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Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.


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Journal ArticleDOI
TL;DR: A mathematical model describing the dynamics of a hematopoietic stem cell population and the analysis of the positive steady state behavior concludes the existence of a Hopf bifurcation and gives criteria for stability switches.
Abstract: We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.

54 citations

Journal ArticleDOI
Xiaoyu Hu1, Chongxin Liu1, Ling Liu1, Junkang Ni1, Shilei Li1 
TL;DR: The results show that the electrical activities, such as quiescent state, spiking and bursting, can be observed when the values of external forcing current beyond certain thresholds are observed.
Abstract: In this paper, the threshold dynamics of Morris–Lecar neuron model is firstly analyzed by bifurcation diagram of interspike interval as a function of external forcing current, and then the discharge series, phase portraits and nullclines of the neuron under different conditions are investigated in a numerical way. The results show that the electrical activities, such as quiescent state, spiking and bursting, can be observed when the values of external forcing current beyond certain thresholds. Finally, based on the 2-D nonlinear differential equations of Morris–Lecar neuron model, a complete electronic implementation of this model is proposed and studied in detail. At the same time, a circuitry realization of the hyperbolic cosine function $$\tau _W (V)$$ in the Morris–Lecar neuron model is put forward and described carefully. The outputs of designed circuits are consistent well with the theoretical predictions, which validate the design methods. Moreover, the circuit presented in this paper can be used as an experimental unit to investigate the dynamics of a single neuron or collective behaviors of a large-scale neural network.

54 citations

Journal ArticleDOI
TL;DR: In this paper, a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form was given.
Abstract: We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.

54 citations

Journal ArticleDOI
TL;DR: In this article, the non-linear Klein-Gordon equation was used to solve the bifurcation problem in the context of nonlinear Klein Gordon Equations (KGE).

54 citations

Journal ArticleDOI
TL;DR: In this paper, an analysis of stability and bifurcation of a Ricker-type competition model of two species is given, and a complete analysis of the center manifold is given.
Abstract: Our main objective is to study a Ricker-type competition model of two species. We give a complete analysis of stability and bifurcation and determine the centre manifolds, as well as stable and unstable manifolds. It is shown that the autonomous Ricker competition model exhibits subcritical bifurcation, bubbles, period-doubling bifurcation, but no Neimark–Sacker bifurcations. We exhibit the region in the parameter space where the competition exclusion principle applies.

54 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023122
2022326
2021187
2020195
2019166
2018220