STRANGE ATTRACTORS OF INFINITESIMAL WIDTHS IN THE BIFURCATION DIAGRAMWITH AN UNUSUAL MECHANISM OF ONSET - NONLINEAR DYNAMICS IN COUPLED FUZZY CONTROL-SYSTEMS - II
Y. Tomonaga et K. Takatsuka, STRANGE ATTRACTORS OF INFINITESIMAL WIDTHS IN THE BIFURCATION DIAGRAMWITH AN UNUSUAL MECHANISM OF ONSET - NONLINEAR DYNAMICS IN COUPLED FUZZY CONTROL-SYSTEMS - II, Physica. D, 111(1-4), 1998, pp. 51-80
A new type of onset of chaos in a multidimensional dissipative system
is reported that has been observed under a circumstance in which a del
icate balance of order formation and chaos is materialized. Two limit
cycles are connected directly to a sequence of strange attractors, the
widths in the bifurcation (Feigenbaum) diagram of which converge to z
ero in a range close to the onset. The measures of the extent of chaos
are also zero there and the trajectories behave regularly in a macros
copic scale. The chaos zones appear alternately with the windows of re
sonance (frequency locking) in the bifurcation diagram, whose number o
f periodicity decreases, as a system parameter is increased, from an i
nfinity in the manner of arithmetical progression 28n + 2, with n bein
g integers down to unity. Concomitantly the extent of chaos develops,
and thereby grows measurable. Also, the ranges of both the chaos and r
esonance zones in the bifurcation diagram become wider. Two Lorenz plo
ts of ''quantum numbers'' n and n + 1 are always necessary to characte
rize the nth chaos zone. For a chaotic state they appear in such a man
ner as to sandwich the identity map y = x. This chaos is readily inter
rupted by a limit cycle, since one of the Lorenz maps is crossed with
y = x by a small change of the control parameter. Moreover, the tangen
tial parts of the individual Lorenz plots to y = x have as many folds
as their quantum numbers. They form clusters, each of which consists o
f many stable points being located densely. At the limit of onset, n =
infinity, both the intervals between these stable points and the fold
s lying between two neighboring stable points converge to zero. This i
s the origin of the unmeasurable degree of chaos.