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1
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84927879189
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Related but different work has been reported by J. M. Deutsch who obtains phase transitions for iterated one-dimensional maps of the form M ( x , n ) = x + εn(x), where εn(x) is random and has zero mean [Phys. Rev. Lett. 52, 1230 (1984)
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3
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84927879187
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Fractal measures resulting from random dynamical systems have been considered by F. Ladrappier and L.-S. Young [, ], and also F. J. Romeiras, C. Grebogi, and E. Ott [, Phys. Rev. A, 41, 784, ]. This paper uses the thermodynamic formalism to study the multifractal properties of random-map snapshot attractors.
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(1989)
Commun. Math. Phys.
, vol.117
, pp. 529
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4
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84927879184
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Romeiras, Grebogi, and Ott, Ref. 2.
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5
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84927879181
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A review and references are contained in J. M. Ottino, Kinematics of Mixing: Stretching Chaos and Transport (Cambridge Univ. Press, Cambridge, 1989).
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7
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84927879168
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This paper also emphasizes the relevance of random maps.
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11
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84927823013
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Equation (1) with θn= 0 is often referred to as the Zaslavsky map and can yield strange attractors
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(1978)
G. M. Zaslavsky, Phys. Lett.
, vol.69 A
, pp. 145
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12
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84927879166
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Laboratory for Plasma Research, University of Maryland, report, 1990 (to be published).
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Yu, L.1
Ott, E.2
Chen, Q.3
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14
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84927879165
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Details concerning how the dotted curve in Fig. 1 is obtained will appear in a future publication (Ref. 8).
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15
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84927879163
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We have said that the chaotic time dependence of the fluid velocity v vec is responsible for the randomness of the map. We stress that the former chaos is distinct from chaos of the particle orbits ( xn, yn) obtained from iteration of the random map. [Indeed the latter may or may not be chaotic (Fig. 1) when the time dependence of v vec is chaotic.]
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16
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0001640825
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edited by, H. O. Peitgen, H. O. Walter, Lecture Notes in Mathematics Vol. 730, (Springer-Verlag, Berlin, It is interesting to note that Ladrappier and Young, Ref. 2) are able to prove that the Lyapunov dimension and the information dimension are equal for random diffeomorphisms of arbitrary dimension, while no such proof exists in the case of deterministic (i.e., nonrandom) diffeomorphisms.
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(1978)
Functional Differential Equations and the Approximation of Fixed Points
, pp. 228
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Kaplan, J.L.1
Yorke, J.A.2
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18
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84927879161
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Very near the transition, s can sometimes become smaller than the computer roundoff. When this happens, the effect of computer discreteness forces the coordinates of all the points in the small cloud of particles to have the same value, and the computed s becomes zero for all subsequent iterates. To avoid this artificial result, small noise of the order of 10-8 (different for each particle) is added to obtain the results in Fig. 4. The results are independent of the noise level over many orders of magnitude provided that the noise is much less (Ref. 8) than exp [ - < ln ( 1 / s ) > ]. (This condition begins to be violated for λ1
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