Comptes Rendus
no. 6-7
Instabilities, patterns
Effect of neutral modes on the order of a transition
Stéphan Fauve1
1 Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 475-487.

Je rappelle comment une expérience réalisée par Yves Couder et son groupe a motivé des travaux théoriques qui ont montré que les modes de phase d’une structure cellulaire engendrée par instabilité peuvent affecter la nature des instabilités secondaires de cette structure, à savoir, le caractère propagatif ou non de l’instabilité secondaire et sa sous-criticalité. Je discute ensuite la sous-criticalité résultant du couplage avec les modes de phase sur d’autres exemples tels que la transition de Peierls en physique de la matière condensée.

Neutral modes related to spontaneous broken symmetries at the onset of a pattern-forming instability can strongly modify the nature of secondary instabilities of the pattern. In particular these neutral modes can change the order of the secondary transition making it first order or subcritical in the language of bifurcation theory. We first discuss this phenomenon in the context of the drift bifurcation from stationary to traveling patterns. We then consider patterns that undergo a spatial period-doubling bifurcation like the Peierls transition in solid state physics.

Publié le :
DOI : 10.5802/crmeca.21
Mots clés : Instability, Symmetry, Neutral modes, Drifting patterns, Peierls transition
Stéphan Fauve 1

1 Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Stéphan Fauve. Effect of neutral modes on the order of a transition. Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 475-487. doi : 10.5802/crmeca.21. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.21/

[1] M. Faraday On the forms and states assumed by fluids in contact with vibrating elastic surfaces, Phil. Trans. R. Soc. Lond., Volume 52 (1831), pp. 319-340

[2] W. Thomson Hydrokinetic solutions and observations, Phil. Mag., Volume 42 (1871), pp. 362-377 | DOI

[3] H. Bénard Etude expérimentale du mouvement des liquides propageant la chaleur par convection. Régime permanent: tourbillons cellulaires, C. R. Acad. Sci. Paris, Volume 130 (1900), pp. 1004-1007

[4] H. Bénard Mouvements tourbillonnaires à structure cellulaire. Etude optique de la surface libre, C. R. Acad. Sci. Paris, Volume 130 (1900), pp. 1065-1068 | Zbl

[5] W. S. Edwards; S. Fauve Structure quasicristalline engendrée par instabilité paramètrique, C. R. Acad. Sci. Paris, Ser. II, Volume 315 (1992), pp. 417-420

[6] W. S. Edwards; S. Fauve Patterns and quasi-patterns in the Faraday experiment, J. Fluid Mech., Volume 278 (1994), pp. 123-148 | DOI | MR

[7] S. Fauve Pattern forming instabilities, Hydrodynamics and Nonlinear Instabilities (C. Godrèche; P. Manneville, eds.), Cambridge University Press, Cambridge, 1998, pp. 387-491 | DOI

[8] M. Rabaud; S. Michalland; Y. Couder Dynamical regimes of directional viscous fingering: spatiotemporal chaos and waves propagation, Phys. Rev. Lett., Volume 64 (1990), pp. 184-187 | DOI

[9] R. W. Walden; P. Kolodner; A. Passner; C. M. Surko Traveling waves and chaos in convection in binary fluid mixtures, Phys. Rev. Lett., Volume 55 (1985), pp. 496-499 | DOI

[10] I. Mutabazi; J. J. Hegseth; C. D. Andereck; J. E. Wesfreid Pattern formation in the flow between two horizontal coaxial cylinders with a partially filled gap, Phys. Rev. A, Volume 38 (1988), pp. 4752-4760 | DOI

[11] A. J. Simon; J. Bechhoefer; A. Libchaber Solitary modes and the eckhaus instability in directional solidification, Phys. Rev. Lett., Volume 61 (1988), pp. 2574-2777 | DOI

[12] G. Faivre; P. de Cheveigné; C. Guthmann; P. Kurowski Solitary tilt waves in thin lamellar eutectics, Europhys. Lett., Volume 9 (1989), pp. 779-784 | DOI

[13] S. Douady; S. Fauve; O. Thual Oscillatory phase modulation of parametrically forced surface waves, Europhys. Lett., Volume 10 (1989), pp. 309-315 | DOI

[14] S. Ciliberto; S. Douady; S. Fauve Investigating space-time chaos in faraday instability by means of the fluctuations of the driving acceleration, Europhys. Lett., Volume 15 (1991), pp. 23-28 | DOI

[15] P. Coullet; R. E. Goldstein; G. H. Gunaratne Parity-breaking transitions of modulated patterns in hydrodynamic systems, Phys. Rev. Lett., Volume 63 (1989), pp. 1954-1957 | DOI

