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Route to chaos in three-dimensional maps of logistic type. (English) Zbl 1132.37309

Summary: A route to chaos is studied in 3-dimensional maps of logistic type. Mechanisms of doubling for invariant closed curves (ICC) are found for specific 3-dimensional maps. These bifurcations cannot be observed for ICC in the 2-dimensional case. When the parameter of the system is modified, localized oscillations occur on the ICC that give rise to weakly chaotic rings, then to chaotic attractors, which finally disappear by contact bifurcations. These maps can be considered as models for the symbiotic interaction of three species.

MSC:

37E05 Dynamical systems involving maps of the interval
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N25 Dynamical systems in biology