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Hayes, S.1
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The Mathematical Theory of Communication (The University of Illinois Press, 1964)
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C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (The University of Illinois Press, 1964).
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Shannon, C.E.1
Weaver, W.2
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8
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85036279775
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It should be noted that [Formula Presented] is not a proof that a system is random since deterministic chaotic systems may also have a topological entropy of [Formula Presented] For instance, the topological entropy of the logistic map at [Formula Presented] is equal to [Formula Presented] but the logistic map is deterministic
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It should be noted that hT=ln 2 is not a proof that a system is random since deterministic chaotic systems may also have a topological entropy of ln 2. For instance, the topological entropy of the logistic map at r=4 is equal to ln 2, but the logistic map is deterministic.
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12
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45949120176
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For chaotic sets arising in two-dimensional invertible maps, every point on the set has a stable and an unstable direction. Distances along the stable (the unstable) direction shrink (expands) on the average exponentially in time. A chaotic set is hyperbolic if there is a stable and an unstable direction at each point of the set, and the angle between them is bounded away from zero. Otherwise the set is nonhyperbolic. In general, nonhyperbolicity is a complicating feature because it can cause fundamental difficulties in the study of the chaotic systems, a known one being the shadowability of numerical trajectories by true trajectories [see, for example, C. Grebogi, S. M. Hammel, and J. A. Yorke, J. Complexity 3, 136 (1987)
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J. Complexity
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Hammel, S.M.2
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C. Grebogi, S. M. Hammel, J. A. Yorke, and T. Sauer, Phys. Rev. Lett. 65, 1527 (1990)
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Phys. Rev. Lett.
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, pp. 1527
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P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics Vol. I (Birkhäuser, Boston, 1980)
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P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics Vol. I (Birkhäuser, Boston, 1980).
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22
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85036226572
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B.-L. Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems (World Scientific, Singapore, 1989)
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B.-L. Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems (World Scientific, Singapore, 1989).
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23
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0003568098
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Springer, New York
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See, for example, K. Aligood, T. Sauer, and J. A. Yorke, Chaos, an Introduction to Dynamical Systems (Springer, New York, 1997).
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Chaos, an Introduction to Dynamical Systems
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Aligood, K.1
Sauer, T.2
Yorke, J.A.3
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85036150773
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See also, J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983)
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See also, J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
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