Transition to Deterministic Chaos in a Hydrodynamic System
Nonlinear hydrodynamic systems often exhibit transition to chaotic behaviour when a properly defined stress parameter is increased. According to an idea due to Landau (1) the route to chaos has been conjectured to consist of a sequence of bifurcation points at which many incommensurate frequencies are gradually introduced. At variance with this, Lorenz (2) and Ruelle and Takens (3) have sugqested that fluid turbulence is associated with the existence of a strange attractor in phase space. Therefore chaotic behaviour is actually caused by finite dimensional deterministic dynamics and is a consequence of the sensitivity of the solutions to the initial conditions.
KeywordsLyapunov Exponent Rayleigh Number Bifurcation Point Chaotic Behaviour Strange Attractor
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- 1.L.D. Landau and E.M. Lifshitz: Fluid Mechanics (Pergamon, Oxford, 1959).Google Scholar
- 4b.A. Libchaber and J. Maurer, J. Phys, (Paris) Coll. C3 41, C3 51 (1980);Google Scholar
- 4c.A. Libchaber and J. Maurer, Nonlinear Phenomena at Phase Transitions, ed. T. Riste (Plenum, 1982).Google Scholar
- (1979).Commun. Math. Phys. 77, 65 (1980).Google Scholar
- 8.At the NATO Advanced Study Workshop Testing Non-Linear Dynamics, Haverf. College, Pa., June 83, we learned of studies of this type by three groups: P. Berge and M. Dubois; J. Guckenheimer, G. Buzyna and R. Pfeffer; A. Brandstater, J. Swift, H. Swinney, A. Wolf, J.D. Farmer, E. Jen and J.P. Crutchfield.Google Scholar
- 9.S. Chandrasekkar, Hydrodynamic and Hydromagnetic Stability, (Clarendon, Oxford, 1961).Google Scholar
- 11.M. Giglio, S. Musazzi and U. Perini, Evolution of Order and Chaos, ed. H. Haken (Springer-Verlag, Berlin Heidelberg New York, 1982).Google Scholar
- 12a.For further reading on this subject see, for exampleGoogle Scholar
- 14.J.L. Kaplan and J.A. Yorke, Functional Differential Equations and Approximations of Fixed Points, eds. H.O. Walther, Lecture Notes in Math. 730 (Springer, 1979).Google Scholar
- 18.A. Ben-Mizrachi, I. Procaccia and P. Granberger, Phys. Rev. A, to appear.Google Scholar