[16] M. E. Brachet; P. Coullet; S. Fauve Propagative phase dynamics in temporally intermittent systems, Europhys. Lett., Volume 4 (1987), pp. 1017-1022 | DOI

[17] K. Seshasayanan; V. Dallas; S. Fauve Bifurcations of a planar parallel flow with Kolmogorov forcing, Phys. Rev. E (2020) arXiv:2004.12418 (submitted)

[18] H. Cummins; L. Fourtune; M. Rabaud Successive bifurcations in directional viscous fingering, Phys. Rev. E, Volume 47 (1993), pp. 1727-1738

[19] S. Fauve; S. Douady; O. Thual Comment on “Parity-breaking transitions of modulated patterns in hydrodynamic systems”, Phys. Rev. Lett., Volume 385, p. 1990

[20] S. Fauve; S. Douady; O. Thual Drift instabilities of cellular patterns, J. Physique II, Volume 1 (1991), pp. 311-322 | DOI

[21] M. R. E. Proctor; C. A. Jones The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance, J. Fluid Mech. (1988), pp. 301-335 (and references therein) | DOI | Zbl

[22] M. Kness; L. S. Tuckerman; D. Barkley Symmetry-breaking bifurcations in one-dimensional excitable media, Phys. Rev. A, Volume 46 (1992), pp. 5054-5062 | DOI

[23] C. Nore; L. S. Tuckerman; O. Daube; S. Xin The 1:2 mode interaction in exactly counter-rotating von Karman swirling flow, J. Fluid Mech., Volume 477 (1988), pp. 51-88 | Zbl

[24] D. Armbruster; J. Guckenheimer; P. Holmes Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica D, Volume 29 (1988), pp. 257-282 (and references therein) | DOI | MR | Zbl

[25] J. Porter; E. Knobloch New type of complex dynamics in the 1:2 spatial resonance, Physica D, Volume 159 (2001), pp. 125-154 | MR | Zbl

[26] L. Fourtune; W.-J. Rappel; M. Rabaud Phase dynamics near a parity breaking instability, Phys. Rev. E, Volume 49 (1994), pp. R3576-3579 | DOI

[27] L. Bellon; L. Fourtune; V. Ter Minassian; M. Rabaud Wave-number selection and parity-breaking bifurcation in directional viscous fingering, Phys. Rev. E, Volume 58 (1998), pp. 565-574

[28] H. Levine; W.-J. Rappel; H. Riecke Resonant interactions and traveling-solidification cells, Phys. Rev. A, Volume 43 (1991), pp. 1122-1125 | DOI

[29] J. Guckenheimer; P. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Vol. 42, Springer Verlag, New York, 1983

[30] P. Coullet; S. Fauve Propagative phase dynamics for systems with galilean invariance, Phys. Rev. Lett., Volume 55 (1985), pp. 2857-2860 | DOI

[31] P. Coullet; G. Iooss Instabilities of one-dimensional cellular patterns, Phys. Rev. Lett., Volume 64 (1990), pp. 866-869 | DOI | MR | Zbl

[32] S. Fauve; E. W. Bolton; M. E. Brachet Nonlinear oscillatory convection: a quantitative phase dynamics approach, Physica D, Volume 29 (1987), pp. 202-214 | MR | Zbl

[33] H. Riecke; H.-G. Paap Parity-breaking and Hopf bifurcations in axisymmetric Taylor vortex flow, Phys. Rev. A, Volume 45 (1992), pp. 8605-8610 | DOI

[34] B. Caroli; C. Caroli; S. Fauve On the phenomenology of tilted domains in lamellar eutectic growth, J. Physique I, Volume 2 (1992), pp. 281-290

[35] L. Pan; J. R. de Bruyn Nonuniform broken-parity waves and the Eckhaus instability, Phys. Rev. E, Volume 49 (1994), pp. 2119-2129

[36] R. Peierls More Surprises in Theoretical Physics, Princeton University Press, Princeton, 1991 | MR

[37] T. Dessup; C. Coste; M. S. Jean Subcriticality of the zigzag transition: a nonlinear bifurcation analysis, Phys. Rev. E, Volume 91 (2015) (032917) | DOI | MR

[38] F. Klasing; T. Frigge; B. Hafke; B. Krenzer; S. Wall; A. Hanisch-Blicharski; M. Horn von Hoegen Hysteresis proves that the In/Si(111) (8×2) to (4×1) phase transition is first-order, Phys. Rev. B, Volume 89 (2014) 121107(R) | DOI

[39] S. Hatta; T. Noma; H. Okuyama; T. Aruga Electrical conduction and metal-insulator transition of indium nanowires on Si(111), Phys. Rev. B, Volume 95 (2017) (195409) | DOI

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