Abstract
This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.
Highlights
Introduction and PreliminariesWe consider difference equation: xn+1 =Bxnxn−1 + F Axn2 + bxn xn−1, n = 0, 1, . . . .By substitution xnB b yn we get yn+1 = B3 b yn AB3 b2 y2n yn−1 + + yn bF yn−1
We show that Equation (1) exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which is locally stable if A < 1, nonhyperbolic if A = 1 when there exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle if A > 1
By center manifold theory, we investigate the stability of the non-hyperbolic equilibrium point x
Summary
We prove that Equation (1) has a unique positive equilibrium that can be locally asymptotically stable, non-hyperbolic, or a saddle point in a particular parametric space. The unique equilibrium point x of Equation (1) is (i) locally asymptotically stable (LAS) if A < 1, (ii) a saddle point (SP) if A > 1, (iii) a non-hyperbolic (NH) if A = 1, with eigenvalues and we have λ−. In view of Theorem 5.9 of [14] the study of the stability of the zero equilibrium of Equation (8), that is the positive non-hyperbolic equilibrium of Equation (7), reduces to the stability of the following equation rn+1 = −rn + γ(rn, sn) = G(rn),. From Theorem 5.9 of [14], the zero equilibrium of Equation (8), that is the positive non-hyperbolic equilibrium of Equation (7) is unstable
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