Stability and bifurcation analysis of two-neuron network with discrete and distributed delays

Karaoğlu, Esra; Yılmaz, Enes; Merdan, H.

doi:10.1016/j.neucom.2015.12.006

Cited by 31 publications

(17 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 4: System (8) is an incommensurate-order neural model, which shares the property of containing two integer-order differential equations and two fractional differential equations. Compared with the integral-order neural models in [14], [15] and the commensurate-order neural networks in [36], [38], the presence of incommensurate order usually makes the analytical work more challenging.…”

Section: The Mathematical Modementioning

confidence: 99%

“…Remark 5: Compared with (14), (15) is more tractable to determine the distribution of roots. As long as (15) owns the positive root z k , (14) has the positive root ω k = z n k . Assuming that (15)…”

Section: Local Stability and Hopf Bifurcationmentioning

confidence: 99%

“…As long as (15) owns the positive root z k , (14) has the positive root ω k = z n k . Assuming that (15)…”

Section: Local Stability and Hopf Bifurcationmentioning

confidence: 99%

“…The direction of Hopf bifurcation and stability of bifurcation periodic solutions were given. Compared with system (2), Karaoglu et al [15] proposed a more general neural network system that has different discrete delays and activation functions, but with only one distributed delay. Additionally, several studies have developed low dimensional systems into high dimensional ones concerning merely discrete delays [16], [17].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dynamics of Fractional-Order Neural Networks With Discrete and Distributed Delays

Xiao

Jiang

et al. 2020

IEEE Access

View full text Add to dashboard Cite

This paper is concerned with the stability and Hopf bifurcation of fractional-order neural networks with discrete and distributed delays. The novelty of this paper is to take into account the discrete time delay and the distributed time delay for fractional-order systems. By introducing two virtual neurons to the original network, a new four-neuron network only involving discrete delays is formed. The sum of discrete delays is adopted as the bifurcation parameter to demonstrate the existence of Hopf bifurcation. It is found that the critical value of bifurcation can be effectively manipulated by choosing appropriate system parameters and order. Finally, numerical simulations are executed to substantiate the theoretical results and describe the relationships between the parameters and the onset of bifurcation. INDEX TERMS Stability, Hopf bifurcation, discrete delay, distributed delay, fractional-order neural networks.

show abstract

Section: The Mathematical Modementioning

confidence: 99%

Section: Local Stability and Hopf Bifurcationmentioning

confidence: 99%

“…As long as (15) owns the positive root z k , (14) has the positive root ω k = z n k . Assuming that (15)…”

Section: Local Stability and Hopf Bifurcationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamics of Fractional-Order Neural Networks With Discrete and Distributed Delays

Xiao

Jiang

et al. 2020

IEEE Access

View full text Add to dashboard Cite

show abstract

“…For the above simulations we consider interesting for further investigation the study of both mixed system, in particular the analysis of all possible bifurcation should be performed in the future. Moreover, in the previous simulations we have only considered the case m = 1 ( as done for example in [6], [7]), while in many cases (see for example [16,15,14]) the values of m may change the qualitative behaviour of the system. In any case, the mixed delay case (both constant and distributed) is able to recover the complex behaviour observed in the constant delay case.…”

Section: Case M = 2 Equation (21) Becomesmentioning

confidence: 99%

Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study

Caraballo¹,

Colucci²,

Guerrini³

2019

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

In this article we consider a model introduced by Ucar in order to simply describe chaotic behaviour with a one dimensional ODE containing a constant delay. We study the bifurcation problem of the equilibria and we obtain an approximation of the periodic orbits generated by the Hopf bifurcation. Moreover, we propose and analyse a more general model containing distributed time delay. Finally, we propose some ideas for further study. All the theoretical results are supported and illustrated by numerical simulations.

show abstract

Bifurcation control of small‐world networks with delays VIA PID controller

Tao

Xiao

Jiang

et al. 2018

Asian Journal of Control

View full text Add to dashboard Cite

In this paper, the problem of bifurcation control for a small‐world network model with time delay is studied. We first put forward a Proportional‐Integral‐Derivative (PID) feedback scheme to control the Hopf bifurcation of the network. The time delay is selected as the bifurcation parameter. The conditions of the stability and Hopf bifurcation are given for the controlled network. By using the center manifold theorem and the normal form theory, the direction and stability of bifurcating periodic solutions are confirmed. The feasible region of the parameters of the controller is determined. It is found that the bifurcation dynamics of the small‐world network are optimized by adjusting the parameters of the PID controller. Finally, a numerical example verifies the effectiveness of the designed PID controller, and the relationships between the onset of the Hopf bifurcation and the control parameters are obtained.

show abstract

Stability and bifurcation analysis of two-neuron network with discrete and distributed delays

Cited by 31 publications

References 24 publications

Dynamics of Fractional-Order Neural Networks With Discrete and Distributed Delays

Dynamics of Fractional-Order Neural Networks With Discrete and Distributed Delays

Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study

Bifurcation control of small‐world networks with delays VIA PID controller

Contact Info

Product

Resources

About