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Volumn 55, Issue 5, 1997, Pages 5350-5360

Chaos and crises in more than two dimensions

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EID: 0008025028     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.55.5350     Document Type: Article
Times cited : (16)

References (39)
  • 9
    • 0003582543 scopus 로고
    • Cambridge University Press, Canada
    • E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Canada, 1993), p. 151.
    • (1993) Chaos in Dynamical Systems , pp. 151
    • Ott, E.1
  • 10
    • 85037187496 scopus 로고    scopus 로고
    • 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)
    • 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).
  • 20
    • 0001870459 scopus 로고
    • 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).
    • (1991) Ann. Math. , vol.133 , pp. 73
    • Benedicks, M.1    Carleson, L.2
  • 21
  • 22
    • 85037230958 scopus 로고    scopus 로고
    • Ergodic Theory of Dynamical Systems (Springer, New York, 1993), p. 201
    • Ergodic Theory of Dynamical Systems (Springer, New York, 1993), p. 201.
  • 34
    • 85037198153 scopus 로고    scopus 로고
    • 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
    • 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.
  • 37
    • 0000498246 scopus 로고
    • 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
    • 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.
    • (1986) Phys. Rev. Lett. , vol.56 , pp. 1011
    • Grebogi, C.1    Ott, E.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.