Abstract
In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.
Highlights
Introduction and PreliminariesConsider the following difference equation: xn+1 cx2n− x2n− 1 + dxn +, f n 0, 1, . . . , (1)where the initial conditions x− 1, x0 ≥ 0 and the parameters satisfy that c, f ≥ 0, d > 0
We start with specific global dynamic scenarios for competitive system (9) that will be applied to equation (7)
Global Dynamics of Equation (1). e global dynamics of equation (1) is quite complicated. us, we provide the following three diagrams that describe all possible bifurcations produced by different values of parameters d and 4fc
Summary
If 4fc 1, equation (1) has the unique equilibrium point E0 which is locally asymptotically stable and the unique minimal period-two solution Px, Py {(1/2c, 0), (0, 1/2c)} which is nonhyperbolic of stable type.
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