scholarly journals Study of the stability of a 3×3 system of difference equations using Centre Manifold Theory

2017 ◽  
Vol 64 ◽  
pp. 185-192 ◽  
Author(s):  
N. Psarros ◽  
G. Papaschinopoulos ◽  
C.J. Schinas
Keyword(s):  
Difference Equations ◽  
Centre Manifold ◽  
System Of Difference Equations ◽  
The Stability ◽  
Manifold Theory

scholarly journals Neimark-Sacker, flip and transcritical bifurcation in a symmetric system of difference equations with exponential terms

2021 ◽  
Author(s):  
Chrysoula Mylona ◽  
Garyfalos Papaschinopoulos ◽  
Christos Schinas
Keyword(s):  
Normal Form ◽  
Difference Equations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Transcritical Bifurcation ◽  
Symmetric System ◽  
Center Manifold Theory ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Manifold Theory

In this paper, we study the conditions under which the following symmetric system of difference equations with exponential terms: \[ x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\] \[ y_{n+1} =a_2\frac{x_n}{b_2+x_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2$, are real constants and the initial values $x_0$, $y_0$ are real numbers, undergoes Neimark-Sacker, flip and transcritical bifurcation. The analysis is conducted applying center manifold theory and the normal form bifurcation analysis.

On the asymptotic stability of linear system of fractional-order difference equations

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Raghib Abu-Saris ◽  
Qasem Al-Mdallal
Keyword(s):  
Asymptotic Stability ◽  
Linear System ◽  
Fractional Order ◽  
Difference Equations ◽  
Equilibrium Solution ◽  
Initial Condition ◽  
System Of Difference Equations ◽  
Order Difference ◽  
Order Linear ◽  
The Stability

AbstractIn this paper we investigate the stability of the equilibrium solution of the νth order linear system of difference equations $(\Delta _{a + \nu - 1}^\nu y)(t) = \Lambda y(t + \nu - 1);t \in \mathbb{N}_a ,a \in \mathbb{R},and\Lambda \in \mathbb{R}^{p \times p} ,$ subject to the initial condition $y(a + \nu - 1) = y - 1,$, where 0 < ν < 1 and y−1 ∈ ℝp.

Semistability of two systems of difference equations using centre manifold theory

2016 ◽  
Vol 39 (18) ◽  
pp. 5216-5222 ◽  
Author(s):  
N. Psarros ◽  
G. Papaschinopoulos ◽  
C. J. Schinas
Keyword(s):  
Difference Equations ◽  
Centre Manifold ◽  
Two Systems ◽  
Manifold Theory

Dynamics of a Modified Predator-Prey System to allow for a Functional Response and Time Delay

2016 ◽  
Vol 6 (4) ◽  
pp. 384-399 ◽  
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
System Theory ◽  
Algebraic System ◽  
Bifurcation Theory ◽  
Dynamical Behaviour ◽  
Centre Manifold ◽  
Predator Prey ◽  
Prey System ◽  
Differential Algebraic System ◽  
The Stability ◽  
Manifold Theory

AbstractA modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

scholarly journals On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Dağistan Simsek ◽  
Bilal Demir ◽  
Cengiz Cinar
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Positive Real

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.

On the Stability of Positive Difference Equations

2009 ◽  
Vol 42 (14) ◽  
pp. 156-161
Author(s):  
M. DiLoreto ◽  
J.J. Loiseau
Keyword(s):  
Difference Equations ◽  
Positive Difference ◽  
The Stability
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