On the Ultimate Dynamics of the Four-Dimensional Rossler System

In this paper, we construct the polytope which contains all compact.-limit sets of the four-dimensional Rossler system which is a generalization of the hyperchaotic Rossler system for the case of positive parameters. Further, we find a few three-dimensional planes containing all compact.-limit sets for bounded positive half-trajectories located in some subdomains in the half-space z > 0. Besides, we analyze one case in which all compact.-limit sets in the half-space z > 0 are contained in one three-dimensional plane. Our approach is based on a combination of the LaSalle theorem and the extreme-based localization method of compact invariant sets.