Invariants and related Liapunov functions for difference equations

Consider the difference equation xn+1 = f(xn), where xn is in Rk and f : D → D is continuous where D ⊂ Rk. Suppose that I : Rk → R is a continuous invariant, that is, I(f(x)) = I(x) for every x ∈ D. We will show that if I attains an isolated minimum or maximum value at the equilibrium (fixed) point p of this system, then there exists a Liapunov function, namely ±(I(x) - I(p)) and so the equilibrium p is stable. This result is then applied to some difference equations appearing in different fields of applications. © 2000 Elsevier Science Ltd. All rights reserved

A coupled system of rational difference equations

We investigate the global stability properties and asymptotic behavior of solutions of the recursive sequence

Necessary and sufficient conditions for the oscillation of a second order linear differential equation

In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation y″ (t) + p(t) y(t) = 0, t \u3e t0, where p is a locally integrable function and either ∫∞t0 p(t) dt ∈ (-∞, ∞) or ∫∞t [Pn-1 (s)]2 Qn-1(s, t) ds ∈ (-∞, ∞), for some n = 1 , 2 , . . . , where P0(t) = ∫∞t p(s) ds, Pn(t) = ∫∞t Pn-1(s)2 Qn-1 (s, t) ds, Qn-1 (s, t) = exp (∑n-1j=0 ∫st Pj (u) du), n = 1 , 2 , . . . . We give some applications which show how these results unify and imply some classical results in oscillation theory

Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = Axn yn/1+yn, y+1 = Byn xn/1+xn, n = 0,1,... where the parameters A and B are in (0, ∞) and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into the basins of attraction of two types of asymptotic behavior

Global behavior of a two-dimensional competitive system of difference equations with stocking

We investigate the global dynamics of solutions of 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 of non-hyperbolic equilibrium points. © 2011 Elsevier Ltd

Global dynamics of a certain two-dimensional competitive system of rational difference equations with quadratic terms

We investigate global dynamics of the following system of difference equations (Formula presented) n where the parameters a, b are positive numbers and the initial conditions x0, y0 are arbitrary nonnegative numbers. We show that this system has an interesting dynamics which depends on the part of parametric space. The obtained ynamics is very different than the dynamics of the corresponding linear fractional system. In particular, we show that the system always exhibit Allee’s effect

Global bifurcation for discrete competitive systems in the plane

A global bifurcation result is obtained for families of competitive systems of difference equations {xn+l = fa(Xn,yn) yn+1 = 9α(Xn,yn) where α is a parameter, fα and gα are continuous real valued functions on a rectangular domain Rα C R2 such that fα(x, y) is non-decreasing in x and non-increasing in y, and gα(x,y) is non-increasing in x and non-decreasing in y. A unique interior fixed point is assumed for all values of the parameter α. As an application of the main result for competitive systems a global period-doubling bifurcation result is obtained for families of second order difference equations of the type x n+l=Fα(xn-l), n = 0,1,... where α is a parameter, Iα is a decreasing function in the first variable and increasing in the second variable, and Iα is a interval in R, and there is a unique interior equilibrium point. Examples of application of the main results are also given

Existence of Nonoscillatory Solution of Second Order Linear Neutral Delay Equation

Consider the neutral delay differential equation with positive and negative coefficients,[formula]wherep∈Rand[formula] Some sufficient conditions for the existence of a nonoscillatory solution of the above equation expressed in terms of ∫∞sQi(s)ds\u3c∞,i=1,2, and certain technical conditions implying thatQ1(s) dominatesQ2(s) are obtained for values ofp≠±1. © 1998 Academic Press

A note on unbounded solutions of a class of second order rational difference equations

We investigate the unbounded solutions of the second order difference equation xn+1 = α + βxn + γx n-1/A + Bxn + Cxn-1, n = 0,1,... where all parameters α, β, γ, A, B, and C and initial conditions x -1,x0 are nonnegative and such that A + Bxn + Cxn-1 \u3e 0 for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to ∞ and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC). © 2006 Taylor & Francis

Global dynamics of generalized second-order Beverton–Holt equations of linear and quadratic type

We inVestigate second-order generaliZed BeVerton–Holt difference equations of the form xn+1 = af(xn, xn−1), n = 0, 1, …, 1 + f(xn, xn−1) Where f is a function nondecreasing in both arguments, the parameter a is a positiVe constant, and the initial conditions x−1 and x0 are arbitrary nonnegatiVe numbers in the domain of f. We Will discuss seVeral interesting eXamples of such equations and present some general theory. In particular, We Will inVestigate the local and global dynamics in the eVent f is a certain type of linear or quadratic polynomial, and We eXplore the eXistence problem of period-tWo solutions