Dynamics of an inertial two-neuron system with time delay

Li, Qun; Liao, Xiaofeng; Liu, Yanbing; Zhou, Shangbo; Guo, Songtao

doi:10.1007/s11071-009-9503-2

Cited by 86 publications

(27 citation statements)

References 44 publications

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“…In this case, Liu et al [30] had made a detailed analysis on the effect of time delay on the Hopf bifurcation. They have clearly shown that the system (1.1) undergo a sequence of Hopf bifurcation when the time delay cross a critical value.…”

Section: Lemma 22mentioning

confidence: 99%

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Bifurcation Behavior for an Electronic Neural Network Model with Two Different Delays

Shao

2014

Neural Process Lett

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In this paper, an electronic neural network model with two different delays is investigated. The conditions for the local stability and the existence of Hopf bifurcation at the equilibrium of the system are derived. By applying the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are obtained. Numerical simulations for justifying the theoretical analysis are also included. Finally, main conclusions are given.

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Section: Lemma 22mentioning

confidence: 99%

“…In details, one can see [1,. In 2009, Liu et al [30] had dealt with the dynamics (including the stability, local Hopf bifurcation and resonant codimensional-two bifurcation) of the following inertial two-neuron system with time delay…”

mentioning

confidence: 99%

Bifurcation Behavior for an Electronic Neural Network Model with Two Different Delays

Shao

2014

Neural Process Lett

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“…In recent years, the stability analysis and the feedback control of time delay systems have been widely investigated since delays are often encountered in various engineering systems (see, i.e., [29][30][31][32][33][34][35]). On the other hand, descriptor systems with or without time delay have been extensively studied in recent years (see, i.e., [11,14,[17][18][19][20][21][22][23][24]27], and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Local stability and Hopf bifurcation of two-dimensional nonlinear descriptor system

Liao

Xie

2015

Nonlinear Dyn

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Descriptor systems, which are also called differential-algebraic systems, singular systems, degenerate systems, constrained systems and so on, have been one of the major research topics in engineering field. However, problems related to the local stability and the Hopf bifurcation of nonlinear descriptor systems with time delay have not been thoroughly investigated. In this paper, we consider the dynamical behavior of twodimensional nonlinear descriptor systems with time delay. First, local stability of the system is analyzed using the location of the roots of the characteristic equation of the corresponding linearized system. It is well known that the equilibrium of the linearized system is locally asymptotically stable if all roots of the corresponding characteristic equation locate in the left half of the complex plane, and they are uniformly bounded away from the imaginary axis. Otherwise, the equilibrium is unstable. The conditions for the existence of Hopf bifurcation are investigated in detail by using the time delay as a bifurcation parameter. Furthermore, based on the descriptor and the "neutral-type" model transformation, some special neutral differential equations can be equivalently transformed into the nonlinear descriptor systems. Finally, the correctness and the effectiveness of the theoretical analysis are justified by numerical examples.

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“…In [15], the stability of bifurcating periodic solutions is discussed for a single delayed inertial neuron model under periodic excitation. Liu et al [16] studied the effect of time delay on dynamical behaviors of a two-neuron inertial system. For a four-neuron delayed inertial system [17], Hopf bifurcation was investigated and chaotic behavior was observed in.…”

Section: Introductionmentioning

confidence: 99%

Double Hopf bifurcation in a four-neuron delayed system with inertial terms

2015

Nonlinear Dyn

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In this paper, a four-neuron delayed system with inertial terms is considered. By studying the distribution of the eigenvalues of the associated characteristic equation, we derive the critical values where double Hopf bifurcation occurs. Then by employing the perturbation-incremental scheme for the system, bifurcation diagrams are obtained. Furthermore, we carry out bifurcation analysis showing that there exist a stable fixed point, two stable periodic solutions, co-existence of a pair of stable periodic solutions and quasi-periodic motion in the neighborhood of the double Hopf critical point. We also find some interesting phenomena that the dynamical period switching occurs in some delayed regions. Finally, some numerical simulations are performed to support the theoretical analysis.

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Dynamics of an inertial two-neuron system with time delay

Cited by 86 publications

References 44 publications

Bifurcation Behavior for an Electronic Neural Network Model with Two Different Delays

Bifurcation Behavior for an Electronic Neural Network Model with Two Different Delays

Local stability and Hopf bifurcation of two-dimensional nonlinear descriptor system

Double Hopf bifurcation in a four-neuron delayed system with inertial terms

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