Transition of fractal dimension in a latticed dynamical system Page: 3 of 11
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Experimental evidence supporting Feigenbaum's route to turbulence2 has
become richer since 1978. In this route, nonlinear systems manifest chaos via
period- doubling bifurcationr.
When the aspect ratio is large, however, very different behavior is found.
As the stress parameter (the Rayleigh number, in the case of Rayleigh-Bfiard
systems) is increased, cascades of instabilities are observed, each step of which adds
new complications to the convective behavior.3,8,10 Unstable patterns are formed
and temporal cbaos8 sets in, with alternating random bursts and quietness: this is
called intermittency.3,8 Noncommensurate frequencies arise in the Fourier
spectrum of the chaotic variable, and entrainment and locking occur as the stress
parameter is varied.3,7,8
In this paper we show that both these routes to turbulence, with all the
properties just described, can be simply simulated with a quadratic map at each site
of a spatial lattice and with a coupling between nearest-neighbor sites. This new
route leads to a fractal dimensions of 2.6 at the oscillatory instability regime and
6.0 at the turbulent regime.
Let u represent the chaotic variable: it may be a velocity component or a
temperature fluctuation of the system being studied. We build a lattice of sites
with a quadratic map u - Q(Q.,u) at each site, and allow interaction between nearest
neighbor sites through a coupling parameter g. We assign a random value of u to
each lattice site. and let the lattice evolve in time steps to n-, n = 1, 2, 3 ..., when
-z is the Poincart time of the system. We find the same behavior for all quadratic
Q(X.u): for example.
Q(A~u) - lw(1-u),
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Duong-van, M. Transition of fractal dimension in a latticed dynamical system, article, March 1, 1986; [Livermore,] California. (digital.library.unt.edu/ark:/67531/metadc1108583/m1/3/: accessed February 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.