Localization of compact invariant sets of the Lorenz’ 1984 model

In 1984 E. Lorenz published a paper [1] in which he proposed “the simplest possible general circulation model”: which is referred to as the Lorenz’1984 model. The existence of chaos was shown in [1, 2] for different values of parameters. Dynamical studies of this system were realized in papers [1, 2]; [3], [4]. This paper is devoted to study of a localization problem of compact invariant sets of the Lorenz’1984 model with help of one approach elaborated in papers of Krishchenko and Starkov, see e.g. [5]. This problem is an important topic in studies of dynamics of a chaotic system because of the interest to a long-time behavior of a system. In this work we establish that all compact invariant sets of the Lorenz’ 1984 model are contained in the set. Further, we improve this localization with help of refining bound η using additional localizations sets. By applying cylindrical coordinates to the Lorenz’ 1984 model we derive yet another localization set of the form. Finally, we discuss how to improve the final localization set and consider one example.