scholarly journals Moment Lyapunov exponent of delay differential equations

2002 ◽  
Vol 30 (6) ◽  
pp. 339-351
Author(s):  
M. S. Fofana
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Fixed Time ◽  
Stability Study ◽  
Unified Framework ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
Delay Differential ◽  
Moment Lyapunov Exponents

The aim of this paper is to establish a connecting thread through the probabilistic concepts ofpth-moment Lyapunov exponents, the integral averaging method, and Hale's reduction approach for delay dynamical systems. We demonstrate this connection by studying the stability of perturbed deterministic and stochastic differential equations with fixed time delays in the displacement and derivative functions. Conditions guaranteeing stable and unstable solution response are derived. It is felt that the connecting thread provides a unified framework for the stability study of delay differential equations in the deterministic and stochastic sense.

P-Bifurcation Aspects of a Modified Duffing Oscillator Near Equilibrium

2001 ◽  
Author(s):  
M. S. Fofana ◽  
Dariusz Szwarc
Keyword(s):  
Time Delay ◽  
Averaging Method ◽  
Duffing Oscillator ◽  
Sufficient Conditions ◽  
Stability Study ◽  
Unified Framework ◽  
Positive Time ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
Delay Differential

Abstract A unified framework for the study of stability of solutions to delay differential equations perturbed deterministically and stochastically has been presented. We review the concepts of Hopf bifurcation, centre manifold, integral averaging method and pth–moment Lyapunov exponent, and then demonstrate their role in the stability study of a modified Duffing oscillator with time delay. Sufficient conditions ensuring Đ- and Þ–bifurcations and the changes in character of the probability density function are established for fixed and positive time delay.

scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

Dynamics of Wild and Sterile Mosquito Population Models with Delayed Releasing

2020 ◽  
Vol 30 (11) ◽  
pp. 2050218
Author(s):  
Li-Ming Cai
Keyword(s):  
Delay Differential Equations ◽  
Time Delays ◽  
Sterile Insect Technique ◽  
Sustained Oscillation ◽  
Hopf Bifurcations ◽  
Control Methods ◽  
Ode Models ◽  
Dynamical Features ◽  
The Stability ◽  
Delay Differential

To reduce the global burden of mosquito-borne diseases, e.g. dengue, malaria, the need to develop new control methods is to be highlighted. The sterile insect technique (SIT) and various genetic modification strategies, have a potential to contribute to a reversal of the current alarming disease trends. In our previous work, the ordinary differential equation (ODE) models with different releasing sterile mosquito strategies are investigated. However, in reality, implementing SIT and the releasing processes of sterile mosquitos are very complex. In particular, the delay phenomena always occur. To achieve suppression of wild mosquito populations, in this paper, we reassess the effect of the delayed releasing of sterile mosquitos on the suppression of interactive mosquito populations. We extend the previous ODE models to the delayed releasing models in two different ways of releasing sterile mosquitos, where both constant and exponentially distributed delays are considered, respectively. By applying the theory and methods of delay differential equations, the effect of time delays on the stability of equilibria in the system is rigorously analyzed. Some sustained oscillation phenomena via Hopf bifurcations in the system are observed. Numerical examples demonstrate rich dynamical features of the proposed models. Based on the obtained results, we also suggest some new releasing strategies for sterile mosquito populations.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

scholarly journals Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Guodong Liu ◽  
Xiaohong Wang ◽  
Xinzhu Meng ◽  
Shujing Gao
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Persistence In Mean ◽  
Predator Prey Model ◽  
Prey System ◽  
Delay Differential ◽  
Lévy Jumps

In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.

Stability Analysis of Fractional Delay Differential Equations by Chebyshev Polynomial

2012 ◽  
Vol 500 ◽  
pp. 586-590
Author(s):  
Xiang Mei Zhang ◽  
Xian Zhou Guo ◽  
Anping Xu
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Delay Differential Equations ◽  
Fractional Derivatives ◽  
Mathieu Equation ◽  
Smooth Coefficients ◽  
Fractional Delay ◽  
The Stability ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The paper is devoted to the numerical stability of fractional delay differential equations with non-smooth coefficients using the Chebyshev collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Chebyshev polynomial of the first kind. Then we solve the stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is examined for a set of case studies that contain the complexities of periodic coefficients, delays and discontinuities.

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