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Bifurcation and transition to turbulence in hydrodynamics

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Bifurcation Theory and Applications

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Bibliography on Bénard convection experiments (small aspect-ratio)

  1. Bergé, P., M. Dubois, P. Manneville, Y. Pomeau, Intermittency in Rayleigh-Bénard convection, J.de Phys. Lettres 41, L341, 1980

    Google Scholar 

  2. Dubois, M., P. Bergé, V. Croquette,Study of non steady convective regimes using Poincaré sections, J.de Phys. Lettres 43, L295, 1982

    Google Scholar 

  3. Giglio, M., S. Musazzi, U. Perini,Transition to chaos via a well ordered sequence of period doubling bifurcations, Phys. Rev. Lett. 47, 243, 1981

    Article  Google Scholar 

  4. Gollub, J.P., S.V. Benson,Many routes to turbulent convection, J.Fluid Mech.,100, 3, 449, 1980

    Article  Google Scholar 

  5. Maurer, J., A. Libchaber, Rayleigh-Bénard experiment in liquid helium; frequency locking and the onset of turbulence, J.de Phys. Lettres 40, L419, 1979

    Google Scholar 

Review papers on this topic (with lot of references inside)

  1. Busse, F.H.,Transition to turbulence in Rayleigh-Bénard convection, in Hydrodynamic Instabilities and the Transition to Turbulence, H.SWINNEY, J.P.GOLLub ed., Topics in Applied Physics 45, 97–137, Springer Verlag 1981

    MathSciNet  MATH  Google Scholar 

  2. Gollub, J.P.,Recent experiments on the transition to turbulent convection, Nonlinear Dynamics and Turbulence, G.BARENBLATT, G.IOOSS, D.D.JOSEPH ed., Pitman 1983

    Google Scholar 

Perioddoubling theory for maps of the interval

  1. Collet, P., J.P. Eckmann,Iterated maps on the interval as dynamical systems, P. Ph.1, Birkhaüser, Boston,1980

    MATH  Google Scholar 

Bibliography on Couette-Taylor problem

  1. Fenstermacher, P.R., H.L. Swinney, J.P. Gollub,Dynamical instabilities and the transition to chaotic Taylor vortex flow, J.Fluid Mech., 94, 103–128, 1979

    Article  Google Scholar 

  2. Gorman, M., L.A. Reith, H.L. Swinney,Modulation patterns, multiple frequencies and other phenomena in circular Couette flow,Nonlinear Dynamics, R.HELLEMAN ed., Ann. N.Y. Acad. Sci. 357, p.10, 1980

    Article  Google Scholar 

  3. Rand, D.,Dynamics and Symmetry: predictions for modulated waves in rotating fluids, Arch. Rat. Mech. Anal. 79,1, 1–38, 1982

    Article  MathSciNet  MATH  Google Scholar 

Review paper (lot of references)

  1. Di Prima, R.C., H.L. Swinney,Instabilities and Transition in flow between concentric rotating cylinders, in Hydrodynamic Instabilities and the Transito Turbulence, H.SWINNEY, J.P.GOLLUB ed., Topics in Applied Physics 45, 139–180, Springer Verlag 1981.

    MathSciNet  MATH  Google Scholar 

Bibliography

  1. Agmon, S.,Lectures on elliptic boundary value problems, Van Nostrand, Math. Studies, 2, Princeton 1965

    Google Scholar 

  2. Brézis, D.,Perturbations singulières et problèmes d'évolution avec défaut d'ajustement, C.R.Acad. Sci. Paris, 276, A, 1597–1600, 1973

    MATH  Google Scholar 

  3. Cattabriga, L.,Su un problema al contorno relativo al sistema di equazioni di Stokes, Rendiconti del Seminario Matematico, Padova, vol.31, 308–340, 1961

