Global Hopf bifurcation of two-parameter flows

In this paper, the dynamics of a phytoplankton–zooplankton system with delay and diffusion are investigated. The positivity and persistence are studied by using the comparison theorem and upper and lower solutions method. The stability of steady states and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And the global existence of positive periodic solutions is established by using the global Hopf bifurcation result given by Wu [1996]. Finally, some numerical simulations are carried out to illustrate the analytical results.

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Local and global Hopf bifurcation analysis on simplified bidirectional associative memory neural networks with multiple delays

Mathematics and Computers in Simulation ◽

10.1016/j.matcom.2018.02.002 ◽

2018 ◽

Vol 149 ◽

pp. 69-90 ◽

Cited By ~ 26

Author(s):

Changjin Xu

Keyword(s):

Neural Networks ◽

Hopf Bifurcation ◽

Associative Memory ◽

Bifurcation Analysis ◽

Multiple Delays ◽

Bidirectional Associative Memory ◽

Global Hopf Bifurcation

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Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501633 ◽

2019 ◽

Vol 29 (12) ◽

pp. 1950163 ◽

Cited By ~ 1

Author(s):

Suqi Ma

Keyword(s):

Hopf Bifurcation ◽

Parameter Space ◽

Time Delays ◽

Neuron Model ◽

Multiple Time ◽

Multiple Time Delays ◽

The Stability ◽

Geometrical Scheme ◽

Delay Differential ◽

Two Parameter

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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Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

Abstract and Applied Analysis ◽

10.1155/2012/909385 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Qiming Liu ◽

Wang Zheng

Keyword(s):

Neural Network ◽

Hopf Bifurcation ◽

Numerical Simulations ◽

Sufficient Conditions ◽

Hopf Bifurcations ◽

Bifurcation Parameter ◽

Explicit Formulas ◽

Global Hopf Bifurcation ◽

Discrete Delays ◽

Bifurcation And Stability

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

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