Hyperchaos and Julia Sets

Abstract

Julia sets are self-similar separatrices of the coast-line type found in noninvertible 2-D maps. The same class of maps also generates hyperchaos (chaos with two mixing directions). Smale’s notion of a “nontrivial basic set” provides a connection. These sets arise when a chaotic (or hyperchaotic) attractor “explodes”. In the case of more than one escape route, this set becomes a “fuzzy boundary” (Mira). Its projection as the map becomes noninvertible (1-D ) is a “Julia set in 1 D ”. In the analogous hyperchaos case the 2-D limiting map contains a classical Julia set of the continuous type. An identically looking set can also be obtained within a non-exploded hyperchaotic attractor, however, as a “cloud”. Julia-like attractors therefore exist. The theory also predicts Mandelbrot sets for 4-D flows. Julia-like behavior is a new, numerically easy-to-test for property o f most nontrivial dynamical systems.

Symmetry and Self-Similarity with Coupled Logistic Maps

“…Further, in accordance with predictions of Roessler et al [4], the map possesses a similar complexity in the parameter space. Neither here nor in the phase space our experiments resulted in classical fractal basin boundaries.…”

“…Roessler et al [4] mention an earlier example of a similar Julia boundary discovered by Mira [15] in the study of two different embedded periodic attractors coexisting in a cubic analogue to Henon's map. Mira called the object "frontiere floue" (fuzzy boundary); both basins were found to accumulate on a Cantor set.…”

“…This Julia-like [4,14] "speed pattern" of the pear loses its clarity, when we increase the parameter It. This is due to the fact, that for h > 0.6 high order periodic, quasiperiodic and chaotic motion appears.…”

“…Mira called the object "frontiere floue" (fuzzy boundary); both basins were found to accumulate on a Cantor set. In [4]. the authors themselves give an almost trivial example of continuous Julia-like boundaries with noninvertible (real) two-dimensional maps; they studied two uncoupled logistic maps in the "exploded" (chaotic) case.…”

“…We have considered the basins of coexisting periodic attractors which form stripe structures in a self-similar manner. Their separatrices are continuous Julia-like [4] sets. Corresponding self-similar features occur, when we study the level sets of equal attraction in the basins of low order cycles.…”