Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form xn+1 = x2n-1/(ax2n + bxnxn-1 + cx2n-1), n = 0, 1, 2, …, where the parameters a, b, and c are positive numbers and the initial conditions x−1 and x0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable

Invariant Manifolds for Competitive Systems in the Plane

Let $T$ be a competitive map on a rectangular region $\mathcal{R}\subset \mathbb{R}^2$, and assume $T$ is $C^1$ in a neighborhood of a fixed point $\bar{\rm x}\in \mathcal{R}$. The main results of this paper give conditions on $T$ that guarantee the existence of an invariant curve emanating from $\bar{\rm x}$ when both eigenvalues of the Jacobian of $T$ at $\bar{\rm x}$ are nonzero and at least one of them has absolute value less than one, and establish that $\mathcal{C}$ is an increasing curve that separates $\mathcal{R}$ into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. Several applications to planar systems of difference equations with non-hyperbolic equilibria are given.Comment: 20 pages, 2 figure

Basins of attraction of equilibrium points of second order difference equations

We investigate the basins of attraction of equilibrium points and period-two solutions of the difference equation of the form x n+1=f(x n,x n-1),n=0,1,⋯, where f is decreasing in the first and increasing in the second variable. We show that the boundaries of the basins of attraction of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points. © 2012 Elsevier Ltd. All rights reserved

Global Behavior of Four Competitive Rational Systems of Difference Equations in the Plane

We investigate the global dynamics of solutions of four distinct competitive rational systems of difference equations in the plane. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or nonhyperbolic equilibrium points. Our results give complete answer to Open Problem 2 posed recently by Camouzis et al. (2009)

Global behavior of two competitive rational systems of difference equations in the plane

We investigate the global dynamics of solutions of two distinct competitive rational systems of difference equations in the plane. We show that the basins of attraction of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of non-hyperbolic equilibrium points. Our results give complete answer to Open Problem 1 posed recently in [3]