Papers by Mustafa Kulenović
Dynamics of Second Order Rational Difference Equations, 2001
Discrete Dynamical Systems and Difference Equations with Mathematica, 2002
W AII I . A 11 «JT / s \ DISCRETE DYNAMICAL SYSTEMS DiFFERENCE EQUATIONS with Mathematics Hsiifti... more W AII I . A 11 «JT / s \ DISCRETE DYNAMICAL SYSTEMS DiFFERENCE EQUATIONS with Mathematics HsiiftiisaS Orlando Merino ^f CHAPMAN & HALLCRC ... DISCRETE DYNAMICAL SYSTEMS and DIFFERENCE EQUATIONS with Mathematica This O 354U-7S6-0GY7
Nonlinear Analysis: Theory, Methods & Applications, 2001
Journal of Dynamics and Differential Equations, 1990
ABSTRACT
Journal of Difference Equations and Applications, 2006
... DOI: 10.1080/10236190500410109 MRS Kulenović a * & Orlando Merino a pages 101-108. Av... more ... DOI: 10.1080/10236190500410109 MRS Kulenović a * & Orlando Merino a pages 101-108. Available online: 20 Aug 2006. ...
Journal of Difference Equations and Applications, 2007
We study global attractivity of the period-two coefficient version of the delay logistic differen... more We study global attractivity of the period-two coefficient version of the delay logistic difference equation, also known as Pielou's equation,whereWe prove that for , zero is the unique equilibrium point. If , then zero is globally asymptotically stable, with basin of attraction given by the nonnegative quadrant of initial conditions. If , then zero is unstable, and a sequence converges
Journal of Difference Equations and Applications, 2009
... Merino b pages 303-323. ... On second-order rational difference equations, part 1. J. Differ.... more ... Merino b pages 303-323. ... On second-order rational difference equations, part 1. J. Differ. Equ. Appl. , 13: 9691004. [Taylor & Francis Online], [Web of Science ®] View all references, 22. Ladas, G. 2008a. On second-order rational difference equations, part 2. J. Differ. Equ. Appl. ...
Journal of Difference Equations and Applications, 2002
ABSTRACT We show that the equation in the title with nonnegative parameters and nonnegative initi... more ABSTRACT We show that the equation in the title with nonnegative parameters and nonnegative initial conditions exhibits a trichotomy character concerning periodicity, convergence, and boundedness which depends on whether the parameter n is equal, less, or greater than the sum of the parameters g and A .
Journal of Difference Equations and Applications, 2004
We investigate the periodic nature, the boundedness character and the global asymptotic stability... more We investigate the periodic nature, the boundedness character and the global asymptotic stability of solutions of the difference equation where the parameter pn is a period-two sequence with positive values and the initial conditions are positive.
Journal of Difference Equations and Applications, 2004
Our aim here is to present a summary of our recent work and a large number of open problems and c... more Our aim here is to present a summary of our recent work and a large number of open problems and conjectures on third order rational difference equations of the form with non-negative parameters and non-negative initial conditions.
Discrete & Continuous Dynamical Systems - B, 2006
ABSTRACT We investigate global behavior of xn+1 = T(xn), n = 0,1,2,... (E) where T : R. → R is a ... more ABSTRACT We investigate global behavior of xn+1 = T(xn), n = 0,1,2,... (E) where T : R. → R is a competitive (monotone with respect to the South-East ordering) map on a set R ⊂ ℝ2 with nonempty interior. We assume the existence of a unique fixed point ē in the interior of R. We give very general conditions which are easily verifiable for (E) to exhibit either competitive-exclusion or competitive-coexistence. More specifically, we obtain sufficient conditions for the interior fixed point ē to be a global attractor when R. is a rectangular region. We also show that when T is strongly monotone in R° (interior of R.), R is convex, the unique interior equilibrium ē is a saddle, and a technical condition is satisfied, the corresponding global stable and unstable manifolds are the graphs of monotonic functions, and the global stable manifold splits the domain into two connected regions, which under additional conditions on R and on T are shown to be basins of attraction of fixed points on the boundary of R. Applications of the main results to specific difference equations are given.
