scholarly journals On the delay-independent stability of a delayed differential equation of 1st order

1989 ◽  
Vol 142 (1) ◽  
pp. 13-25 ◽  
Author(s):  
Takashi Amemiya
Keyword(s):  
Differential Equation ◽  
Delayed Differential Equation

Analysis of a Crowley-Martin Type Chemostat with Delayed Growth Response and Pulsed Input

2014 ◽  
Vol 556-562 ◽  
pp. 4333-4337
Author(s):  
Ming Juan Sun ◽  
Hua Xin Zhao ◽  
Qing Lai Dong
Keyword(s):  
Differential Equation ◽  
Growth Response ◽  
Global Attractivity ◽  
Computational Techniques ◽  
Chemostat Model ◽  
Delayed Differential Equation ◽  
Delayed Growth

In this paper, we introduce and study a Crowley-Martin type Chemostat model with delayed growth response and pulsed input. We get that the existence and the global attractivity of a ‘microorganism-extinction’periodic solution. We prove that the system is permanent under appropriate conditions, by use of new computational techniques for impulsive and delayed differential equation.

scholarly journals Effects of Time Delay and Noise on Asymptotic Stability in Human Quiet Standing Model

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Caihong Wang ◽  
Jian Xu
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Quiet Standing ◽  
Delayed Differential Equation ◽  
Noise Perturbation ◽  
Small Delay ◽  
Stochastic Average Method ◽  
Small Change ◽  
The Stability ◽  
Necessary And Sufficient

A human quiet standing stability is discussed in this paper. The model under consideration is proposed to be a delayed differential equation (DDE) with multiplicative white noise perturbation. The method of the center manifold is generalized to reduce a delayed differential equation to a two-dimensional ordinary differential equation, to study delay-induced instability or Hopf bifurcation. Then, the stochastic average method is employed to obtain the Itô equation. Thus, the top Lyapunov exponent is calculated and the necessary and sufficient condition of the asymptotic stability in views of probability one is obtained. The results show that the exponent is related to not only the strength of noise but also the delay, namely, the reaction speed of brain. The effect of the strength of noise on the human quiet standing losing stability is weak for a small delay. With the delay increasing, such effect becomes stronger and stronger. A small change in the strength of noise may destabilize the quiet standing for a large delay. It implies that a person with slow reaction is easy to lose the stability of his/her quiet standing.

scholarly journals On Ulam Stability of a Generalized Delayed Differential Equation of Fractional Order

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Janusz Brzdęk ◽  
Nasrin Eghbali ◽  
Vida Kalvandi
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
Fractional Order ◽  
Nonlinear Operator ◽  
Main Tool ◽  
Ulam Stability ◽  
Speed Of Convergence ◽  
Delayed Differential Equation ◽  
Pointwise Limit

AbstractWe investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution $$\phi $$ ϕ generates the exact solution as a pointwise limit of the sequence $$\varLambda ^n\phi $$ Λ n ϕ with some integral (possibly, nonlinear) operator $$\varLambda $$ Λ . We estimate the speed of convergence and the distance between those approximate and exact solutions. Moreover, we provide some exemplary calculations, involving the Chebyshev and Bielecki norms and some semigauges, that could help to obtain reasonable outcomes for such estimations in some particular cases. The main tool is the Diaz–Margolis fixed point alternative.

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