STRANGE ATTRACTORS OF INFINITESIMAL WIDTHS IN THE BIFURCATION DIAGRAMWITH AN UNUSUAL MECHANISM OF ONSET - NONLINEAR DYNAMICS IN COUPLED FUZZY CONTROL-SYSTEMS - II

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.