Codimension-1 and Codimension-2 Bifurcations and Stability of Certain Second-Order Rational Difference Equation with Arbitrary Parameters

Motivated by previous investigations that analyzed the boundedness of positive solutions, global stability, and the occurrence of Neimark-Sacker bifurcation in specific parameter cases, this paper comprehensively investigates the dynamics of certain second-order rational difference equation with four positive parameters and positive initial conditions. We provide a complete topological classification of fixed (equilibrium) points and examine the local behavior of orbits in the neighborhood of these points, which, to our knowledge, has not been previously studied in the entire admissible parameter space. Our research has discovered highly complex and rich dynamic behavior, ranging from the occurrence of supercritical and sub-critical Neimark-Sacker bifurcations in different parameter spaces to the appearance of codimension-2 bifurcations in the case of 1:1 strong resonance. A very interesting situation appears when one of the equilibria is nonhyperbolic in a specific parameter space; direct calculations have shown that both the first and second Lyapunov coefficients are equal to zero, implying that this equilibrium is a Hopf point of codimension at least 3. This strongly suggests the complex behavior of the studied equation, which the numerical simulations have also confirmed.