Abstract
The “strange attractors” plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction. In view of this we study a new definition of attractors based on ideas of Conley. We argue that the attractors observed in the presence of small random perturbations correspond to this new definition.
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Ahlers, G.: Low-temperature studies of the Rayleigh-Bénard instability and turbulence. Phys. Rev. Lett.33, 1185–1188 (1974)
Bourbaki, N.: Topologie générale (fascicule de résultats). Paris: Hermann 1953
Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin, Heidelberg, New York: Springer 1975
Bowen, R.: On axiomA diffeomorphisms. CBMS Regional Conference, Ser. 35. Providence RI: Am. Math. Soc. 1978
Bowen, R., Ruelle, D.: The ergodic theory of AxiomA flows. Invent. math.29, 181–202 (1975)
Campanino, M., Epstein, H.: On the existence of Feigenbaum's fixed point. Commun. Math. Phys.79, 261–302 (1981)
Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Progress in physics, Vol. 1. Boston: Birkhäuser 1980
Collet, P., Eckmann, J.-P., Lanford, O.E.: Universal properties of maps of an interval. Commun. Math. Phys.76, 211–254 (1980)
Conley, Ch.: Isolated invariant sets and the Morse index. CMBS Regional Conference, Ser. 38. Providence RI: Am. Math. Soc. 1978
Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys.19, 25–52 (1978)
Feigenbaum, M.J.: The universal metric properties of nonlinear transformations. J. Stat. Phys.21, 669–706 (1979)
Feigenbaum, M.J.: The transition to a periodic behaviour in turbulent systems. Commun. Math. Phys.77, 65–86 (1980)
Fox, R.F.: Hydrodynamic fluctuation theories. J. Math. Phys.19, 1993–1999 (1978)
Gollub, J.P., Swinney, H.L.: Onset of turbulence in a rotating fluid. Phys. Rev. Lett.35, 927–930 (1975)
Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys.50, 69–77 (1976)
Hopf, E.: A mathematical example displaying the features of turbulence. Commun. Pure Appl. Math.1, 303–322 (1948)
Hurley, M.: Attractors: persistence and density of their basins. (to appear)
Kifer, Yu.I.: On small random perturbations of some smooth dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat.38, 1091–1115 (1974). English translation: Math. USSR Izv.8, 1083–1107 (1974)
Kifer, Yu.I.: Stochastic stability of the topological pressure. J. Anal. Math.38, 255–286 (1980)
Landau, L.D.: Turbulence. Dokl. Akad. Nauk SSSR44, 339–342 (1944)
Lanford, O.E.: A computer-assisted proof of the Feigenbaum conjectures (to appear)
Levy, Y.: Some remarks about computer studies of dynamical systems (to appear)
Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci.20, 130–141 (1963)
Misiurewicz, M.: Structure of mappings of an interval with zero entropy. Publ. Math. IHES53, 5–16 (1981)
Pesin, Ya.B.: Invariant manifold families which correspond to non-vanishing characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat.40, 1332–1379 (1976). English translation: Math. USSR Izv.10, 1261–1305 (1975)
Pugh, C.C., Shub, M.: Differentiability and continuity of invariant manifolds (to appear)
Ruelle, D.: Am. J. Math.98, 619–654 (1976)
Ruelle, D.: Dynamical systems with turbulent behavior. Mathematical problem in theoretical Physics. Lecture Notes in Physics, Vol. 80, pp. 341–360. Berlin, Heidelberg, New York: Springer 1978
Ruelle, D.: Thermodynamic formalism. Encyclopedia of Math. and its Appl. Vol. 5. Reading, Mass.: Addison-Wesley 1978
Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math. IHES50, 275–306 (1979)
Ruelle, D.: Characteristic exponents and invariant manifolds in Hilbert space. Ann. Math. (to appear)
Ruelle, D.: Differentiable dynamical systems and the problem of turbulence. Bull. Am. Math. Soc. (New Series)5, 29–42 (1981)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys.20, 167–192 (1971). Note concerning our paper “On the nature of turbulence”. Commun. Math. Phys.23, 343–344 (1971)
Sinai, Ya.G.: Gibbsian measures in ergodic theory. Usp. Mat. Nauk27, 21–64 (1972). English translation: Russ. Math. Surv.27, 21–69 (1972)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747–817 (1967)
Smale, S.: The Θ-stability theorem. Global analysis. Proc. Symp. Pure Math., Vol. 14, pp. 289–298. Providence, RI: Am. Math. Soc. 1970
Williams, R.F.: Expanding attractors. Publ. Math. IHES43, 169–203 (1974)
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Ruelle, D. Small random perturbations of dynamical systems and the definition of attractors. Commun.Math. Phys. 82, 137–151 (1981). https://doi.org/10.1007/BF01206949
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DOI: https://doi.org/10.1007/BF01206949