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 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 The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
M. R. S. Kulenović ◽  
Connor O’Loughlin ◽  
E. Pilav
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Invariant Manifolds ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Positive Equilibrium ◽  
Equilibrium Solutions ◽  
The Difference

We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.

scholarly journals Global Asymptotic Stability for Linear Fractional Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
A. Brett ◽  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Real Numbers ◽  
Fractional Difference ◽  
Fractional Difference Equation ◽  
The Difference

Consider the difference equation xn+1=(α+∑i=0kaixn-i)/(β+∑i=0kbixn-i),  n=0,1,…, where all parameters α,β,ai,bi,  i=0,1,…,k, and the initial conditions xi,  i∈{-k,…,0} are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

scholarly journals Bifurcation analysis and chaos control of a second-order exponential difference equation

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 5003-5022 ◽  
Author(s):  
Q. Din ◽  
E.M. Elabbasy ◽  
A.A. Elsadany ◽  
S. Ibrahim
Keyword(s):  
Difference Equation ◽  
Chaos Control ◽  
Control Method ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Period Doubling ◽  
Second Order ◽  
Period Doubling Bifurcation ◽  
Positive Real ◽  
Second Order Difference Equation

The aim of this article is to study the local stability of equilibria, investigation related to the parametric conditions for transcritical bifurcation, period-doubling bifurcation and Neimark-Sacker bifurcation of the following second-order difference equation xn+1 = ?xn + ?xn-1 exp(-?xn-1) where the initial conditions x-1, x0 are the arbitrary positive real numbers and ?,? and ? are positive constants. Moreover, chaos control method is implemented for controlling chaotic behavior under the influence of Neimark-Sacker bifurcation and period-doubling bifurcation. Numerical simulations are provided to show effectiveness of theoretical discussion.

scholarly journals Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-22
Author(s):  
M. R. S. Kulenović ◽  
S. Moranjkić ◽  
M. Nurkanović ◽  
Z. Nurkanović
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Local Asymptotic Stability ◽  
Stable Periodic ◽  
Rational Difference Equation ◽  
Unknown Period

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12,  n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

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.

scholarly journals Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

scholarly journals Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation xn+1=α+βxn+γxn-1/A+Bxn+Cxn-1, n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).

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.

scholarly journals Asymptotic Stability for a Class of Nonlinear Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Chang-you Wang ◽  
Shu Wang ◽  
Zhi-wei Wang ◽  
Fei Gong ◽  
Rui-fang Wang
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Positive Real ◽  
Fractional Difference ◽  
Fractional Difference Equation

We study the global asymptotic stability of the equilibrium point for the fractional difference equationxn+1=(axn-lxn-k)/(α+bxn-s+cxn-t),n=0,1,…, where the initial conditionsx-r,x-r+1,…,x1,x0are arbitrary positive real numbers of the interval(0,α/2a),l,k,s,tare nonnegative integers,r=max⁡⁡{l,k,s,t}andα,a,b,care positive constants. Moreover, some numerical simulations are given to illustrate our results.

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