Dynamics of a Two-Dimensional Delayed Small-World Network Under Delayed Feedback Control

Xu, Xu; Hu, Haiyan; Wang, Huailei

doi:10.1142/s021812740601677x

Cited by 17 publications

(5 citation statements)

References 23 publications

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“…Since Watts and Strogatz's pioneering work in 1998, the novel concept and intriguing behavior of the small-world have drawn substantial attention to the network comprising a large number of sites with a small number of short-cut connections inserted between randomly chosen pairs of sites [20,21]. Furthermore, many practical networks, such as power networks, financial networks, and brain networks, frequently behave like small-world networks [22,23]. Therefore, investigating the small-world effect of each short-cut link introduced to a delayed NN is quite essential.…”

Section: Introductionmentioning

confidence: 99%

Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves

Mo,

Huang,

et al. 2024

Math Methods in App Sciences

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Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes.

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Section: Introductionmentioning

confidence: 99%

Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves

Mo,

Huang,

et al. 2024

Math Methods in App Sciences

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“…In addition, a great number of real networks, such as power networks, financial networks and neural networks, often behave like small-world networks [27,28]. Usually, it is very difficult to gain an insight into the dynamics of small-world networks, especially when they involve time delays in signal transmissions.…”

Section: Introductionmentioning

confidence: 99%

Hopf bifurcation analysis of a four-neuron network with multiple time delays

Mao

2008

Nonlinear Dyn

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This paper reveals the dynamics of a neural network of four neurons with multiple time delays and a short-cut connection through a combined study of theoretical analysis, numerical simulations, and experiments. The first step of the study is to derive the sufficient conditions for the stability and instability of the network equilibrium, and the second step is to determine the properties of the periodic response bifurcating from a Hopf bifurcation of the network equilibrium on the basis of the normal form and the center manifold reduction. Afterwards, the study turns to the validation of theoretical results through numerical simulations and a circuit experiment. The case studies show that both numerical simulations and circuit experiment get a nice agreement with theoretical results.

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“…Some common networks such as power grids, financial networks, and neural networks behave like small-world networks. Therefore, numerous studies [2][3][4][5][6][7][8][9][10][11] on the small-world networks have been witnessed since the pioneering work of Watts and Strogatz [12].…”

Section: Introductionmentioning

confidence: 99%

Effects of small world connection on the dynamics of a delayed ring network

Xiao-li

Wang

2008

Nonlinear Dyn

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This paper presents a detailed analysis on the dynamics of a ring network with small world connection. On the basis of Lyapunov stability approach, the asymptotic stability of the trivial equilibrium is first investigated and the delay-dependent criteria ensuring global stability are obtained. The existence of Hopf bifurcation and the stability of periodic solutions bifurcating from the trivial equilibrium are then analyzed. Further studies are paid to the effects of small world connection on the stability interval and the stability of periodic solution. In particular, some complex dynamical phenomena due to short-cut strength are observed numerically, such as: period-doubling bifurcation and torus breaking to chaos, the coexistence of multiple periodic solutions, multiple quasi-periodic solutions, and multiple chaotic attractors. The studies show that small world connection may be used as a simple but efficient "switch" to control the dynamics of a system.

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Dynamics of a Two-Dimensional Delayed Small-World Network Under Delayed Feedback Control

Cited by 17 publications

References 23 publications

Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves

Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves

Hopf bifurcation analysis of a four-neuron network with multiple time delays

Effects of small world connection on the dynamics of a delayed ring network

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