Stability analysis in a second-order differential equation with delays

Wang, Qiubao

doi:10.1080/00207160.2012.680447

Cited by 3 publications

(1 citation statement)

References 16 publications

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“…But in the whole paper, they did not discuss the effects of the time delay τ on the bacterial density. In our recent works [24,25], we discussed the stability and the Hopf bifurcation of some models with delay-dependent parameters. Moreover, it is well known that the delayed logistic differential equation…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis in a Delay Differential Equations, Which Confers a Strong Allee Effect in Escherichia Coli

Wang

Tian

2015

Mathematical Modelling and Analysis

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The paper addresses the bifurcations for a delay differential model with parameters which confers a strong Allee effect in Escherichia coli. Stability and local Hopf bifurcations are analyzed when the delay τ or σ as parameter. It is also found that there is a non-resonant double Hopf bifurcation occur due to the vanishing of the real parts of two pairs of characteristic roots. We transform the original system into a finite dimensional system by the center manifold theory and simplify the system further by the normal form method. Then, we obtain a complete bifurcation diagram of the system. Finally, we provide numerical results to illustrate our conclusions. There are many interesting phenomena, such as attractive quasi-periodic solution and three-dimensional invariant torus.

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Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis in a Delay Differential Equations, Which Confers a Strong Allee Effect in Escherichia Coli

Wang

Tian

2015

Mathematical Modelling and Analysis

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Bifurcation analysis in a predator–prey model for the effect of delay in prey

Wang

2016

Int. J. Biomath.

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In this paper, we study dynamics in a predator–prey model with delay, in which predator can be infected, with particular attention focused on nonresonant double Hopf bifurcation. By using center manifold reduction methods, we obtain the equivalent normal forms near a double Hopf critical point in this system. Moreover, bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

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On the Existence and Stability of Periodic Solutions of Airy’s Equation with Elastic Coefficients

Obasi

Osu

Ogbogbo

2021

NJPAS

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In this paper, the existence and stability of periodic solutions of a certain second order differential equation with elastic coefficient were investigated using power series method, eigenvalue approach and lyapunov direct method. Existence of analytical solution which is independent of time was achieved using the power series method. Eigenvalue approach and Lyapunov direct method were used to investigate the stability of the resulting solution. Periodic solution was obtained using the eigenvalues of the resulting matrix. The first stability method further examined stability of the equilibrium point by considering the intervals around the origin and it’s discriminate. The equilibrium points for the intervals and the discriminate were unstable because the real part of the characteristics root is zero. Unstable equilibrium point was also obtained for the second stability method using the energy function and time derivative around the equilibrium point. The two unstable results indicated that there were highly instability regions with a strictly positive elastic coefficient. The highly instability regions were confirmed by the presence of elastic coefficient which reduces oscillation with an increase in amplitude. Furthermore, numerical simulations for existence and stability of Airy’s equation at different values of the elastic coefficient were illustrated in order to demonstrate the behaviour of the solutions which extends some results in literature.

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Stability analysis in a second-order differential equation with delays

Cited by 3 publications

References 16 publications

Bifurcation Analysis in a Delay Differential Equations, Which Confers a Strong Allee Effect in Escherichia Coli

Bifurcation Analysis in a Delay Differential Equations, Which Confers a Strong Allee Effect in Escherichia Coli

Bifurcation analysis in a predator–prey model for the effect of delay in prey

On the Existence and Stability of Periodic Solutions of Airy’s Equation with Elastic Coefficients

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