Stability and Oscillation of the Solutions for a Coupled FHN Model with Time Delays

Lin, Yuanhua

doi:10.1007/978-3-319-09330-7_21

Cited by 3 publications

(2 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Buric and Todorovic [4] have analyzed examples of systems of identical FHN neurons arranged in linear or circular lattices, with uni-or bi-directional symmetrical coupling by some typical coupling models. Lin [30] studied a network of FHN neurons with delayed coupling and different synaptic strength of self-connection. They have described stability properties of emerging periodic patterns.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Tehrani¹,

Razvan²

2016

Preprint

View full text Add to dashboard Cite

This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with quadratic term and delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics. Bifurcation diagrams are obtained numerically or analytically from the mathematical model, and the parameter regions of different behaviors are clarified. The neural system exhibits a unique rest point or three ones by employing the saddle-node bifurcation, when strong coupling is applied in the system. Also the trivial rest point shows transcritical bifurcation with one of the new rest points. The asymptotic stability and possible Hopf and Bautin bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Fold cycle, torus, fold-torus, and big homoclinic bifurcations of limit cycles, together with 1 : 1 and 1 : 4 resonances are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf, Hopf-transcritical, and double-zero bifurcation points can be obtained from the intersection of different bifurcation branches. Various patterns of multistability have been observed, both for small and large values of delay. The system may exhibit one synchronous together with one or two anti-phase periodic activities, two synchronous and two anti-phase periodic solutions, one synchronous and one anti-phase periodic solution also one equilibrium, and one anti-phase periodic solution and non-trivial eguilibria, which occur due to Hopf, fold cycle and torus bifurcations. Also one synchronous periodic solution and one torus due to fold cycle, torus and Chenciner bifurcations can be seen. By increasing

show abstract

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Tehrani¹,

Razvan²

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…The appearance and stability of spatio-temporal periodic patterns in unidirectional rings of coupled FHN systems interacting via excitatory delayed chemical synapses is described in [48]. A network of connected FHN neurons with delayed coupling and different synaptic strength of self-connection is studied in [49]. The stability and oscillation of the solutions is investigated by constructing of Liapunov functional and applying the Chafees limit cycle criterion.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation structure of two coupled FHN neurons with delay

Tehrani

Razvan

2015

Mathematical Biosciences

View full text Add to dashboard Cite

This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.

show abstract

Triple-Zero Bifurcation of Van Der Pol Oscillator with Delay Feedback

Ma¹

2021

IJMNTA

View full text Add to dashboard Cite

A van der Pol equation underlying state feedback control is investigated and the triple-zero bifurcation arises at the bifurcation point which is of codimension three singularity. By applying Schmidt-Lyapunov reduction method combined with center manifold analytical technique, the near approximating formal norm is derived at the triple-zero point. Hence after, as varying parameters continuously, the numerical simulation produces homoclinic bifurcation solutions appearing in system. In addition, the numerical simulation also exhibits the produced double-period limit cycle with chosen bifurcation parameters and the routes to chaos via period-doubling bifurcation are also verified.

show abstract

Stability and Oscillation of the Solutions for a Coupled FHN Model with Time Delays

Cited by 3 publications

References 12 publications

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Bifurcation structure of two coupled FHN neurons with delay

Triple-Zero Bifurcation of Van Der Pol Oscillator with Delay Feedback

Contact Info

Product

Resources

About