Basins of attraction of equilibrium and boundary points of second-order difference equations

Abstract

Dedicated to Gerasimos Ladas on occasion of his retirement.We investigate the global behaviour of the difference equation of the formwith non-negative parameters and initial conditions such thatWe give a precise description of the basins of attraction of different equilibrium points, and show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are in fact the global stable manifolds of neighbouring saddle or non-hyperbolic equilibrium points. Different types of bifurcations when one or more parameters b; d; e; f are 0 are explained.

Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms

“…A consequence of Theorem 1 is that every bounded solution of (6) converges to either an equilibrium or a period-two solution or to a point on the boundary where the equation is not defined, see [11]. Thus the most important question becomes determining the basins of attraction of these solutions.…”

Global dynamics and bifurcation of a perturbed Sigmoid Beverton–Holt difference equation

“…The consequence of Theorem is that every bounded solution of converges to either an equilibrium or a period‐two solution or to the point on the boundary where equation is not defined . Thus, the most important question becomes determining the basins of attraction of these solutions.…”

“…Equation is a special case of equations and Some special cases of Eq. have been considered in the series of papers . Some special second‐order quadratic fractional difference equations have appeared in analysis of competitive and anti‐competitive systems of linear fractional difference equations in the plane .…”

“…Some special cases of Eq. (2) have been considered in the series of papers [1][2][3][4][5][6][7][8]. Some special second-order quadratic fractional difference equations have appeared in analysis of competitive and anti-competitive systems of linear fractional difference equations in the plane [9][10][11][12].…”

Global Dynamics of Some Fractional Difference Equations

“…We investigate the global behaviour of the equation , , 1 , 0 , Hrustić et al [13] studied the global dynamics and the bifurcations of a certain second-order rational difference equation with quadratic terms Kulenović et al [16] investigated the solutions of the difference equation Other related results on rational difference equations can be found in (([4]- [9]), [11], [14,15], [18,19], [21]).…”