Strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset Nonlinear dynamics in coupled fuzzy control systems. II

Abstract

A new type of onset of chaos in a multidimensional dissipative system is reported that has been observed under a circumstance in which a delicate 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 zero 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 macroscopic scale. The chaos zones appear alternately with the windows of resonance (frequency locking) in the bifurcation diagram, whose number of periodicity decreases, as a system parameter is increased, from an infinity in the manner of arithmetical progression 28 n+2, with n being integers down to unity. Concomitantly the extent of chaos develops, and thereby grows measurable. Also, the ranges of both the chaos and resonance zones in the bifurcation diagram become wider. Two Lorenz plots of “quantum numbers” n and n + 1 are always necessary to characterize the nth chaos zone. For a chaotic state they appear in such a manner as to sandwich the identity map y = x. This chaos is readily interrupted 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 tangential parts of the individual Lorenz plots to y = x have as many folds as their quantum numbers. They form clusters, each of which consists of many stable points being located densely. At the limit of onset, n = ∞, both the intervals between these stable points and the folds lying between two neighboring stable points converge to zero. This is the origin of the unmeasurable degree of chaos.