Abstract
During the 60s and 70s, the idea was put forward that turbulence corresponds to hyperbolicity (i.e. exponential separation of most nearby orbits) for the governing equations [A1,L,RT].
This is a preview of subscription content, log in via an institution to check access.
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
Arnol’d VI, Sur la géometrie des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaites, Ann Inst Fourier Grenoble 16 (1966) 319–361
Arnol’d VI, Geometric methods in the theory of ordinary differential equations (Springer, 1983)
Baesens C, Guckenheimer J, Kim S, MacKay RS, Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos, subm. to Physica D
Baesens C, Guckenheimer J, Kim S, MacKay RS, Simple resonance regions for torus diffeomorphisms, an IMA Conf Proc, to appear
Battelino PM, Grebogi C, Ott E, Yorke JA, Chaotic attractors on a 3-torus and torus breakup, Physica D 39 (1989) 299–314
Bogoljubov NN, Mitropolski Ju A, Samoilenko AM, Methods of accelerated convergence in nonlinear mechanics (Springer, 1976 )
Bohr T, Rand DA, A mechanism for the generation of turbulent spots, Physica D, to appear
Broomhead DS, Jones R, King GP, Topological dimension and local coordinates from time series data, J Phys A 20 (1987) L563–9
Broomhead DS, King GP, Extracting qualitative dynamics from experimental data, Physica D 20 (1986) 217–236
Bunimovich LA, Strange attractors, Chapter 8.2, in Dynamical Systems II, ed Sinai Ya G (Springer, 1989 )
Bunimovich LA, Sinai Ya G, Space-time chaos in coupled map lattices, Nonlinearity 1 (1988) 491–516
Chenciner A, Iooss G, Bifurcations de tores invariants, Arch Rat Mech Anal 69 (1979) 109–198
Glazier JA, Libchaber A, Quasiperiodicity and dynamical systems: an experi-mentalist’s view, IEEE Trans Circ Sys 35 (1988) 790–809
Gollub J, Benson SV, Many routes to turbulent convection, J Fluid Mech 100 (1980) 449–470
Gollub JP, Libchaber A, Laboratory experiments on the transition to chaos, in Chaotic behaviour of deterministic systems, eds Iooss G, Helleman RHG, Stora R (N. Holland, 1983 ) 591–607
Gollub JP, Swinney HL, Onset of turbulence in a rotating fluid, Phys Rev Lett 35 (1975) 927–930
Grebogi C, Ott E, Yorke JA, Are three-frequency quasiperiodic orbits to be expected in typical nonlinear dynamical systems? Phys Rev Lett 51 (1983) 339–342;
Grebogi C, Ott E, Yorke JA, Attractors on an n-torus: Quasiperiodicity versus chaos, Physica D 15 (1985) 354–373
Grebogi C, Ott E, Yorke JA, Crises, sudden changes in chaotic attractors, and transient chaos, Physica D 7 (1983) 181–200
Guckenheimer J, Holmes P, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, 1983 )
Herman MR, Sur la conjugaison differentiable des difféomorphismes du cercle a des rotations, Publ Math IHES 49 (1979) 5–234
Iooss G, Los J, Quasigenericity of bifurcations to high-dimensional invariant tori for maps, Commun Math Phys 119 (1988) 453–500
Kaneko K, Collapse of tori and genesis of chaos (World Sci, 1985)
Langford WF, Periodic and steady-state mode interactions lead to tori, SIAM J Appl Math 37 (1979) 22–48
Linsay PS, Cumming AW, Three-frequency quasiperiodicity, phase-locking and the onset of chaos, Physica D 40 (1989) 196–217
Lorenz EN, Deterministic nonperiodic flow, J Atmos Sci 20 (1963) 130–141
MacKay RS, notes written with the assitance of Oliveira A, Torus Maps, to appear in the proceedings of the Como School on Chaos, Order and Patterns, ed Cvitanovic P (Plenum)
MacKay RS, Muldoon MR, Topology from a time series, in preparation
Maurer J, Libchaber A, Effect of the Prandtl number on the onset of turbulence in liquid 4He, J Physique Lett 41 (1980) L515–8
Newell AC, Rand DA, Russell D, Turbulent transport and the random occurrence of coherent events, Physica D 33 (1988) 281–303
Newhouse SE, Ruelle D, Takens F, Occurrence of strange Axiom-A attractors near quasiperiodic flows on Tm, m >_ 3, Commun Math Phys 64 (1978) 35–40
Pesin Ya B, General theory of smooth hyperbolic dynamical systems, Chapter 7, in Dynamical Systems II, ed Sinai Ya G (Springer, 1989 )
Ruelle D, Chaotic evolution and Strange attractors (CUP, 1989)
Ruelle D, Takens F, On the nature of turbulence, Commun Math Phys 20 (1971) 167–192; and Commun Math Phys 23 (1971) 343–4
Ruelle D, Chaotic evolution and Strange attractors (CUP, 1989)
Ruelle D, Takens F, On the nature of turbulence, Commun Math Phys 20 (1971) 167–192?
Ruelle D, Takens F, Commun Math Phys 23 (1971) 343–4
Sell G, Arch Rat Mech Anal 69 (1979) 199-
Sell G, Resonance and bifurcation in Hopf-Landau dynamical systems, in Nonlinear Dynamics and Turbulence, eds Barenblatt GI, Iooss G, Joseph DD (Pitman, 1983 ) 305–313
Swinney HL, Observations of order and chaos in nonlinear systems, Physica D 7 (1983) 3–15
Walden RW, Kolodner P, Passner A, Surko CM, Nonchaotic Rayleigh-Bénard convection with four and five incommensurate frequencies, Phys Rev Lett 53 (1984) 242–5
Zeeman EC, Bifurcation, catastrophe, and turbulence, in New Directions in Applied Mathematics, eds Hilton PJ, Young GS (Springer, 1982 ) 109–153
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
MacKay, R.S. (1991). An Appraisal of the Ruelle-Takens Route to Turbulence. In: Jiménez, J. (eds) The Global Geometry of Turbulence. NATO ASI Series, vol 268. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3750-2_21
Download citation
DOI: https://doi.org/10.1007/978-1-4615-3750-2_21
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6670-6
Online ISBN: 978-1-4615-3750-2
eBook Packages: Springer Book Archive