Zero-Hopf singularity in bidirectional ring network model with delay

He, Xing; Li, Chuandong; Huang, Tingwen; Huang, Junjian

doi:10.1007/s11071-014-1612-x

Cited by 9 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One topic is aimed at the study of the local and global stability of the equilibrium, the existence and stability of periodic solutions yielded from Hopf bifurcation, as reported in (Xu and Li 2012;Yang and Ye 2009;Xu et al 2011;Cao 2003;Sun et al 2007;Zheng et al 2008;Liu and Cao 2011;Yang et al 2014), and another topic is aimed at the study of more complex dynamics mainly involving some degenerate bifurcations existing in the neighborhood of the codimensional singularity. Such issues have been addressed by several articles (Ding et al 2012;He et al 2012aHe et al , b, 2013He et al , 2014Li and Wei 2005;Dong and Liao 2013;Song and Xu 2012;Campbell and Yuan 2008;Guo et al 2008;Yang 2008;Liu 2014;Garliauskas 1998;Kepler et al 1990). For instance, Ding et al (2012) have considered the zero-Hopf bifurcation of a generalized Gopalsamy neural network model, the normal forms of near a zero-Hopf critical point were deduced by using multiple time scales and center manifold reduction methods respectively.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, Ding et al (2012) have considered the zero-Hopf bifurcation of a generalized Gopalsamy neural network model, the normal forms of near a zero-Hopf critical point were deduced by using multiple time scales and center manifold reduction methods respectively. The similar work has been carried out by He et al (2014), but contraposing a different neural network. The authors in (Dong and Liao 2013;He et al 2012a;Li and Wei 2005;Xu 2012, 2013;Campbell and Yuan 2008;Guo et al 2008;Yang 2008) have devoted to the analysis of the Bogdanov-Takens (B-T) bifurcation with codimension two.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

High codimensional bifurcation analysis to a six-neuron BAM neural network

Liu

et al. 2015

Cogn Neurodyn

View full text Add to dashboard Cite

In this article, the high codimension bifurcations of a six-neuron BAM neural network system with multiple delays are addressed. We first deduce the existence conditions under which the origin of the system is a Bogdanov-Takens singularity with multiplicities two or three. By choosing the connection coefficients as bifurcation parameters and using the formula derived from the normal form theory and the center manifold, the normal forms of Bogdanov-Takens and triple zero bifurcations are presented. Some numerical examples are shown to support our main results.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High codimensional bifurcation analysis to a six-neuron BAM neural network

Liu

et al. 2015

Cogn Neurodyn

View full text Add to dashboard Cite

show abstract

“…Orosz [41] developed an effective decomposition method for investigating the dynamics in the vicinity of steady and oscillatory cluster states and showed the coexistence of multiple stable and unstable steady and oscillatory cluster states. For more related studies on the dynamics of coupled systems with time delays, the reader is referred to the work in [14,[42][43][44][45][46][47], and some references cited therein. In previous studies, the authors often limited their research to the models of two or three coupled sub-systems and/or the models with a special structure since such models are fundamental and relatively simple.…”

Section: Introductionmentioning

confidence: 99%

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

Mao

Wang

2015

Nonlinear Dyn

View full text Add to dashboard Cite

This paper reveals the dynamical behaviors of a coupled network consisting of four neural subnetworks and multiple two-way couplings of neurons between the individual sub-networks. Different types of time delays are introduced into the internal connections within the individual sub-networks and the couplings between the sub-networks. Stability switches of the network equilibrium and Hopf bifurcation are analyzed by discussing the associated characteristic equation. The conditions for the existence of different patterns of oscillations are discussed. By using the Lyapunov's second method, the global stability of the network equilibrium is studied. Numerical simulations are performed to validate the theoretical results, and rich dynamical behaviors are observed, such as synchronous oscillations, multiple stability switches between the rest state and oscillations, phase-locked oscillations, asynchronous oscillations, and the coexistence of different patterns of bifurcated oscillations.

show abstract

“…In fact, by considering the combined effects of two independent varying parameters, the occurrence of bifurcation in such a network is certain codimension-2 bifurcations such as double Hopf bifurcation [18,19], Bogdanov-Takens singularity [20] and Hopf-pitchfork bifurcation [21][22][23][24]. So far, there are only a few articles on double Hopf bifurcation of delayed neural networks.…”

Section: Introductionmentioning

confidence: 99%

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

2015

Nonlinear Dyn

View full text Add to dashboard Cite

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.

show abstract

Zero-Hopf singularity in bidirectional ring network model with delay

Cited by 9 publications

References 29 publications

High codimensional bifurcation analysis to a six-neuron BAM neural network

High codimensional bifurcation analysis to a six-neuron BAM neural network

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

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

Contact Info

Product

Resources

About