scholarly journals Basin of Attraction through Invariant Curves and Dominant Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ziyad AlSharawi ◽  
Asma Al-Ghassani ◽  
A. M. Amleh
Keyword(s):  
Stable Equilibrium ◽  
Basin Of Attraction ◽  
Invariant Curves ◽  
Equilibrium Solutions ◽  
Invariant Region ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Stability ◽  
The Basin Of Attraction

We study a second-order difference equation of the formzn+1=znF(zn-1)+h, where bothF(z)andzF(z)are decreasing. We consider a set of invariant curves ath=1and use it to characterize the behaviour of solutions whenh>1and when0<h<1. The caseh>1is related to the Y2K problem. For0<h<1, we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.

scholarly journals Revisiting Samuelson's Models, Linear and Nonlinear, Stability Conditions and Oscillating Dynamics

2020 ◽  
Author(s):  
Fabio TRAMONTANA ◽  
Laura Gardini
Keyword(s):  
Analytic Solution ◽  
Oscillatory Behavior ◽  
Complete Analysis ◽  
Third Order ◽  
Order Difference ◽  
Wide Range ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Stability ◽  
Range Of Values

Abstract In this work we reconsider the dynamics of a few versions of the classical Samuelson's multiplier- accelerator model for national economy. First we recall that the classical one with constant governmental expenditure, represented by a linear second-order difference equation, is able to generate oscillations con- verging to the equilibrium for a wide range of values of the parameters, and give its analytic solution for all the possible cases. A delayed version proposed in the recent literature, represented by a linear third-order di¤erence equation, is also considered. We show that also this model is able to produce converging oscilla- tions, and give a complete analysis of the stability region of the equilibrium. A new simple nonlinear model is proposed, showing that it keeps oscillatory behavior, although coupled with other dynamics related to global effects. Our analysis confirms that the seminal work of Samuelson and simple modifications of it, may give powerful tools in the study of the business cycles.

scholarly journals Revisiting Samuelson’s models, linear and nonlinear, stability conditions and oscillating dynamics

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Fabio Tramontana ◽  
Laura Gardini
Keyword(s):  
Difference Equation ◽  
Oscillatory Behavior ◽  
Complete Analysis ◽  
Third Order ◽  
Order Difference ◽  
Wide Range ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Stability ◽  
Range Of Values

AbstractIn this work, we reconsider the dynamics of a few versions of the classical Samuelson’s multiplier–accelerator model for national economy. First we recall that the classical one with constant governmental expenditure, represented by a linear second-order difference equation, is able to generate oscillations converging to the equilibrium for a wide range of values of the parameters, and give its analytic solution for all the possible cases. A delayed version proposed in the recent literature, represented by a linear third-order difference equation, is also considered. We show that also this model is able to produce converging oscillations, and give a complete analysis of the stability region of the equilibrium. A new simple nonlinear model is proposed, showing that it keeps oscillatory behavior, although coupled with other dynamics related to global effects. Our analysis confirms that the seminal work of Samuelson and simple modifications of it, may give powerful tools in the study of the business cycles.

scholarly journals Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 790
Author(s):  
Tarek F. Ibrahim ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Elliptic Equilibrium ◽  
Order Difference Equation ◽  
The Difference ◽  
The Stability ◽  
Theoretical Results

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n + 1 = α t n + β t n 2 − t n − 1 , n = 0 , 1 , 2 , … , where are t − 1 , t 0 , α ∈ R , α ≠ 0 , β > 0 . By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.

Effect of Delay on the Boundary of the Basin of Attraction in a Self-Excited Single Graded-Response Neuron

1997 ◽  
Vol 9 (2) ◽  
pp. 319-336 ◽  
Author(s):  
K. Pakdaman ◽  
C. P. Malta ◽  
C. Grotta-Ragazzo ◽  
J.-F. Vibert
Keyword(s):  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Transmission Delays ◽  
The Past ◽  
Graded Response ◽  
The Stability ◽  
Delay Dependent ◽  
The Basin Of Attraction

Little attention has been paid in the past to the effects of interunit transmission delays (representing a xonal and synaptic delays) ontheboundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the “location” of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria. The dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.

scholarly journals Analytic solutions of nonlinear Cournot duopoly game

2005 ◽  
Vol 2005 (2) ◽  
pp. 119-133
Author(s):  
Akio Matsumoto ◽  
Mami Suzuki
Keyword(s):  
Stable Equilibrium ◽  
Analytic Solutions ◽  
Stable Equilibrium Point ◽  
Cournot Duopoly ◽  
Cournot Model ◽  
Order Difference ◽  
Reaction Functions ◽  
Duopoly Model ◽  
Second Order Difference Equation ◽  
Order Difference Equation

We construct a Cournot duopoly model with production externality in which reaction functions are unimodal. We consider the case of a Cournot model which has a stable equilibrium point. Then we show the existence of analytic solutions of the model. Moreover, we seek general solutions of the model in the form of nonlinear second-order difference equation.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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