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.