Discrete & Continuous Dynamical Systems - B, 2005
... MRS Kulenovic and Orlando Merino Department of Mathematics University of Rhode Island Kingsto... more ... MRS Kulenovic and Orlando Merino Department of Mathematics University of Rhode Island Kingston, Rhode Island 02881-0816 (Communicated by Linda Allen) ... It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not ...
Discrete and Continuous Dynamical Systems - Series B, 2009
A global bifurcation result is obtained for families of competitive systems of difference equatio... more A global bifurcation result is obtained for families of competitive systems of difference equations $x_{n+1} = f_\alpha(x_n,y_n) $ $y_{n+1} = g_\alpha(x_n,y_n)$ where $\alpha$ is a parameter, $f_\alpha$ and $g_\alpha$ are continuous real valued functions on a rectangular domain $\mathcal{R}_\alpha \subset \mathbb{R}^2$ such that $f_\alpha(x,y)$ is non-decreasing in $x$ and non-increasing in $y$, and $g_\alpha(x, y)$ is non-increasing in $x$ and non-decreasing in $y$. A unique interior fixed point is assumed for all values of the parameter $\alpha$. 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+1} = F_\alpha(x_n, x_{n-1}), \quad n=0,1, \ldots $ where $\alpha$ is a parameter, $F_\alpha:\mathcal{I_\alpha}\times \mathcal{I_\alpha} \rightarrow \mathcal{I_\alpha}$ is a decreasing function in the first variable and increasing in the second variable, and $\mathcal{I_\alpha}$ is a interval in $\mathbb{R}$, and there is a unique interior equilibrium point. Examples of application of the main results are also given.
Bulletin of Mathematical Biology, 1987
A linearized oscillation theorem due to Kulenovi6, Ladas and Meimaridou (1987, Quart. appl. Math.... more A linearized oscillation theorem due to Kulenovi6, Ladas and Meimaridou (1987, Quart. appl. Math. XLV, 155-164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980, Nature, Lond. 287, 17-21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.
Applicable Analysis, 1992
We established sufficient conditions for the global attractivity of the positive equilibrium of t... more We established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation [Ndot](t) ≡ −δN(t) + PN(t–τ)e which was used by Gurney, Blythe and Nisbet [1] in describing the dynamics of Nicholson's blowflies
International Journal of Bifurcation and Chaos, 2010
Let T be a competitive map on a rectangular region [Formula: see text], and assume T is C1 in a n... more Let T be a competitive map on a rectangular region [Formula: see text], and assume T is C1 in a neighborhood of a fixed point [Formula: see text]. The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from [Formula: see text] when both eigenvalues of the Jacobian of T at [Formula: see text] are nonzero and at least one of them has absolute value less than one, and establish that [Formula: see text] is an increasing curve that separates [Formula: see text] 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. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coe...
Computers & Mathematics with Applications, 2002
this paper, some new oscillation criteria are given for the second-order nonlinear differential e... more this paper, some new oscillation criteria are given for the second-order nonlinear differential equation [rWW))cp (z'(t))]' + c(Q&(t)
Discrete Dynamics in Nature and Society, 2020
In this paper, certain dynamic scenarios for general competitive maps in the plane are presented ... more 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.
Mathematics, 2018
We investigate the nonautonomous difference equation with real initial conditions and coefficient... more We investigate the nonautonomous difference equation with real initial conditions and coefficients g i , i = 0, 1 which are in general functions of n and/or the state variables x n , x n−1 ,. . ., and satisfy g 0 + g 1 = 1. We also obtain some global results about the behavior of solutions of the nonautonomous non-homogeneous difference equation where g i , i = 0, 1, 2 are functions of n and/or the state variables x n , x n−1 ,. . ., with g 0 + g 1 = 1. Our results are based on the explicit formulas for solutions. We illustrate our results by numerous examples.
Uploads
Papers by Mustafa Kulenović