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A test for a conjecture on the nature of attractors for smooth dynamical systems
1. Afraĭmovič, V. S. , Bykov, V. V. , and Silnikov, L. P. , “ The origin and structure of the Lorenz attractor,” Dokl. Akad. Nauk SSSR 234, 336–339 (1977).
2. Alves, J. F. , Freitas, J. M. , Luzzatto, S. , and Vaienti, S. , “ From rates of mixing to recurrence times via large deviations,” Adv. Math. 228, 1203–1236 (2011).
6. Benedicks, M. and Young, L.-S. , “ Markov extensions and decay of correlations for certain Hénon maps,” Astérisque 261, 13–56 (2000).
7. Bowen, R. , Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Vol. 470 Lecture Notes in Math. (Springer, Berlin, 1975).
17. Gottwald, G. A. and Melbourne, I. , “ On the implementation of the 0-1 test for chaos,” SIAM J. Appl. Dyn. Syst. 8, 129–145 (2009).
18. Gottwald, G. A. and Melbourne, I. , “ On the validity of the 0-1 test for chaos,” Nonlinearity 22, 1367–1382 (2009).
21. Guckenheimer, J. and Williams, R. F. , “ Structural stability of Lorenz attractors,” Inst. Hautes Études Sci. Publ. Math. 50, 59–72 (1979).
23. Jakobson, M. V. , “ Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,” Commun. Math. Phys. 81, 39–88 (1981).
25. Lorenz, E. , “ Predictability—A problem solved,” in Predictability, edited by T. Palmer (European Centre for Medium-Range Weather Forecast, Shinfield Park, Reading, 1996).
32. Melbourne, I. and Nicol, M. , “ A vector-valued almost sure invariance principle for hyperbolic dynamical systems,” Ann. Prob. 37, 478–505 (2009).
34. Palis, J. , “ A global view of dynamics and a conjecture on the denseness of finitude of attractors,” Astérisque 261, 335–347 (2000).
37. Ratner, M. , “ The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature,” Israel J. Math. 16, 181–197 (1973).
39. Ruelle, D. , Thermodynamic Formalism, Vol. 5 Encyclopedia of Math. and its Applications (Addison Wesley, Massachusetts, 1978).
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Dynamics arising persistently in smooth dynamical systems ranges from regular dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov, uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The latter include many classical examples such as Lorenz and Hénon-like attractors and enjoy strong statistical properties. It is natural to conjecture (or at least hope) that most dynamical systems fall into these two extreme situations. We describe a numerical test for such a conjecture/hope and apply this to the logistic map where the conjecture holds by a theorem of Lyubich, and to the 40-dimensional Lorenz-96 system where there is no rigorous theory. The numerical outcome is almost identical for both (except for the amount of data required) and provides evidence for the validity of the conjecture.
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