کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4623990 | 1339529 | 2006 | 25 صفحه PDF | دانلود رایگان |
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.
Journal: Journal of Mathematical Analysis and Applications - Volume 323, Issue 1, 1 November 2006, Pages 228-252