scholarly journals Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Toufik Khyat ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

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.

scholarly journals Global Period-Doubling Bifurcation of Quadratic Fractional Second Order Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Senada Kalabušić ◽  
M. R. S. Kulenović ◽  
M. Mehuljić
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Local Stability ◽  
Initial Conditions ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Global Dynamics ◽  
Second Order Difference Equation ◽  
The Difference

We investigate the local stability and the global asymptotic stability of the difference equationxn+1=αxn2+βxnxn-1+γxn-1/Axn2+Bxnxn-1+Cxn-1,n=0,1,…with nonnegative parameters and initial conditions such thatAxn2+Bxnxn-1+Cxn-1>0, for alln≥0. We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, whereβ=B=0, in which case we show that such equation exhibits a global period doubling bifurcation.

scholarly journals Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
S. Jašarević Hrustić ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Initial Conditions ◽  
Second Order ◽  
Global Dynamics ◽  
Order Difference ◽  
Local Dynamics ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation:xn+1=Axn2+Exn-1+F/axn2+exn-1+f,  n=0,1,2,…, where the parametersA,E,F,a,e, andfare nonnegative numbers with conditionA+E+F>0,a+e+f>0, and the initial conditionsx-1,x0are arbitrary nonnegative numbers such thataxn2+exn-1+f>0,  n=0,1,2,….

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals The global behavior of a quadratic difference equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6203-6210
Author(s):  
Vahidin Hadziabdic ◽  
Midhat Mehuljic ◽  
Jasmin Bektesevic ◽  
Naida Mujic
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Julia Set ◽  
Second Order ◽  
Global Behavior ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

In this paper we will present the Julia set and the global behavior of a quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1 + bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.

scholarly journals DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT

2004 ◽  
Vol 15 (09) ◽  
pp. 959-965 ◽  
Author(s):  
KAZUHIRO HIKAMI
Keyword(s):  
Difference Equation ◽  
Jones Polynomial ◽  
Order Difference ◽  
Colored Jones Polynomial ◽  
First Order ◽  
Torus Knot ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Hypergeometric Type ◽  
The Difference

We prove that the N-colored Jones polynomial for the torus knot [Formula: see text] satisfies the second order difference equation, which reduces to the first order difference equation for a case of [Formula: see text]. We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for [Formula: see text].

Fuzzy Difference Equations: The Initial Value Problem

2001 ◽  
Vol 5 (6) ◽  
pp. 315-325 ◽  
Author(s):  
James J. Buckley ◽  
◽  
Thomas Feuring ◽  
Yoichi Hayashi ◽  
◽  
...  
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
National Income ◽  
Linear Difference Equation ◽  
Second Order ◽  
Solution Methods ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
The Difference

In this paper we study fuzzy solutions to the second order, linear, difference equation with constant coefficients but having fuzzy initial conditions. We look at two methods of solution: (1) in the first method we fuzzify the crisp solution and then check to see if it solves the difference equation; and (2) in the second method we first solve the fuzzy difference equation and then check to see if the solution defines a fuzzy number. Relationships between these two solution methods are also presented. Two applications are given: (1) the first is about a second order difference equation, having fuzzy initial conditions, modeling national income; and (2) the second is from information theory modeling the transmission of information.

On the solutions of a higher order difference equation

2020 ◽  
Vol 27 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Raafat Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Forbidden Set ◽  
Order Difference Equation ◽  
The Difference

AbstractIn this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equationx_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots,where{a,b,c}are positive real numbers and the initial conditions{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}}are real numbers. We show that when{a=b=c}, the behavior of the solutions depends on whetherkis even or odd.

On the solutions of a second-order difference equation in terms of generalized Padovan sequences

2018 ◽  
Vol 68 (3) ◽  
pp. 625-638 ◽  
Author(s):  
Yacine Halim ◽  
Julius Fergy T. Rabago
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Initial Conditions ◽  
Solution Stability ◽  
Two Dimensional ◽  
Real Numbers ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractThis paper deals with the solution, stability character and asymptotic behavior of the rational difference equation$$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$where ℕ0= ℕ ∪ {0},α,β,γ∈ ℝ+, and the initial conditionsx–1andx0are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by$$\begin{array}{} \displaystyle x_{n+1} = \frac{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0. \end{array}$$

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

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. Garić-Demirović ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equation ◽  
Unique Feature ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Minimal Period ◽  
Fractional Difference ◽  
The Difference

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…,where the parameters  a,  b, and  c  are positive numbers and the initial conditionsx-1andx0are 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.

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