An iterative method to compute a closest saddle node or Hopf bifurcation instability in multidimensional parameter space

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

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Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate

Complexity ◽

10.1155/2020/7310347 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16

Author(s):

Ding Fang ◽

Yongxin Zhang ◽

Wendi Wang

Keyword(s):

Hopf Bifurcation ◽

Numerical Simulations ◽

Bifurcation Analysis ◽

Epidemic Model ◽

Multiple Equilibria ◽

Homoclinic Bifurcation ◽

Propagation Model ◽

Bifurcation Curve ◽

Saddle Node Bifurcation ◽

Saddle Node

An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

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Bifurcation analysis method of nonlinear traffic phenomena

International Journal of Modern Physics C ◽

10.1142/s0129183115501119 ◽

2015 ◽

Vol 26 (10) ◽

pp. 1550111 ◽

Cited By ~ 3

Author(s):

Wenhuan Ai ◽

Zhongke Shi ◽

Dawei Liu

Keyword(s):

Hopf Bifurcation ◽

Traffic Flow ◽

Bifurcation Analysis ◽

Phase Plane ◽

Flow Model ◽

Analysis Method ◽

Equilibrium Solutions ◽

Saddle Node Bifurcation ◽

Traffic Flow Model ◽

Saddle Node

A new bifurcation analysis method for analyzing and predicting the complex nonlinear traffic phenomena based on the macroscopic traffic flow model is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis. Although the substitution seems to be simple, it can extend the range of the variable to infinity and build a relationship between the traffic congestion and the unstable system in the phase plane. So the problem of traffic flow could be converted into that of system stability. The analysis identifies the types and stabilities of the equilibrium solutions of the new model and gives the overall distribution structure of the nearby equilibrium solutions in the phase plane. Then we deduce the existence conditions of the models Hopf bifurcation and saddle-node bifurcation and find some bifurcations such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles and Bogdanov–Takens bifurcation. Furthermore, the Hopf bifurcation and saddle-node bifurcation are selected as the starting point of density temporal evolution and it will be helpful for improving our understanding of stop-and-go wave and local cluster effects observed in the free-way traffic.

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COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

International Journal of Modern Physics B ◽

10.1142/s0217979211101673 ◽

2011 ◽

Vol 25 (29) ◽

pp. 3977-3986 ◽

Cited By ~ 25

Author(s):

HUAGUANG GU ◽

HUIMIN ZHANG ◽

CHUNLING WEI ◽

MINGHAO YANG ◽

ZHIQIANG LIU ◽

...

Keyword(s):

Hopf Bifurcation ◽

Bifurcation Point ◽

Firing Pattern ◽

Neuronal System ◽

Neural Firing ◽

Coherence Resonance ◽

Saddle Node Bifurcation ◽

Hopf Bifurcation Point ◽

Firing Patterns ◽

Saddle Node

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

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A planar model system for the saddle–node Hopf bifurcation with global reinjection

Nonlinearity ◽

10.1088/0951-7715/17/4/001 ◽

2004 ◽

Vol 17 (4) ◽

pp. 1119-1151 ◽

Cited By ~ 12

Author(s):

Bernd Krauskopf ◽

Bart E Oldeman

Keyword(s):

Hopf Bifurcation ◽

Model System ◽

Planar Model ◽

Saddle Node

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Bifurcation Analysis of the Wound Rotor Induction Motor

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415501631 ◽

2015 ◽

Vol 25 (12) ◽

pp. 1550163 ◽

Cited By ~ 1

Author(s):

R. Salas-Cabrera ◽

A. Hernandez-Colin ◽

J. Roman-Flores ◽

N. Salas-Cabrera

Keyword(s):

Hopf Bifurcation ◽

Induction Motor ◽

Bifurcation Analysis ◽

Open Loop ◽

Experimental Results ◽

Saddle Node Bifurcation ◽

Saddle Node ◽

Bifurcation Phenomena ◽

Three Phase ◽

The Individual

This work deals with the bifurcation phenomena that occur during the open-loop operation of a single-fed three-phase wound rotor induction motor. This paper demonstrates the occurrence of saddle-node bifurcation, hysteresis, supercritical saddle-node bifurcation, cusp and Hopf bifurcation during the individual operation of this electromechanical system. Some experimental results associated with the bifurcation phenomena are presented.

