In this paper we explore additional properties of a two-dimensional map denoted by \( T: (x^{\prime }, y^{\prime }) = \left( -\frac{ax}{1 + y^2},\, x + by\right) ,\) which depend on the parameters \(a\) and \(b\) . While this map has been examined in numerous articles, many aspects of its dynamical analysis remain unexplored. Specifically, we delve into the invertibility of the map and demonstrate that it conforms to a \(Z_0-Z_2\) type through critical curve computations. Notably, the map exhibits simplicity through symmetry, which enables us to derive explicit closed-form expressions for both the stable and saddle period-two orbits. This closed-form solution, absent in earlier studies, allows us to precisely locate the repeller 2-cycle and paves the way for a detailed investigation of the bifurcations associated with these cycles. Furthermore, we observe various codimension-one bifurcations including saddle-node, period-doubling, Neimark–Sacker, and subcritical pitchfork bifurcations and present two-parameter Lyapunov charts that illustrate the presence of hyperchaos with two positive Lyapunov exponents. By bridging these gaps in the literature, our analysis offers new insights into the routes to chaos and the complex interplay between periodic and chaotic behavior in non-invertible systems. Additionally, we find that the curves \( b = -\tfrac{7a+8}{a} \) and \( b = 2a + 1 \) intersect at the point \((-2, -3)\) , indicating a codimension-2 bifurcation. This intersection marks the simultaneous occurrence of two distinct bifurcations affecting the system’s dynamics, leading to richer bifurcation structures in its vicinity.
