scholarly journals Center Manifold Reduction of the Hopf-Hopf Bifurcation in a Time Delay System

2013 ◽  
Vol 39 ◽  
pp. 57-65 ◽  
Author(s):  
Christoffer Heckman ◽  
Jakob Kotas ◽  
Richard Rand
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Delay System ◽  
Time Delay System ◽  
Center Manifold Reduction ◽  
Manifold Reduction

scholarly journals Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jinbin Wang ◽  
Rui Zhang ◽  
Lifenq Ma
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Two Dimensional ◽  
Main Attention ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Physical Phenomena ◽  
Manifold Reduction ◽  
Theoretical Results

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

2020 ◽  
Vol 30 (09) ◽  
pp. 2050130 ◽  
Author(s):  
Shangzhi Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Banach Spaces ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delay Effect ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Equivariant Hopf Bifurcation ◽  
Manifold Reduction

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

scholarly journals Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Luca Guerrini ◽  
Giovanni Molica Bisci
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Production Function ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

Stability and Oscillation Phenomena Analysis in Microbial Continuous Fermentation Time-Delay System

2013 ◽  
Vol 483 ◽  
pp. 337-341
Author(s):  
Yan Wang ◽  
Xiao Hong Li ◽  
Xiao Ming Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Continuous Time ◽  
Continuous Fermentation ◽  
Delay System ◽  
Cellular Growth ◽  
Time Delay System ◽  
Fermentation Time ◽  
Continuous Time Delay ◽  
Feed Medium

Based on 1,3-propanediol production from continuous fermentation of glycerol by Klebsiellapneumoniae, the dynamic behaviors of a kinetics system with continuous time-delay are analyzed. At first, a six-dimension kinetics system concerning the intracellular substance is proposed by introducing weak nuclear continuous time-delay into the specific cellular growth rate. Second, taking the inverse of average time-delay as parameter, the local stability of the positive equilibriums and the existence of Hopf bifurcation are discussed. Finally, the period solution of biomass and the parameter region of Hopf bifurcation are pictured numerically taking some special number in substrate concentration in feed medium and dilution rate.

FOLD–HOPF BIFURCATION ANALYSIS FOR A COUPLED FITZHUGH–NAGUMO NEURAL SYSTEM WITH TIME DELAY

2010 ◽  
Vol 20 (12) ◽  
pp. 3919-3934 ◽  
Author(s):  
BIN ZHEN ◽  
JIAN XU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Analysis ◽  
Signal Transmission ◽  
Neural System ◽  
Center Manifold Reduction ◽  
Delayed Coupling ◽  
Normal Form Method ◽  
Manifold Reduction ◽  
System With Time Delay

A FitzHugh–Nagumo (FHN) model with delayed coupling is considered. For a critical case when the corresponding characteristic equation has a single zero root and a pair of purely imaginary roots, a complete bifurcation analysis is presented by employing the center manifold reduction and the normal form method. The Fold–Hopf bifurcation diagrams are provided to illustrate the correctness of our theoretical analysis. Whether almost periodic motion and bursting behavior occur in the FHN neural system with delayed coupling depends on the time delay in the signal transmission between the neurons.

scholarly journals Dynamic properties of the coupled Oregonator model with delay

2013 ◽  
Vol 18 (3) ◽  
pp. 377-397
Author(s):  
Xiang Wu ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dynamic Properties ◽  
Hopf Bifurcations ◽  
Center Manifold Reduction ◽  
Multiple Periodic Solutions ◽  
Normal Form Method ◽  
The Stability ◽  
Oregonator Model ◽  
Manifold Reduction

This work explores a coupled Oregonator model. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. We also discussed the Z2 equivariant property and the existence of multiple periodic solutions. Numerical simulations are presented to illustrate the results in Section 5.

Coexistence of Periodic Oscillations Induced by Predator Cannibalism in a Delayed Diffusive Predator–Prey Model

2019 ◽  
Vol 29 (07) ◽  
pp. 1950089
Author(s):  
Daifeng Duan ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Reduction ◽  
Periodic Oscillations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Cannibalism ◽  
Manifold Reduction

This paper is concerned with the effect of predator cannibalism in a delayed diffusive predator–prey system. We aim for the case where the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, leading to Hopf–Hopf bifurcation. An approach of center manifold reduction is applied to derive the normal form for such nonresonant Hopf–Hopf bifurcations. We find that the system exhibits very rich dynamics, including the coexistence of periodic and quasi-periodic oscillations. Numerically, we show that Hopf–Hopf bifurcation is induced if the strength of the predator cannibalism term belongs to an appropriate interval.

STABILITY AND HOPF BIFURCATION OF A DELAYED NETWORK OF FOUR NEURONS WITH A SHORT-CUT CONNECTION

2008 ◽  
Vol 18 (10) ◽  
pp. 3053-3072 ◽  
Author(s):  
XIAOCHEN MAO ◽  
HAIYAN HU
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Center Manifold ◽  
Network Equilibrium ◽  
Stability Switches ◽  
Center Manifold Reduction ◽  
The Stability ◽  
Manifold Reduction ◽  
Theoretical Results ◽  
Good Agreement

This paper reveals the dynamics of a delayed neural network of four neurons, with a short-cut connection through a theoretical analysis and some case studies of both numerical simulations and experiments. It presents a detailed analysis of the stability and the stability switches of the network equilibrium, as well as the Hopf bifurcation and the bifurcating periodic responses on the basis of the normal form and the center manifold reduction. Afterwards, the study focuses on the validation of theoretical results through numerical simulations and circuit experiments. The numerical simulations and the circuit experiments not only show good agreement with theoretical results, but also show abundant effects of the short-cut connection on the network dynamics.

LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL

2004 ◽  
Vol 14 (11) ◽  
pp. 3909-3919 ◽  
Author(s):  
YONGLI SONG ◽  
JUNJIE WEI ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Global Hopf Bifurcation ◽  
Local Hopf Bifurcation ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

In this paper, we consider the following nonlinear differential equation [Formula: see text] We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay τ. Finally, numerical simulation results are given to support the theoretical predictions.

Sign in / Sign up

Export Citation Format

Share Document