Abstract
A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Abraham, R., Marsden, J.: Foundations of mechanics. New York: Benjamin 1967.
Bass, J.: Fonctions stationnaires. Fonctions de corrélation. Application à la représentation spatio-temporelle de la turbulence. Ann. Inst. Henri Poincaré. Section B5, 135–193 (1969).
Brunovsky, P.: One-parameter families of diffeomorphisms. Symposium on Differential Equations and Dynamical Systems. Warwick 1968–69.
Hirsch, M., Pugh, C. C., Shub, M.: Invariant manifolds. Bull. A.M.S.76, 1015–1019 (1970).
—— —— -- Invariant manifolds. To appear.
Hopf, E.: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig94, 1–22 (1942).
Kelley, A.: The stable, center-stable, center, center-unstable, and unstable manifolds. Published as Appendix C of R. Abraham and J. Robbin: Transversal mappings and flows. New York: Benjamin 1967.
Landau, L. D., Lifshitz, E. M.: Fluid mechanics. Oxford: Pergamon 1959.
Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math.63, 193–248 (1934).
Moser, J.: Perturbation theory of quasiperiodic solutions of differential equations. Published in J. B. Keller and S. Antman: Bifurcation theory and nonlinear eigenvalue problems. New York: Benjamin 1969.
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747–817 (1967).
Thom, R.: Stabilité structurelle et morphogénèse. New York: Benjamin 1967.
Williams, R. F.: One-dimensional non-wandering sets. Topology6, 473–487 (1967).
Berger, M.: A bifurcation theory for nonlinear elliptic partial differential equations and related systems. In: Bifurcation theory and nonlinear eigenvalue problems. New York: Benjamin 1969.
Fife, P. C., Joseph, D. D.: Existence of convective solutions of the generalized Bénard problem which are analytic in their norm. Arch. Mech. Anal.33, 116–138 (1969).
Krasnosel'skii, M.: Topological methods in the theory of nonlinear integral equations. New York: Pergamon 1964.
Rabinowitz, P. H.: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rat. Mech. Anal.29, 32–57 (1968).
Velte, W.: Stabilität und Verzweigung stationärer Lösungen der Navier-Stokesschen Gleichungen beim Taylorproblem. Arch. Rat. Mech. Anal.22, 1–14 (1966).
Yudovich, V.: The bifurcation of a rotating flow of fluid. Dokl. Akad. Nauk SSSR169, 306–309 (1966).
Author information
Authors and Affiliations
Additional information
The research was supported by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).
Rights and permissions
About this article
Cite this article
Ruelle, D., Takens, F. On the nature of turbulence. Commun.Math. Phys. 20, 167–192 (1971). https://doi.org/10.1007/BF01646553
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01646553