Abstract
We use the Rayleigh-Bénard convection and a Chemical Reaction problem in order to introduce various concepts in Nonlinear Physics. The nonlinear phenomena described occur in many other fields, as for example Optics, Acoustics, etc.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
(Lord) Rayleigh: Phil. mag. 32, 529 (1916)
S. Chandrasekar: Hydrodynamic and hydromagnetic stability (Dover, New York 1981)
J. Boussinesq: Theorie analytique de la chaleur, Vol. 2 (Paris, Gauthier Villars 1903)
A. Oberdeck: Annalen der Physik und Chemie 7, 271 (1879)
F.H. Busse: Rep. Prog. Phys. 41, 1929–1967 (1978)
H. Haken: Synergetics, (Springer-Verlag, New York 1968)
L. Landau: Dokl. Akad. Nauk SSSR 44, 311 (1944);
L. Landau and E.M. Lifshitz: Fluid mechanics, (Perqamon Press, Oxford 1959)
J. Guckenheimer and P. Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Appl. Math. Sci. 42 (Springer-Verlag, 1983)
J. Carr: Application of the center manifold theory, Lecture Notes, Lefshetz Center for Dynamical Systems, Brown Univ., Providence RI (1975)
J.T. Stuart: J. Fluid Mech. 9, 353 (1960)
P. Coullet and E.A. Spiegel: SIAM J. Appl. Math. 43, 776 (1983)
H. Poincaré: Les Methodes nouvelles de la mecanique celeste: III (Dover, New York 1957)
A. Lindstedt: Memoires de l’academie de Saint-Petersbourg (1882)
L. Gorkov: JETP 6, 311 (1958)
W.V.R. Malkus and G. Verones: J. Fluid. Mech. 4, 225 (1958)
G. Iooss and D.D. Joseph: Elementary Stability and Bifurcation Theory (Springer-Verlag, New York, 1980)
E. Knoblock and J. Guckenheimer: Phys. Rev. A 27, 1 (1983)
A.C. Newell and J.A. Whitehead: J. Fluid Mech. 38, 279 (1969)
L.A. Segel: J. Fluid Mech. 38, 203 (1969)
Y. Pomeau and P. Manneville: J. Phys. (Paris) Lett. 40, 609 (1979)
W. Eckauss: Studies on nonlinear stability theory, Springer Tracts in natural Philosophy, Vol. 6 (Springer, Berlin 1965)
A. Pocheau and V. Croquette: J. Physique 45, 35 (1984)
Y. Kuramoto: Prog. Theor. Phys. 71, 1182 (1984)
L. Kramer and W. Zimmermann: On the Eckauss instability for spatially Periodic Patterns, Preprint 84
P. Coullet and S. Fauve: Rayleigh-Bénard Convection: a typical nonlinear system, to appear in Workshop on Combustion, Flames and Fires held at Les Houches March 1984
E. Siggia and A. Zippelius: Phys. Rev. Lett. 47, 835 (1981)
Hydrodynamic stability and the transition to turbulence. Ed. by H. Swinney and J.P. Gollub, Topics in Applied Physics 45 (Sprinaer-Verlag 1981)
F.H. Busse: J. Fluid Mech. 52, 97 (1972)
J.E. Marsden and M. McCracken: The Hopf bifurcation and its applications (Springer-Verlag, Berlin 1976)
V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, Berlin 1982)
V.I. Arnold: Singularity Theory: Selected Papers, London Mathematical Society Lecture Notes, Vol. 53 (Cambridge University Press, Cambridge 1981)
R. Thom: Structural Stability and Morphogenesis, W.A. Benjamin (Reading MA 1975)
