Localization of Invariant Compact Sets of Dynamical Systems

Recently, a number of problems that can be called localization problems have been considered in qualitative theory of differential equations [1–15]. Their statements slightly vary, but geometrically one seeks subsets with some properties in the phase space. (For example, these subsets should contain certain solutions of a system of differential equations.) Such problems include the construction of positively invariant sets containing some separatrices of the Lorenz system [1]; analysis of the asymptotic behavior of solutions of the Lorenz system; finding sets that contain an attractor of the Lorenz system [2–5, 14]; finding sets containing all periodic trajectories [6–13] or separatrices and other trajectories [10, 11]. Such sets are naturally said to be localizing, and it is obviously of interest to consider methods and results that permit one to find the sharpest possible localizing sets for any structure in the phase space. The present paper focuses attention on the localization of invariant compact sets of a system of differential equations in the phase space of the system, that is, on the problem of finding subsets of the phase space that contain all invariant compact sets of the system. Invariant compact sets include equilibria, periodic trajectories, separatrices, limit cycles, invariant tori, and other sets as well as their finite unions. These sets and their properties largely determine the structure of the phase space and the qualitative behavior of solutions of a system of differential equations. In Section 1, we introduce the notation and present the main results obtained in [6–8] for the problem of localization of periodic trajectories. Finding localizing sets for a specific autonomous system of differential equations with the use of the method in [6–8] requires finding the least upper and greatest lower bounds of a function on some set. The latter problem has a special form, and its properties are discussed in Section 2. In Section 3, we show that the localizing sets constructed in [6–8] actually contain all invariant compact sets; we state a theorem that justifies our method for finding localizing sets for invariant compact sets of the system. As a corollary, we obtain a sufficient condition for the absence of invariant compact sets in a subset of the phase space; this condition can be viewed as an analog of the well-known Bendixson theorem for two-dimensional systems. In Section 4, we construct localizing sets for invariant compact sets of the Lorenz system, and in Section 5, we discuss the efficiency of various approaches to localization of the attractor. Section 6 gives an iterative procedure for the localization of invariant compact sets and discusses results obtained by this procedure.