The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (a, b), when a third parameter c varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (x, y).

The theory of critical curves determines the main characteristics of a discrete dynamical system in two dimensions. One important property that has garnered recent attention is the problem of chaos synchronization, along with the location of its chaotic attractors, basin boundaries, and bifurcation mechanisms. Varying the parameters of the maps reveals the instrumental role that these curves play, where the bifurcation leads to complex topological structures of the basins occurs by contact with the basin boundaries, resulting in the appearance or disappearance of some components of the basin. This study focuses on the properties of a discrete dynamical system consisting of two symmetrically coupled non-invertible maps, specifically those with an invariant one-dimensional submanifold (or one-dimensional maps). These maps exhibit a complex structure of basins with the coexistence of symmetric chaotic attractors, riddled basins, blow-out, on-off intermittency, and, most significantly, the appearance of chaotic synchronization with a correlation between all the characteristics. The numerical method of critical curves can be used to demonstrate a wide range of dynamic scenarios and explain the bifurcations that lead to their occurrence. These curves play a crucial role in a system of two symmetrically coupled maps, and their significance will be discussed.