Abstract
We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12, n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.
Highlights
Introduction and PreliminariesIn this paper, we investigate the local and global dynamics of the following difference equation: xn+1 =Bxnxn−1 + F bxnxn−1 + cx2n−1 n = 0, 1, . . . (1)where the parameters B, F, b, c are positive real numbers and initial conditions x−1 and x0 are arbitrary positive real numbers
We prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability
We show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period
Summary
We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1 = (Bxnxn−1 + F)/(bxnxn−1 + cx2n−1), n = 0, 1, . Several global asymptotic results for some special cases of Equation (2) were obtained in [1,2,3,4,5,6,7,8,9,10,11]. In some cases when the associated map changes its monotonicity with respect to the first variable in an invariant interval, we will use Theorems 1 and 2 below in order to obtain the convergence results.
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