    MathSciNet  MATH  Google Scholar 

  4. Fujita, H. & T. Kato,On the Navier-Stokes initial value problem I, Arch.Rat. Mech. Anal. 16, 4, 269–315, 1964

    Article  MathSciNet  MATH  Google Scholar 

  5. Iooss,G.,Bifurcation et stabilité, Pub. Math. Orsay 31, 1973 (chap.VIIbis), C.R. Acad. Sci. Paris, 271, A, 187–190, 1970

    Google Scholar 

  6. Iooss, G.,Théorie non lineaire de la stabilité des écoulements laminaires dans le cas de l'échange des stabilités, Arch. Rat. Mech. Anal 40, 166–208 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iooss, G.,Sur la deuxième bifurcation d'une solution stationnaire de systèmes du type Navier-Stokes, Arch. Rat. Mech. Anal. 64, 4, 339–369, 1977

    Article  MathSciNet  MATH  Google Scholar 

  8. Iooss, G., & R. Lozi,Convection entre deux plaques planes en rotation, et effet dynamo résultant d'une bifurcation secondaire, J. de Mécanique, 16, 5, 675–703, 1977

    MathSciNet  MATH  Google Scholar 

  9. Kato, T.,Perturbation theory for linear operators, Springer Verlag, Berlin 1966

    Book  MATH  Google Scholar 

  10. Ladyzhenskaya, O.A.,The mathematical theory of viscous incompressible flow, New York, Gordon & Breach, 1963

    MATH  Google Scholar 

  11. Pazy, A.,On the differentiability and compactness of semi-groups of linear operators, J.Math. Mech. 17, 12, 1131–1141, 1968

    MathSciNet  MATH  Google Scholar 

  12. Témam, R.,Navier-Stokes equations, North Holland, Amsterdam, 1977.

    MATH  Google Scholar 

Bibliography

  1. Iooss, G., Bifurcation et Stabilité, Pub. Math. Orsay 31,(chap.X), 1973

    Google Scholar 

  2. Iooss, G., & D.D. Joseph, Elementary stability and bifurcation theory, Undergraduate Texts in Math., Springer Verlag, New York, 1980

    Book  MATH  Google Scholar 

  3. Kato,T., see ref. [5] of chapter II.

    Google Scholar 

Bibliographical remark for the infinite dimensional case

  1. Iudovich, V.I.,Apparition d'auto-oscillations dans un fluide (in Russian), Prikl. Mat. Mek. 35, 638–655, 1971

    MathSciNet  Google Scholar 

  2. Iudovich, V.I.,Etude des auto-oscillations d'un milieu continu, intervenant lors de la perte de stabilité d'un mode stationnaire (in Russian), Prikl. Mat. Mek. 36, 450–459, 1972

    Google Scholar 

  3. Sattinger, D.H.,Bifurcation of periodic solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 41, 66–80, 1971

    Article  MathSciNet  MATH  Google Scholar 

  4. Sattinger,D.H.,Topic in Stability and Bifurcation theory, Lecture Notes in Math, 309, Springer,1973

    Google Scholar 

  5. Iooss, G.,Existence et Stabilité de la solution périodique secondaire intervenant dans les problèmes d'évolution du type Navier-Stokes, Arch. Rat. Mech. Anal. 47, 301–329, 1972

    Article  MathSciNet  MATH  Google Scholar 

  6. Marsden,J., M.Mc Cracken,The hopf bifurcation and its applications, Applied Math., 19, Springer Verlag, 1976

    Google Scholar 

  7. Crandall, M.G., & P.H. Rabinovitz,The Hopf bifurcation theorem in infinite dimensions, Arch. Rat. Mech. Anal. 67, 53–72, 1978

