scholarly journals Dynamics of a Nonstandard Finite-Difference Scheme for a Limit Cycle Oscillator with Delayed Feedback

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yuanyuan Wang ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Limit Cycle ◽  
Euler Method ◽  
Delayed Feedback ◽  
Dimensional System ◽  
Two Dimensional ◽  
Limit Cycle Oscillator ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

We consider a complex autonomously driven single limit cycle oscillator with delayed feedback. The original model is translated to a two-dimensional system. Through a nonstandard finite-difference (NSFD) scheme we study the dynamics of this resulting system. The stability of the equilibrium of the model is investigated by analyzing the characteristic equation. In the two-dimensional discrete model, we find that there are stability switches on the time delay and Hopf bifurcation when the delay passes a sequence of critical values. Finally, computer simulations are performed to illustrate the theoretical results. And the results show that NSFD scheme is better than the Euler method.

DEVELOPMENT OF NEW HARMONIC EULER USING NONSTANDARD FINITE DIFFERENCE TECHNIQUE FOR SOLVING STIFF PROBLEMS

2015 ◽  
Vol 77 (20) ◽  
Author(s):  
Nurhafizah Moziyana Mohd Yusop ◽  
Mohammad Khatim Hasan
Keyword(s):  
Finite Difference ◽  
Numerical Algorithm ◽  
Mesh Size ◽  
High Accuracy ◽  
Euler Method ◽  
Stiff Problems ◽  
Harmonic Mean ◽  
Stiff Problem ◽  
Finite Difference Technique ◽  
Nonstandard Finite Difference

Solving stiff problem always required very tiny size of meshes if it is solved via traditional numerical algorithm. Using insufficient of mesh size, will triggered instabilities. In this paper, we develop an algorithm applying Harmonic Mean on Euler method to solve the stiff problems. The main purpose of this paper is to discuss the improvement of Harmonic Euler using Nonstandard Finite Difference (NSFD). The combination of these methods can provide new advantages that Euler method could offer. Four set of stiff problems are solved via three schemes, i.e. Harmonic Euler, Nonstandard Harmonic Euler and Nonstandard EO with Harmonic Euler. Findings show that both nonstandard schemes produce high accuracy results.

scholarly journals An unconditionally positive and global stability preserving NSFD scheme for an epidemic model with vaccination

2014 ◽  
Vol 24 (3) ◽  
pp. 635-646 ◽  
Author(s):  
Deqiong Ding ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulation ◽  
Global Stability ◽  
Difference Equations ◽  
Epidemic Model ◽  
Lyapunov Functions ◽  
Numerical Solutions ◽  
Time Step ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

Abstract In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivity of the solutions and the stability of the equilibria, without being restricted by the time step. Our proof of global stability utilizes the method of Lyapunov functions. Numerical simulation illustrates the effectiveness of our results

On the stability of steady-states of a two-dimensional system of ferromagnetic nanowires

2017 ◽  
Vol 23 (2) ◽  
Author(s):  
Sharad Dwivedi ◽  
Shruti Dubey
Keyword(s):  
Finite Number ◽  
Steady States ◽  
Dimensional System ◽  
Two Dimensional ◽  
Ferromagnetic Nanowires ◽  
Two Dimensional System ◽  
The Stability

AbstractWe investigate the stability features of steady-states of a two-dimensional system of ferromagnetic nanowires. We constitute a system with the finite number of nanowires arranged on the

Aggregation and stability in parasite—host models

Parasitology ◽  
1992 ◽  
Vol 104 (2) ◽  
pp. 199-205 ◽  
Author(s):  
F. R. Adler ◽  
M. Kretzschmar
Keyword(s):  
Equilibrium Distribution ◽  
Empirical Studies ◽  
Dimensional System ◽  
Two Dimensional ◽  
Full System ◽  
Dimensional Approximation ◽  
Approximate System ◽  
The Mean ◽  
Two Dimensional System ◽  
The Stability

SUMMARYThis paper generalizes the two-dimensional approximation of models of macroparasites on homogeneous populations developed by Anderson & May (1978), focusing on how the dispersion (the variance to mean ratio) of the equilibrium distribution of parasites on hosts is related to the stability of the equilibrium. We show in the approximate system that the equilibrium is stabilized not by aggregation, but by dispersion which increases as a function of the mean. Computer simulations indicate, however, that this analysis fails to capture properly the dynamics of the full system, raising the question of whether any two-dimensional system could produce an adequate approximation. We discuss the relevance of our results to several empirical studies which have examined the relation of dispersion to the mean.

scholarly journals An Accurate Predictor-Corrector-Type Nonstandard Finite Difference Scheme for an SEIR Epidemic Model

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Asma Farooqi ◽  
Riaz Ahmad ◽  
Rashada Farooqi ◽  
Sayer O. Alharbi ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Continuous System ◽  
Euler Method ◽  
State Variables ◽  
Nonstandard Finite Difference Scheme ◽  
Long Time ◽  
Predictor Corrector ◽  
Nonstandard Finite Difference

The present work deals with the construction, development, and analysis of a viable normalized predictor-corrector-type nonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles. The proposed numerical scheme double refines the solution and gives realistic results even for large step sizes, thus making it economical when integrating over long time periods. Moreover, it is dynamically consistent with a continuous system and unconditionally convergent and preserves the positive behavior of the state variables involved in the system. Simulations are performed to guarantee the results, and its effectiveness is compared with well-known numerical methods such as Runge–Kutta (RK) and Euler method of a predictor-corrector type.

scholarly journals A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Dumitru Baleanu ◽  
Sadegh Zibaei ◽  
Mehran Namjoo ◽  
Amin Jajarmi
Keyword(s):  
Finite Difference ◽  
Numerical Simulations ◽  
Active Control ◽  
Chaotic System ◽  
Equilibrium Points ◽  
Control Technique ◽  
New Model ◽  
Fractional Chaotic System ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

AbstractThe aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

scholarly journals STABILITY OF THE REST POINTS FRACTIONAL OSCILLATOR FITZHUGH-NAGUMO

2019 ◽  
pp. 63-70
Author(s):  
О.Д. Липко
Keyword(s):  
Finite Difference ◽  
Qualitative Analysis ◽  
Numerical Method ◽  
Limit Cycle ◽  
Finite Difference Schemes ◽  
Asymptotically Stable ◽  
Phase Trajectories ◽  
Fractional Oscillator ◽  
The Stability ◽  
Unstable Focus

В работе с помощью качественного анализа были исследованы на устойчивость точки покоя дробного осциллятора ФитцХью-Нагумо в соизмеримом и несоизмеримом случаях. Для соответвующей точки покоя, с помощью численного метода теории конечно-разностных схем, была построена фазовая траектория. Показано, что точки покоя могут быть как асимптотически устойчивыми, что соответствуют устойчивым фокусам, так и являться асимптотически неустойчивыми (неустойчивыми фокусами), причем для них фазовые таректории, как правило, выходят на предельный цикл. In this paper, using the qualitative analysis, we studied the stability of the point of rest of the fractional oscillator FitzHugh-Nagumo in commensurate and incommensurable cases. For the corresponding point of rest, using the numerical method of the theory of finite difference schemes, phase trajectories were constructed. It is shown that quiescent points can be both asymptotically stable, which correspond to stable focus, and are asymptotically unstable (unstable focus), and for them the phase trajectories usually go to the limit cycle.

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