<p>In the theory of dynamical systems, the basin of attraction (BA) serves a pivotal role in analysis, design and control. In many cases, the BA could be embedded in a high-dimensional phase space. This makes such BA a suitable choice for a practical study of a high-dimensional object. The layout (boundaries) of a BA normally is shown in 2D i.e. via the intersection of the separatrix manifold and a typical 2D section. Of major interest, from practical point of view, is finding the general shape of the separatrix and hence, its 2D projection/intersection. Here, taking the BA of a 4D system, we study how the peculiarities of the 4D-to-2D projection serves in finding the expression for an important substructure of the BA. We notice the so-called “compact regions” as some characteristic substructures of the BA and suggest an analytical approach to detect such regions. As an important application in sensitivity analysis, we then show how the technique easily captures such compact regions in the perturbed versions of the system too. Then we discuss that the compact regions are, in general, the regions of the highest entropy and investigate how to drive the system towards such regions; an important control strategy in applications where the most diverse/versatile behavior of the system is desired. Alongside, we also present another significant finding; we coin the term “double perspective” to discuss how the demonstration of a 4D object is related to its 2D projections and how this is exemplified by the compact regions of a BA.</p>