Y. Kuramoto and T. Yamada: Prog. Theor. Phys. 56, 679 (1976)
H.T. Moon, P. Huerre and L.G. Redekopp: Physica D 7, 135 (1983)
C. Bretherton and E.A. Spiegel: Phys. Lett. 96A, 152 (1983)
Y. Kuramoto: Prog. Theor. Phys. Suppl. 64, 346 (1978)
Y. Pomeau, A. Pumir and P. Pelce: Intrinsic stochasticity with many degree of freedom, Preprint (1984)
H.R. Brand, P.C. Hohenberg and V. Steinberg: Phys. Rev. A 27, 591 (1983)
A. Arneodo, P. Coullet, C. Tresser and E.A. Spiegel: Asymptotic chaos (to appear in Physica D)
P. Coullet: Phys. Rep. 103, 95 (1984)
P. Coullet, C. Elphick and E. Tirapegui: Normal forms for periodic perturbations of singular vector field, Preprint (1984)
G. Ahlers and R.P. Behringer: Phys. Rev. Lett. 40, 712 (1978)
J. Maurer and A. Libchaber: J. Phys. Letters 40, 505 (1979)
M. Dubois and P. Berge: Phys. Lett. 72A, 53 (1980)
J.P. Gollub and S.V. Benson: J. Fluid Mech. 100, 449 (1980)
D. Ruelle and F. Takens: Comm. Math. Phys. 20, 167 (1971)
V.I. Arnold: Ordinary differential equations (M.I.T. Press, Cambridge MA 1973)
F. Takens: in Dynamical systems and turbulence, Ed. by D.A. Rand and L.S. Young, Lecture notes in Mathematics 898, 366 (Springer-Verlag, Berlin 1980)
G. Iooss: Bifurcation of maps and applications, Mathematical studies Vol. 36 (North Holland, Amsterdam 1979)
P. Coullet and A. Libchaber: The Mathematics of time in a convection experiment (unpublished)
V.I. Arnold: A.M.S. Transl. Ser. 2, 46, 213 (1965)
A. Denjoy: J. Math. 17, 333 (1932)
M.R. Herman: Publ. Math. IHES, 49, 669 (1982)
A. Libchaber and J. Maurer: in Nonlinear Phenomena at Phase-transition and Instabilities, Ed. by T. Riste (Plenum, New York) 259
J. Maurer and A. Libchaber: J. Phys. Lettres 40, 419 (1979)
M. Giglio, S. Musazzi and V. Perini: Phys. Rev. Lett. 47, 243 (1981)
A. Libchaber, G. Laroche and S. Fauve: J. Phys. Lettres 43, 211 (1982)
P. Coullet, E. Tresser and A. Arneodo: Phys. Lett. 77A, 327 (1980)
P. Collet and J.P. Eckmann: Iterated map on the interval as Dynamical Systems, Progress on Physics Vol. I. (Birckhauser-Boston 1980)
M. Henon: Comm. Math. Phys. 50, 69 (1976)
S.M. Ulam and J. von Neumann: Bull. Amer. Math. Soc. 53, 1120 (1947)
B. Derrida, A. Gervois and Y. Pomeau: Ann. Inst. Henri Poincaré, Vol. XXIX, 3 305 (1978)
P. Coullet and C. Tresser: in Field Theory, Quantization and Statistical Physics, Ed. by E. Tirapegui (Reidel 1981)
C. Tresser and P. Coullet: CR. Acad. Sc. Paris 287A, 577 (1978)
P. Coullet and C. Tresser: J. Phys. Lettres 41, 225 (1980)
B.A. Huberman and J. Rudnick: Phys. Rev. Lett. 45, 154 (1980)
J. Crutchfield, M. Nauenberg and J. Rudnick: Phys. Rev. Lett. 46, 933 (1981);
B. Shraiman, C.E. Wayne and P.C. Martin: Phys. Rev. Lett., 46, 935 (1981)
M.J. Feigenbaum: J. Stat. Phys. 19, 25 (1978)
P. Coullet and C Tresser: J. Phys. Coll. (Paris) C5, 39 (1978)
P. Collet, J.P. Eckmann and O.E. Lanford: Comm. Math. Phys. 76, 211 (1980)
