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The route to chaos for the Kuramoto-Sivashinsky equationThe results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609... . Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.
Document ID
19910004329
Acquisition Source
Legacy CDMS
Document Type
Preprint (Draft being sent to journal)
Authors
Papageorgiou, Demetrios T. (New Jersey Inst. of Tech. Newark., United States)
Smyrlis, Yiorgos (California Univ. Los Angeles., United States)
Date Acquired
September 6, 2013
Publication Date
October 1, 1990
Subject Category
Fluid Mechanics And Heat Transfer
Report/Patent Number
ICASE-90-78
NAS 1.26:187461
NASA-CR-187461
AD-A228590
Accession Number
91N13642
Funding Number(s)
CONTRACT_GRANT: N00014-86-K-0691
CONTRACT_GRANT: NAS1-18605
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.

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