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1
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0000252317
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S. Dawson, C. Grebogi, T. Sauer and J.A. Yorke, Phys. Rev. Lett. 73, 1927 (1994).
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(1994)
Phys. Rev. Lett.
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, pp. 1927
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Dawson, S.1
Grebogi, C.2
Sauer, T.3
Yorke, J.A.4
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9
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0003582543
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Cambridge University Press, Canada
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E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Canada, 1993), p. 151.
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(1993)
Chaos in Dynamical Systems
, pp. 151
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Ott, E.1
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10
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85037187496
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T. Tél, in Directions in Chaos, Vol. 3, edited by H. Bai-lin (World Scientific, Singapore, 1990), p. 149; T. Tél, in STATPHYS 19, edited by H. Bai-lin (World Scientific, Singapore, 1996)
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T. Tél, in Directions in Chaos, Vol. 3, edited by H. Bai-lin (World Scientific, Singapore, 1990), p. 149; T. Tél, in STATPHYS 19, edited by H. Bai-lin (World Scientific, Singapore, 1996).
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20
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0001870459
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Here we use the poorly defined concept of an attractor being ``smooth'' along the unstable directions so as to avoid getting into very deep mathematical ideas whose full discussion goes beyond the scope of this paper. The attractor being smooth means that there is an invariant probability measure that has absolutely continuous conditional measures on unstable manifolds, i.e., an SBR measure. For more details about what this means we refer the reader to 17 . Actually, the existence of an SBR measure for Hénon maps has only been proved for certain regions of parameter values by Benedicks and Young in 16, building upon the work of M. Benedicks and L. Carleson, Ann. Math. 133, 73 (1991).
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(1991)
Ann. Math.
, vol.133
, pp. 73
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Benedicks, M.1
Carleson, L.2
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21
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33845317713
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M. Benedicks and L.-S. Young, Inv. Math. 112, 541 (1993).
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(1993)
Inv. Math.
, vol.112
, pp. 541
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Benedicks, M.1
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22
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85037230958
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Ergodic Theory of Dynamical Systems (Springer, New York, 1993), p. 201
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Ergodic Theory of Dynamical Systems (Springer, New York, 1993), p. 201.
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24
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0023310831
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C. Grebogi, E. Kostelich, E. Ott and J.A. Yorke, Physica D 25, 347 (1987).
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(1987)
Physica D
, vol.25
, pp. 347
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Grebogi, C.1
Kostelich, E.2
Ott, E.3
Yorke, J.A.4
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25
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44049111231
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R. Romeiras, C. Grebogi, E. Ott and W.P. Dayawansa, Physica D 58, 165 (1992).
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(1992)
Physica D
, vol.58
, pp. 165
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Romeiras, R.1
Grebogi, C.2
Ott, E.3
Dayawansa, W.P.4
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34
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85037198153
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Actually, the mathematical model is an idealization of the real system and usually not all points in the phase space correspond to states that can be physically achieved
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Actually, the mathematical model is an idealization of the real system and usually not all points in the phase space correspond to states that can be physically achieved.
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37
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0000498246
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Physica D 24, 243 (1987). Although we cannot guarantee what happens with the whole basin between (Formula presented) and (Formula presented), we can say that at least a neighborhood of (Formula presented) which lies inside (Formula presented) is a smooth deformation of a neighborhood of (Formula presented) in (Formula presented). In many cases this is not just a small neighborhood, but the bulk of the basins
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Actually basins and in particular basin boundaries, can also undergo dramatic changes at certain parameter values as described in C. Grebogi, E. Ott, and J.A. Yorke, Phys. Rev. Lett. 56, 1011 (1986); Physica D 24, 243 (1987). Although we cannot guarantee what happens with the whole basin between (Formula presented) and (Formula presented), we can say that at least a neighborhood of (Formula presented) which lies inside (Formula presented) is a smooth deformation of a neighborhood of (Formula presented) in (Formula presented). In many cases this is not just a small neighborhood, but the bulk of the basins.
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(1986)
Phys. Rev. Lett.
, vol.56
, pp. 1011
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Grebogi, C.1
Ott, E.2
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