Dynamical behavior of a two dimensional difference equation system with quadratic term

This article investigates the dynamical behavior of a two-dimensional asymmetric system of fractional difference equations given by μ n + 1 = 1 + ℏ μ n ν n − 1 2 , ν n + 1 = 1 + ℏ ν n μ n − 1 2 , n ∈ ℕ , where ℏ > 0 is a parameter and the initial values μ i , ν i ( i = − 1 , 0 ) are positive real numbers. By employing linear stability analysis and eigenvalue localization within the unit disk, we rigorously establish the existence and stability of equilibrium points. For 0 < ℏ ≤ 3 4 , the system exhibits a unique symmetric equilibrium ( ξ , ξ ) that is globally asymptotically stable. When 3 4 < ℏ < 1 , two distinct asymmetric equilibria emerge, both of which retain local and global asymptotic stability. Furthermore, the boundedness and persistence of the solutions are demonstrated for all 0 < ℏ <