Hopf bifurcation in a reaction–diffusion model with Degn–Harrison reaction scheme

Dong, Yaying; Li, Shanbing; Zhang, Shunli

doi:10.1016/j.nonrwa.2016.07.002

Cited by 15 publications

(7 citation statements)

References 26 publications

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“…which forms a global attractor if 0 ≤ k ≤ 1 u 2 . Some further results on the local asymptotic stability of the equilibrium and the systems bifurcation characteristics were reported in Li et al 14 A similar system with slight modifications was studied in Donga et al 15 Perhaps, the latest work related to the stability of the Degn-Harrison model is that of Lisena,5 where it was shown that for any solution (u, v) of the ODE problem, there exists a constant T > 0, which may depend on u 0 and v 0 , such that…”

Section: Introductionmentioning

confidence: 80%

“…Before we present our analysis and numerical examples, let us recall the most relevant results reported in the literature in relation to the proposed system . The Jacobian of the Degn‐Harrison model is given by

J = (\begin{matrix} F_{0} & - G_{0} \\ 1 + F_{0} & - G_{0} \end{matrix}),

where

F_{0} = - \frac{a + α^{2} k (a - 2 b)}{α (1 + k α^{2})} and G_{0} = \frac{α}{1 + k α^{2}},

with

α = a - b .

In Peng et al, the system was shown to have the unique constant steady state

(u^{*}, v^{*}) = (α, \frac{b}{α} (1 + α^{2} k)),

which forms a global attractor if

0 \leq k \leq \frac{1}{ũ^{2}} .

Some further results on the local asymptotic stability of the equilibrium and the systems bifurcation characteristics were reported in Li et al A similar system with slight modifications was studied in Donga et al Perhaps, the latest work related to the stability of the Degn‐Harrison model is that of Lisena, where it was shown that for any solution

((), u, v)

of the ODE problem, there exists a constant T > 0, which may depend on u 0 and v 0 , such that

(u, v) \in ℜ = [\overset{}{u}]

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the local and global asymptotic stability of the Degn‐Harrison reaction‐diffusion model

Abbad

Bendoukha

Abdelmalek

2018

Math Methods in App Sciences

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Section: Introductionmentioning

confidence: 80%

J = (\begin{matrix} F_{0} & - G_{0} \\ 1 + F_{0} & - G_{0} \end{matrix}),

where

F_{0} = - \frac{a + α^{2} k (a - 2 b)}{α (1 + k α^{2})} and G_{0} = \frac{α}{1 + k α^{2}},

with

α = a - b .

In Peng et al, the system was shown to have the unique constant steady state

(u^{*}, v^{*}) = (α, \frac{b}{α} (1 + α^{2} k)),

which forms a global attractor if

0 \leq k \leq \frac{1}{ũ^{2}} .

((), u, v)

of the ODE problem, there exists a constant T > 0, which may depend on u 0 and v 0 , such that

(u, v) \in ℜ = [\overset{}{u}]

…”

Section: Introductionmentioning

confidence: 99%

On the local and global asymptotic stability of the Degn‐Harrison reaction‐diffusion model

Abbad

Bendoukha

Abdelmalek

2018

Math Methods in App Sciences

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“…More recently, sufficient conditions for the global asymptotic stability of the unique constant steady state have been obtained in [12] and more relaxed sufficient conditions have been then derived in [13]. The existence of Hopf bifurcation and the corresponding normal form have been also determined in [14,15].…”

Section: Introductionmentioning

confidence: 99%

A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results

Abbad

Abdelmalek

Bendoukha

et al. 2021

Nonlinear Analysis: Real World Applications

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In this paper we study the Degn-Harrison system with a generalized reaction term. Once proved the global existence and boundedness of a unique solution, we address the asymptotic behaviour of the system. The conditions for the global asymptotic stability of the steady state solution are derived using the appropriate techniques based on the eigen-analysis, the Poincaré-Bendixson theorem and the direct Lyapunov method. Numerical simulations are also shown to corroborate the asymptotic stability predictions. Moreover, we determine the constraints on the size of the reactor and the diffusion coefficient such that the system does not admit non-constant positive steady state solutions.

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“…The Degn-Harrison system (1) has been studied extensively in the literature, but most of the researches focus on the dynamics of this model including the local and global asymptotic stability of the steady-state solutions [25], [26], Turing instability [27], [28] and Hopf bifurcation [29], [30]. However, as far as we know, this is the first work deal with control synchronization of the model (1).…”

Section: Introductionmentioning

confidence: 99%

Synchronization Methods for the Degn-Harrison Reaction-Diffusion Systems

et al. 2020

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Hopf bifurcation in a reaction–diffusion model with Degn–Harrison reaction scheme

Cited by 15 publications

References 26 publications

On the local and global asymptotic stability of the Degn‐Harrison reaction‐diffusion model

On the local and global asymptotic stability of the Degn‐Harrison reaction‐diffusion model

A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results

Synchronization Methods for the Degn-Harrison Reaction-Diffusion Systems

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