STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (04) ◽  
pp. 1355-1366 ◽  
Author(s):  
WENWU YU ◽  
JINDE CAO
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Neuron System ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach ◽  
System With Time Delay

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

scholarly journals Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

2021 ◽  
Vol 31 (08) ◽  
pp. 2150143
Author(s):  
Zunxian Li ◽  
Chengyi Xia
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Cellular Neural Networks ◽  
Bifurcation Theorem ◽  
Decoupling Method ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Approximate Expressions ◽  
Zero Equilibrium

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

Hopf Bifurcation for Implicit Neutral Functional Differential Equations

1993 ◽  
Vol 36 (3) ◽  
pp. 286-295 ◽  
Author(s):  
Tomasz Kaczynski ◽  
Huaxing Xia
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Bifurcation Theorem ◽  
Functional Differential

AbstractAn analog of the Hopf bifurcation theorem is proved for implicit neutral functional differential equations of the form F(xt, D′(xt, α), α) = 0. The proof is based on the method of S1-degree of convex-valued mappings. Examples illustrating the theorem are provided.

Harmonic balance and the Hopf bifurcation

1977 ◽  
Vol 82 (3) ◽  
pp. 453-467 ◽  
Author(s):  
D. J. Allwright
Keyword(s):  
Hopf Bifurcation ◽  
Harmonic Balance ◽  
Infinite Order ◽  
Transfer Functions ◽  
Block Diagram ◽  
Nyquist Diagram ◽  
Bifurcation Theorem ◽  
Control Loops ◽  
Existence And Stability ◽  
Linear Elements

SummaryA form of the Hopf bifurcation theorem specially suited to systems in block-diagram form has been developed, which allows one to deal with high or infinite order linear elements solely in terms of their transfer functions. The results are proved by the method of harmonic balance, and, for general non-linear systems, lead to criteria for the existence and stability of bifurcated orbits generalizing those derived by various authors for systems of ordinary differential equations. In the particular case of control loops with a single non-linearity, a simple addition to the Nyquist diagram of the loop determines the amplitude and frequency of bifurcated orbits, and whether they occur when the equilibrium is stable or unstable. The analysis is independent of the central manifold theorem, and of Floquet theory.

Sign in / Sign up

Export Citation Format

Share Document