Rapid fluctuations of chaotic maps on RN
The iterates fn of a chaotic map f display heightened oscillations (or fluctuations) as n → ∞. If f is a chaotic interval map in one dimension, then it is now known that the total variation of fn on that interval grows exponentially with respect to n [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. However, the characterization of chaotic behavior of maps in multi-dimensional spaces is generally much more challenging. Here, we generalize the definition of bounded variations for vector-valued maps in terms of the Hausdorff measure and then use it to study what we call rapid fluctuations on fractal sets in multi-dimensional chaotic discrete dynamical systems. The relations among rapid fluctuations, strict turbulence and positive entropy are established for Lipschitz continuous systems on general N-dimensional Euclidean spaces. Applications to planar monotone or competitive systems, and triangular systems on the square are also given. © 2005 Elsevier Inc. All rights reserved.
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