M. Campanino and H. Epstein: Comm. Math. Phys. 79, 261 (1981)
P. Collet, J.P. Eckmann and H. Koch: J. Stat. Phys. 25, 1 (1981)
S.E. Newhouse: Topology 13, 9 (1974)
A. Arneodo, P. Coullet, C. Tresser, J. Maurer, A. Libchaber and D. D’Humieres: Physica D 6, 385 (1983)
R.S. Mac Kay and C Tresser: J. Phys. Lettres 45, 741, (1984)
P. Grassberger and I. Procaccia: Physica D 9, 189 (1983)
B. Malraison, P. Atten, P. Berge and M. Dubois: J. Phys. Lettres 44, 897 (1983)
A. Brandstäter, J. Swift, H.L. Swinney, A. Wolf and J.D. Farmer: Phys. Rev. Lett. 51, 1442 (1983)
P. Coullet: in Stochastic phenomena and chaotic behaviour in complex systems, ed. by P. Schuster (Springer-Verlag 1984)
A. Arneodo, P. Coullet and E.A Spiegel: Phys. Lett. 92A, 369 (1982)
A. Arneodo, P. Coullet, C. Tresser and E.A. Spiegel: Asymptotic chaos, to appear in Physica D.
L.P. Shilnikov: Sov. Math. Dokl. 6, 163 (1965)
L.P. Shilnikov: Math. U.S.S.R. Sbornik 10, 91 (1970)
A. Arneodo, P. Coullet and C. Tresser: Commun. Math. Phys. 79, 573 (1981)
A. Arneodo, P. Coullet and C. Tresser: J. Stat. Phys. 27, 171 (1982)
D.R. Moore, J. Toome, E. Knobloch and N.O. Weiss: Nature 303, 663 (1983)
P. Glendinning and C. Sparrow: J. Stat. Phys. 5, 645 (1984)
P. Gaspard, G. Nicolis: J. Stat. Phys. 31, 499 (1983);
P. Gaspard, R. Kapral and G. Nicolis: J. Stat. Phys. 5, 697 (1984)
C. Tresser, P. Coullet and A. Arneodo: J. Phys. Lettres 41, 243 (1980)
P. Berge, M. Dubois, P. Manneville and Y. Pomeau: J. Phys. Lettres 41, 341 (1980)
P. Manneville and Y. Pomeau: Phys. Letters 75A, 1 (1979)
Y. Pomeau and P. Manneville: Comm. Math. Phys. 74, 189 (1980)
J.E. Hirsch, M. Nauenberg and D.J. Scalapino: Phys. Lett. 87A, 391 (1982)
J.E. Hirsch, B.A. Huberman and D.J. Scalapino: Phys. Rev. A 25, 519 (1982)
S.E. Newhouse, J. Palis and F. Takens: Publ. Math. IHES, 57, 5 (1983)
M.J. Feigenbaum, L.P. Kadanoff and S.J. Shenker: Physica 5D, 370 (1982)
D. Rand, S. Ostlund, J. Sethna and E. Siggia: Phys. Rev. Letters 49, 182 (1982);
S. Ostlund, D. Rand, J. Sethna and E. Siggia: Physica D 8, 303 (1983)
P. Coullet: in Chaos and Statistical Methods, Ed. by Kuramoto (Springer-Verlag (1984)
P. Collet, P. Coullet and C. Tresser: Scenarios under constraint, Preprint (1984)
E.N. Lorenz: J. Atmos. Sci. 20, 130 (1963)
A. Arneodo, P. Coullet and C. Tresser: Phys. Lett. 81A, 197 (1981)
Y. Kuramoto and S. Koga: Phys. Lett. 92A, 1 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Coullet, P. (1985). Nonlinear Phenomena in Dissipative Systems. In: Claro, F. (eds) Nonlinear Phenomena in Physics. Springer Proceedings in Physics, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93289-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-93289-2_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-93291-5
Online ISBN: 978-3-642-93289-2
eBook Packages: Springer Book Archive