Computation of the Stability Condition for the Hopf Bifurcation of Diffeomorphisms on $\mathbb{R}^2 $

1978 ◽  
Vol 34 (1) ◽  
pp. 167-175 ◽  
Author(s):  
Yieh-Hei Wan
Keyword(s):  
Hopf Bifurcation ◽  
Stability Condition ◽  
The Stability

Stability of the Jeffcott Rotor with Local Rubbing

2011 ◽  
Vol 55-57 ◽  
pp. 933-936
Author(s):  
He Li ◽  
Xiao Zhe Chen ◽  
Bang Chun Wen
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Stability Condition ◽  
Center Manifold ◽  
System Stability ◽  
Double Zero ◽  
Stability And Bifurcation ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Bifurcation Behavior

The stability and bifurcation behavior of Jeffcott rotor with local rubbing are investigated in terms of Hartman-Grobman theorem in this paper. The case with double zero real part of eigenvalues is analyzed by means of the theory of center manifold and n-dimension Hopf bifurcation. Along with discussion for the effects of parameters on system stability and bifurcation behavior, numerical simulation of rotor locus is conducted and the stability condition is derived.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1447
Author(s):  
Jose P. Suárez ◽  
Agustín Trujillo ◽  
Tania Moreno
Keyword(s):  
Data Structure ◽  
Stability Condition ◽  
Open Problem ◽  
Integer Arithmetic ◽  
Exact Number ◽  
Similarity Class ◽  
The Stability ◽  
High Level ◽  
The Cost

Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes.

scholarly journals Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

2019 ◽  
Vol 29 (11) ◽  
pp. 1950144 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Center Manifold Reduction ◽  
Existence Of Positive Solutions ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
The Stability

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify 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.

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