scholarly journals Delay-induced blow-up in a planar oscillation model

2021 ◽  
Author(s):  
Alexey Eremin ◽  
Emiko Ishiwata ◽  
Tetsuya Ishiwata ◽  
Yukihiko Nakata
Keyword(s):  
Delay Differential Equations ◽  
Blow Up ◽  
Stable Limit Cycle ◽  
Delay System ◽  
Stable Limit ◽  
Small Delay ◽  
Delay Differential ◽  
Infinitely Many Periodic Solutions ◽  
Theoretical Results ◽  
Stable Unit

AbstractIn this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits blow-up solutions, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call “delay-induced blow-up” phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this paper is an example that arbitrary small delay can be responsible for a drastic change of the dynamics. We show numerical examples to illustrate our theoretical results.

Bogdanov–Takens and Triple Zero Bifurcations of Coupled van der Pol–Duffing Oscillators with Multiple Delays

2017 ◽  
Vol 27 (09) ◽  
pp. 1750133 ◽  
Author(s):  
Xia Liu ◽  
Tonghua Zhang
Keyword(s):  
Delay Differential Equations ◽  
Time Delays ◽  
Normal Forms ◽  
Multiple Delays ◽  
Third Order ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Form Theory ◽  
Delay Differential ◽  
Theoretical Results

In this paper, the Bogdanov–Takens (B–T) and triple zero bifurcations are investigated for coupled van der Pol–Duffing oscillators with [Formula: see text] symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B–T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results.

Dependence Analysis of the Solutions on the Parameters of Fractional Neutral Delay Differential Equations

2016 ◽  
Vol 8 (5) ◽  
pp. 772-783
Author(s):  
Shuiping Yang
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Initial Function ◽  
Hand Function ◽  
Dependence Analysis ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
Delay Differential ◽  
Theoretical Results

AbstractIn this paper, we discuss the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional neutral delay differential equations (FNDDEs). The corresponding theoretical results are given respectively. Furthermore, we present some numerical results that support our theoretical analysis.

Dependence Analysis of the Solutions on the Parameters of Fractional Delay Differential Equations

2011 ◽  
Vol 3 (5) ◽  
pp. 586-597 ◽  
Author(s):  
Shuiping Yang ◽  
Aiguo Xiao ◽  
Xinyuan Pan
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Initial Function ◽  
Hand Function ◽  
Dependence Analysis ◽  
Numerical Examples ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations ◽  
Theoretical Results

AbstractIn this paper, we investigate the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional delay differential equations (FDDEs) with the Caputo fractional derivative. Some results including an estimate of the solutions of FDDEs are given respectively. Theoretical results are verified by some numerical examples.

scholarly journals Stability and Hopf-Bifurcating Periodic Solution for Delayed Hopfield Neural Networks withnNeuron

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Haji Mohammad Mohammadinejad ◽  
Mohammad Hadi Moslehi
Keyword(s):  
Neural Networks ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Neural System ◽  
Hopfield Neural Networks ◽  
Local Asymptotic Stability ◽  
Trivial Equilibrium ◽  
Delay Differential ◽  
Bifurcating Periodic Solution ◽  
Theoretical Results

We consider a system of delay differential equations which represents the general model of a Hopfield neural networks type. We construct some new sufficient conditions for local asymptotic stability about the trivial equilibrium based on the connection weights and delays of the neural system. We also investigate the occurrence of an Andronov-Hopf bifurcation about the trivial equilibrium. Finally, the simulating results demonstrate the validity and feasibility of our theoretical results.

scholarly journals Optimal Control of a Time Delayed HIV-1 Infection Model

2019 ◽  
Vol 12 (2) ◽  
pp. 506-518
Author(s):  
Nigar Ali ◽  
Muhammad Ikhlaq Chohan ◽  
Gul Zaman
Keyword(s):  
Optimal Control ◽  
Delay Differential Equations ◽  
Control Function ◽  
Infection Model ◽  
Virus Concentration ◽  
Hiv Infection Model ◽  
Delay Differential ◽  
Objective Functional ◽  
Theoretical Results ◽  
Hiv 1

In this paper, an optimal control problem of HIV infection model of delay differential equations is taken into account. Then we set a control function which represents the efficiency of reverse transcriptase inhibitors. Objective functional is constructed to minimize the virus concentration as well as treatment costs.Adjoint system is derived using Pontryagins Maximum Principle. Optimality system is calculated and numerical simulation is carried out to illustrate the theoretical results. Finally, conclusion is drawn

scholarly journals Stability of Runge-Kutta Methods for Neutral Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Liping Wen ◽  
Xiong Liu ◽  
Yuexin Yu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Stability ◽  
Neutral Delay ◽  
Numerical Examples ◽  
Neutral Delay Differential Equations ◽  
Runge Kutta ◽  
Delay Differential ◽  
Stability Results ◽  
Theoretical Results

This paper is concerned with the numerical stability of a class of nonlinear neutral delay differential equations. The numerical stability results are obtained for(k,l)-algebraically stable Runge-Kutta methods when they are applied to this type of problem. Numerical examples are given to confirm our theoretical results.

OSCILLATIONS IN DELAY DIFFERENTIAL EQUATION MODEL OF REPRODUCTIVE HORMONES IN MEN WITH COMPUTER SIMULATIONS

1994 ◽  
Vol 02 (01) ◽  
pp. 73-90 ◽  
Author(s):  
PRITHA DAS ◽  
A.B. ROY
Keyword(s):  
Stable Limit Cycle ◽  
Equation Model ◽  
Reproductive Hormones ◽  
Phase Portraits ◽  
Stable Limit ◽  
Delay Model ◽  
Local Asymptotic Stability ◽  
The Stability ◽  
Linearized Model ◽  
Delay Differential

We produce here a delay model to explain the control of testosterone secretion. We have modified our earlier model by incorporating one negative feedback function which explains the inhibition of the pituitary secretion of the hormone LH (Luteinizing hormone) by the local testosterone concentration. We have derived the conditions for local asymptotic stability and switching to instability of the steady state. The length of the delay preserving the stability has also been derived. Lastly the conditions for instability and bifurcation results have been derived for the linearized model. Phase portraits of the original nonlinear model showing stable limit cycle have been simulated.

scholarly journals Almost Surely Asymptotic Stability of Numerical Solutions for Neutral Stochastic Delay Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Zhanhua Yu ◽  
Mingzhu Liu
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Euler Method ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Backward Euler ◽  
Delay Differential ◽  
Theoretical Results

We investigate the almost surely asymptotic stability of Euler-type methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.

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