Bounds for the domain containing all compact invariant sets of the system modeling dynamics of acoustic gravity waves

In this paper, we study the localization problem of compact invariant sets of the system modeling the dynamics of acoustic gravity waves constructed by Stenflo (the LorenzStenflo system). We discuss relations between compact localization domains and trapping domains with the help of extended invariance principle due to Rodrigues et al. Based on this analysis, we compute the trapping domain for the system modeling the dynamics of acoustic gravity waves. By using the first order extremum conditions and a comparison with localization results obtained earlier for the Lorenz system we find that all compact invariant sets of the LorenzStenflo system are located in the intersection of one-parameter set of ellipsoids with a few domains bounded by some quadratic surfaces. Further, we derive polytopic bounds for the locus of compact invariant sets. Finally, we present the general formulae validating the application of rational localizing functions and use these formulae for the LorenzStenflo system and for the Lorenz system.