## 1 Introduction  
Historically, humanity has tried to understand, explain and represent its surrounding environment by using several tools familiar with its perception of reality. Numerous examples of exceptional theories and enlightenment of natural incidents have been explained by instruments provided by different civilizations as the ancient Egyptian, Mesopotamian, Greek, Chinese, Maya, etc. [1]. Up to date, those tools in combination with theories and information accumulated through the years has allowed us to represent and analyze complex physical events in a manageable manner by using, among others, mathematical models. A model, in general terms, is a mathematics representation to study a phenomenon in the best conditions of space, time and cost [2]. Indeed, modeling complexity increases in terms of analysis information needs. For instance, there are several levels of modeling as macroscopic, microscopic (non-spatial, spatial), and submicroscopic. The interested reader is referred for more details about modeling to reference [3]. Particularly, in control theory, two basic and widely used visual analysis tools are based on Poincare maps and phase portraits. A discrete time system, derived from the original continuous system with the main aim to reduce its complexity is referred as Poincare map [PERSON] (1968). On the other hand, a pictographic tool used to represent the instantaneous state of two interest variables in any autonomous system is referred to as phase portrait; this paper deals with the second kind of representation. A regular phase portrait is a two-dimensional, qualitative representation of the system dynamics with an implicit order reduction, information related to time is lost and only the instantaneous behavior is presented for a set of initial conditions [PERSON] (1968). Such initial conditions are represented unambiguously as points in an \(\mathbb{R}^{2}\) plane. From each point, an arrow directed towards the corresponding derivative with size proportional to its magnitude is drawn (e.g., Figure 1). This representation can be extended to \(\mathbb{R}^{3}\) phase portraits; however, some information about the state variables is lost. The three-dimensional phase portrait is turned two-dimensional by a snapshot in order to be printed out and one must select the best view of the 3D-to-2D converted phase portrait or look for an additional degree-of-freedom for the plot, as color or time (by generating a video). Some authors have preferred to graph the solution of the system for a representative initial condition instead of making a phase portrait e.g., Figure 3. For nonlinear systems, an unexpected dynamic behavior can be lost sight since the point of view of the snapshot or the selection of the initial condition rely on the appreciations capacity. Some authors have also intuitively presented multiple phase-portraits for 3 and 4 state variables, by pair-combining [PERSON] (1968); [PERSON] (1968); [PERSON] (1968); however, there is still the question of whether it is possible to extend this idea to higher order systems and its mathematical basis. A brief historical background of phase portrait for dimensions greater than two is given as follows. A first effort to represent phase plane of higher order was analyzed in [PERSON] (1968). In that study, was proposed the use of a \(x^{n}\) vs. \(x\) plane analysis, where the Poincare's fundamental phase plane is contemplated as the simplest case of the general proposed method. A \(\mathbb{R}^{3}\) dimensional case was analyzed in [PERSON] (1968). The authors in that study used as an example, a harmonic oscillator, and also used a Wigner Transformation as an order reduction tool to simplify the system complexity. Another interesting approach was proposed in [PERSON] (1968). In that study, an oriented linear graph for piecewise-linear system called \"generalized phase portrait\" was proposed. A three-phase system was analyzed, but examples for a larger dimension were not reported.