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Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2

Journal of Fluid Mechanics ◽

10.1017/s0022112001007285 ◽

2002 ◽

Vol 455 ◽

pp. 263-281 ◽

Cited By ~ 38

Author(s):

F. MARQUES ◽

J. M. LOPEZ ◽

J. SHEN

Keyword(s):

Hopf Bifurcation ◽

Parameter Space ◽

Swirling Flow ◽

Three Dimensional ◽

Ekman Layer ◽

Mode Interaction ◽

Dimensional Structure ◽

Navier Stokes ◽

Double Hopf Bifurcation ◽

Rotating Wave

A double Hopf bifurcation has been found of the flow in a cylinder driven by the rotation of an endwall. A detailed analysis of the multiple solutions in a large region of parameter space, computed with an efficient and accurate three-dimensional Navier-Stokes solver, is presented. At the double Hopf point, an axisymmetric limit cycle and a rotating wave bifurcate simultaneously. The corresponding mode interaction generates an unstable two-torus modulated rotating wave solution and gives a wedge-shaped region in parameter space where the two periodic solutions are both stable. By exploring in detail the three-dimensional structure of the flow, we have identified the two mechanisms that compete in the neighbourhood of the double Hopf point. Both are associated with the jet that is formed when the Ekman layer on the rotating endwall is turned by the stationary sidewall.

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APPLICATION OF A TWO-DIMENSIONAL HINDMARSH–ROSE TYPE MODEL FOR BIFURCATION ANALYSIS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500557 ◽

2013 ◽

Vol 23 (03) ◽

pp. 1350055 ◽

Cited By ~ 11

Author(s):

SHYAN-SHIOU CHEN ◽

CHANG-YUAN CHENG ◽

YI-RU LIN

Keyword(s):

Hopf Bifurcation ◽

Bifurcation Analysis ◽

Sufficient Conditions ◽

Type Model ◽

Two Dimensional ◽

Saddle Node Bifurcation ◽

Trapping Region ◽

Saddle Node ◽

The Stability ◽

Two Equilibria

In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh–Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: (1) the number and stability of the equilibria are analyzed in detail using a table to demonstrate the matter in which the stability of the equilibrium changes and to determine which two equilibria collapse through the saddle-node bifurcation; (2) the sufficient conditions for an Andronov–Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and (3) we elaborately evaluate the sufficient conditions for the Bogdanov–Takens (BT) and Bautin bifurcations. Several numerical simulations for these conditions are also presented. In particular, two types of bistable behaviors are numerically demonstrated: the BT and Bautin bifurcations. Notably, all of the bifurcation curves in the domain of the remaining parameters are similar when the time scale is large. Additionally, to show the potential for a limit cycle, the existence of a trapping region is demonstrated. These results present a variety of diverse behaviors for this model. The results of this study will be helpful in assessing suitable parameters for fitting the resemblances of experimental observations.

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Bifurcation Analysis of a Two-Dimensional Neuron Model under Electrical Stimulation

Mathematical Problems in Engineering ◽

10.1155/2021/9314736 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Chunhua Yuan ◽

Xiangyu Li

Keyword(s):

Electrical Stimulation ◽

Hopf Bifurcation ◽

Dynamic Characteristics ◽

Neuron Model ◽

Equilibrium Points ◽

Model Parameters ◽

Two Dimensional ◽

Saddle Node Bifurcation ◽

Saddle Node ◽

Firing Characteristics

The two-dimensional neuron model can not only reproduce abundant firing patterns, but also satisfy the research of dynamical behavior because of its nonlinear characteristics. It is the most simplified model that includes the fast and slow variables required for neuron firing. In this paper, the dynamic characteristics of two-dimensional neuron model are described by both analytical and numerical methods, and the influence of model parameters and external stimuli on dynamic characteristics is described. The firing characteristics of the Prescott model under external electrical stimulation are studied, and the influence of electrophysiological parameters on the firing characteristics is analyzed. The saddle-node bifurcation and Hopf bifurcation characteristics are studied through the distribution of equilibrium points. It is found that there are critical saddle-node bifurcation and critical Hopf bifurcation in the Prescott model. And the value range of the key parameters that cause the critical bifurcation of the model is obtained by analytical methods.

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