    Article  MathSciNet  Google Scholar 

Bibliography

  1. Di Prima, R.C. & G.J. Habetler, Arch. Rat. Mech. Anal., 34, 218–227, 1969

    Article  Google Scholar 

  2. Di Prima, R.C., H.L. Swinney, (see ref.[10] of chap.I)

    MathSciNet  MATH  Google Scholar 

  3. Iooss, G., (see [6] at chap.II)

    Article  MathSciNet  MATH  Google Scholar 

  4. Joseph,D.D.,Stability of fluid motions I and II, Springer Tracts in Natural Philosophy, New York 1976

    Google Scholar 

  5. Kirchgassner, K.&H. Kielhöfer,Stability and bifurcation in fluid dynamics, Rocky Mountains J.of Math. 3,2, 275–318, 1973

    Article  MathSciNet  MATH  Google Scholar 

  6. Andereck, C.D., R. Dickman, H.L. Swinney,New flows in a circular Couette system with co-rotating cylinders, Phys.Fluids 26,6, 1395–1401, 1983

    Article  Google Scholar 

For the Bénard problem, about the question of breaking of symmetries

  1. Golubitsky,M., J.W.Swift, E.Knobloch,Symmetries and Pattern Selection in Rayleigh-Bénard convection, (preprint)

    Google Scholar 

  2. Buzano,E., & M.Golubitsky, Phil. Trans. Roy. Soc., 1983

    Google Scholar 

  3. Chossat,P.Le problème de Bénard dans une couche sphérique, Thèse d'Etat, Univ. de Nice, 1982

    Google Scholar 

A more general exposition is in

  1. Sattinger,D.,Group theoretic methods in Bifurcation theory, Lecture Notes in Maths. 762, Springer 1979.

    Google Scholar 

Bibliography

  1. Arnold, V.I., Chapitres supplémentaires de la Théorie des Equations Différentielles ordinaires, MIR, Moscou, 1980

    MATH  Google Scholar 

  2. Aronson, D.G., M.A. Chory, G.R. Hall, R.P. Mc Gehee,Bifurcations from an invariant circle for two-parameters families of maps of the plane, Comm.Math.Phys. 83, 303, 1982

    Article  MathSciNet  MATH  Google Scholar 

  3. Herman, M.,Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ.,I.H.E.S., 49, 1979

    Google Scholar 

  4. Iooss,G.,Bifurcation of maps and applications, North Holland Math.Studies 36, 1979

    Google Scholar 

  5. Iooss, G., A. Arnéodo, P. Coullet, C. Tresser,Simple computation of bifurcating invariant circles for mappings, Lect.Notes in Math. 898,192–211, 1980

    Article  MathSciNet  MATH  Google Scholar 

  6. Iudovich, V.I.,On the stability of self-oscillations of a liquid, Dokl.Acad. Nauk SSSR 195,3, 574–576 (1970), (in Russian)

    Google Scholar 

  7. Marsden,J., M.Mc Cracken,The Hopf bifurcation and its applications, Applied Math 19, Springer Verlag, 1976

    Google Scholar 

  8. Neimark, J.,On some cases of periodic motions depending on parameters, Dokl. Akad. Nauk. SSR 129, 736–739, (1959), (in Russian)

    MathSciNet  Google Scholar 

  9. Rand, D.,Dynamics and Symmetry: Predictions for Modulated Waves in rotating fluids, Arch. Rat. Mech. Anal. 79,1, 1–38, (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Renardy, M.,Bifurcation from rotating waves, Arc.Rat.Mech.Anal. 79,1, 49–84, (1982)

    MathSciNet  MATH  Google Scholar 

  11. Ruelle, D. & F. Takens,On the nature of turbulence, Comm. Math.Phys. 20, 167–192, 1971

    Article  MathSciNet  MATH  Google Scholar 

  12. Sacker,R.J.,On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, New York Univ. IMM.NYU, 333, 1964

    Google Scholar 

  13. See ref.[5] of chapter I

    Google Scholar 

  14. See ref.[2] of chapter I.

    Google Scholar 

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Luigi Salvadori

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Iooss, G. (1984). Bifurcation and transition to turbulence in hydrodynamics. In: Salvadori, L. (eds) Bifurcation Theory and Applications. Lecture Notes in Mathematics, vol 1057. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098596

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