In this paper we study a multiparametric nonlinear system with a transcritical bifurcation in a region of points of ${\mathbb{R}}^3$. The parametric regions that constitute the boundaries where important qualitative changes occur in the dynamics of the system are determined. The equilibrium points in each of the regions are also established and classified. Finally, the stability of the equilibrium points at infinity of the system obtained from the Poincare compactification is classified, and the global phase portrait of the system is made.

In this work, the notion of function of bounded variation in 2-normed spaces was established (Definition 4.2), the set of functions of bounded $(2,k)$-variation was endowed with a norm (Theorem 4.5), and it was proved that such a set is a Banach space (Theorem 4.6). In addition, the fundamental properties of the functions of bounded $(2,k)$-variation in the formalism of 2-Hilbert and 2-normed spaces were studied (see Theorems 4.1, 4.3, 4.4). Also, it was shown how to endow a 2-normed space with a function of bounded $(2,k)$-variation from a classical Hilbert space (Proposition 4.1). A series of examples and counterexamples are presented that enrich the results obtained in this work (4.1 and 4.2).