scholarly journals Bifurcation Analysis in a Kind of Fourth-Order Delay Differential Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-20
Author(s):  
Xiaoqian Cui ◽  
Junjie Wei
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Stability Switches ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Delay Differential

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the analytic results.

scholarly journals Stability Switches and Hopf Bifurcations in a Second-Order Complex Delay Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
M. Roales ◽  
F. Rodríguez
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Second Order ◽  
Delay Equation ◽  
Hopf Bifurcations ◽  
Order Complex ◽  
Stability Switches ◽  
Delay Differential ◽  
Complex Coefficients

The existence of stability switches and Hopf bifurcations for the second-order delay differential equation x′′t+ax′t-τ+bxt=0,  t>0, with complex coefficients, is studied in this paper.

scholarly journals Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Zeqing Liu ◽  
Jingjing Zhu ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Approximate Solutions ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

scholarly journals Stability and Hopf Bifurcation Analysis in a Delayed Myc/E2F/miR-17-92 Network Involving Interlinked Positive and Negative Feedback Loops

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Guiyuan Wang ◽  
Zhuoqin Yang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Negative Feedback ◽  
Feedback Loop ◽  
Sufficient Conditions ◽  
Center Manifold Theorem ◽  
Negative Feedback Loops ◽  
Feedback Strength ◽  
Normal Form Method ◽  
Delay Differential

MiR-17-92 plays an important role in regulating the levels of the Myc/E2F protein. In this paper, we consider a coupling network between Myc/E2F/miR-17-92 delayed negative feedback loop and Myc/E2F positive feedback loop described by a two-dimensional delay differential equation. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing time delay as well as negative feedback strength associated with oscillations as bifurcation parameters, respectively. Furthermore, direction and stability of Hopf bifurcation of time delay are studied by using the normal form method and center manifold theorem. Finally, several numerical simulations are performed to verify the results we obtained.

MULTISCROLL CHAOTIC ATTRACTORS FROM A HYSTERESIS BASED TIME-DELAY DIFFERENTIAL EQUATION

2010 ◽  
Vol 20 (10) ◽  
pp. 3275-3281 ◽  
Author(s):  
SELÇUK KILINÇ ◽  
MÜŞTAK E. YALÇIN ◽  
SERDAR ÖZOGUZ
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Delay Differential Equation ◽  
Order Differential Equation ◽  
Chaotic Attractors ◽  
Nonlinear Characteristic ◽  
Circuit Implementation ◽  
First Order ◽  
State Variable ◽  
Delay Differential

In this paper, the generation of multiscroll chaotic attractors derived from a time-delay differential equation is presented. The proposed system is represented by only one first-order differential equation including time-delayed state variable, and employs hysteresis function as the nonlinear characteristic. The generalization of the introduced system is based on adding multihysteresis nonlinear characteristic which leads to n-scroll chaotic attractors. The circuit implementation of the proposed system and some experimental results referring to two-, three-, four-, and five-scroll chaotic attractors are reported.

scholarly journals New Improved Results for Oscillation of Fourth-Order Neutral Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2388
Author(s):  
Osama Moaaz ◽  
Rami Ahmad El-Nabulsi ◽  
Ali Muhib ◽  
Sayed K. Elagan ◽  
Mohammed Zakarya
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Oscillation Criterion ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Riccati Substitution ◽  
Delay Differential ◽  
New Criterion

In this study, a new oscillation criterion for the fourth-order neutral delay differential equation ruxu+puxδu‴α′+quxβϕu=0,u≥u0 is established. By introducing a Riccati substitution, we obtain a new criterion for oscillation without requiring the existence of the unknown function. Furthermore, the new criterion improves and complements the previous results in the literature. The results obtained are illustrated by an example.

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