131MR1755001
  18. Karin Gatermann, Reiner Lauterbach, Automatic classification of normal forms, Nonlinear Anal. 34 (1998), 157-190 Zbl0947.34023MR1635741
  19. Jean-Jacques Gervais, Bifurcations of subharmonic solutions in reversible systems, J. Differential Equations 75 (1988), 28-42 Zbl0664.34051MR957006
  20. M. Golubitsky, D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 (1979), 21-98 Zbl0409.58007MR508917
  21. Martin Golubitsky, David Chillingworth, Bifurcation and planar pattern formation for a liquid crystal, Bifurcation, symmetry and patterns (Porto, 2000) (2003), 55-66, Birkhäuser, Basel Zbl1187.82132MR2014355
  22. Martin Golubitsky, Mark Roberts, A classification of degenerate Hopf bifurcations with O ( 2 ) symmetry, J. Differential Equations 69 (1987), 216-264 Zbl0635.34036MR899161
  23. Martin Golubitsky, David Schaeffer, Bifurcations with O ( 3 ) symmetry including applications to the Bénard problem, Comm. Pure Appl. Math. 35 (1982), 81-111 Zbl0492.58012MR637496
  24. Martin Golubitsky, Ian Stewart, David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, 69 (1988), Springer-Verlag, New York Zbl0691.58003MR950168
  25. Ali Lari-Lavassani, Yung-Chen Lu, Equivariant multiparameter bifurcation via singularity theory, J. Dynam. Differential Equations 5 (1993), 189-218 Zbl0778.58014MR1223447
  26. E. J. N. Looijenga, Isolated singular points on complete intersections, 77 (1984), Cambridge University Press, Cambridge Zbl0552.14002MR747303
  27. Jean Martinet, Déploiements versels des applications différentiables et classification des applications stables, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975) (1976), 1-44. Lecture Notes in Math., Vol. 535, Springer, Berlin Zbl0362.58004MR649264
  28. John N. Mather, Stability of C ∞ mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. (1968), 279-308 Zbl0159.25001MR275459
  29. David Mond, Ciclos Evanescentes para las applicaciones Analíticas, (1990) 
  30. David Mond, James Montaldi, Deformations of maps on complete intersections, Damon’s 𝒦 V -equivalence and bifurcations, Singularities (Lille, 1991) 201 (1994), 263-284, Cambridge Univ. Press, Cambridge Zbl0847.58007MR1295079
  31. James Montaldi, The path formulation of bifurcation theory, Dynamics, bifurcation and symmetry (Cargèse, 1993) 437 (1994), 259-278, Kluwer Acad. Publ., Dordrecht Zbl0811.58015MR1305382
  32. J. H. Rieger, Versal topological stratification and the bifurcation geometry of map-germs of the plane, Math. Proc. Cambridge Philos. Soc. 107 (1990), 127-147 Zbl0696.58009MR1021879
  33. J. H. Rieger, 𝒜 -unimodal map-germs into the plane, Hokkaido Math. J. 33 (2004), 47-64 Zbl1152.58314MR2034807
  34. J. H. Rieger, M. A. S. Ruas, Classification of 𝒜 -simple germs from k n to k 2 , Compositio Math. 79 (1991), 99-108 Zbl0724.58008MR1112281
  35. Mark Roberts, Characterisations of finitely determined equivariant map germs, Math. Ann. 275 (1986), 583-597 Zbl0582.58003MR859332
  36. Mark Roberts, A note on coherent G -sheaves, Math. Ann. 275 (1986), 573-582 Zbl0579.32013MR859331
  37. Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265-291 Zbl0496.32007MR586450
  38. Bernard Tessier, The hunting of invariants in the geometry of the discriminant, Real and Complex Singularities, Oslo 1976 (1977), 565-677, Sijthoff and Noordhoff, Alphen aan den Rijn Zbl0388.32010MR568901
  39. A. Vanderbauwhede, Bifurcation of subharmonic solutions in time-reversible systems, Z. Angew. Math. Phys. 37 (1986), 455-478 Zbl0603.58013MR854464
  40. A. Vanderbauwhede, Subharmonic branching in reversible systems, SIAM J.Math.Anal. 21 (1990), 954-979 Zbl0707.34038MR1052881

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