scholarly journals Reversed S-Shaped Bifurcation Curve for a Neumann Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Hui Xing ◽  
Hongbin Chen ◽  
Ruofei Yao
Keyword(s):  
Degree Theory ◽  
Neumann Boundary ◽  
Topological Degree Theory ◽  
Indefinite Weight ◽  
Three Solutions ◽  
Bifurcation Theorem ◽  
Eigenvalue Comparison ◽  
The Stability ◽  
Exact Multiplicity Of Solutions ◽  
Exact Multiplicity

We study the bifurcation and the exact multiplicity of solutions for a class of Neumann boundary value problem with indefinite weight. We prove that all the solutions obtained form a smooth reversed S-shaped curve by topological degree theory, Crandall-Rabinowitz bifurcation theorem, and the uniform antimaximum principle in terms of eigenvalues. Moreover, we obtain that the equation has exactly either one, two, or three solutions depending on the real parameter. The stability is obtained by the eigenvalue comparison principle.

scholarly journals Multiplicity Results to a Conformable Fractional Differential Equations Involving Integral Boundary Condition

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Shuman Meng ◽  
Yujun Cui
Keyword(s):  
Degree Theory ◽  
Stieltjes Integral ◽  
Topological Degree Theory ◽  
Initial Value ◽  
Three Solutions ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
The Given

In this article, by using topological degree theory couple with the method of lower and upper solutions, we study the existence of at least three solutions to Riemann-Stieltjes integral initial value problem of the type Dαx(t)=f(t,x),  t∈[0,1], x(0)=∫01x(t)dA(t), where Dαx(t) is the standard conformable fractional derivative of order α, 0<α≤1, and f∈C([0,1]×R,R). Simultaneously, the fixed point theorem for set-valued increasing operator is applied when considering the given problem.

scholarly journals Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super–sublinear case

2016 ◽  
Vol 146 (3) ◽  
pp. 449-474 ◽  
Author(s):  
Alberto Boscaggin ◽  
Guglielmo Feltrin ◽  
Fabio Zanolin
Keyword(s):  
Differential Equation ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Positive Periodic Solutions ◽  
Indefinite Weight ◽  
Nonlinear Odes ◽  
Superlinear Growth ◽  
Image Position ◽  
Neumann Boundary Value Problems

We study the periodic and Neumann boundary-value problems associated with the second-order nonlinear differential equationwhere is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions whenand λ > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

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 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 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.

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.

scholarly journals Stability and Bifurcation Analysis for a Class of Generalized Reaction-Diffusion Neural Networks with Time Delay

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Tianshi Lv ◽  
Qintao Gan ◽  
Qikai Zhu
Keyword(s):  
Neural Networks ◽  
Local Field ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Static Neural Networks ◽  
Stability And Bifurcation ◽  
Uniform Steady State ◽  
The Stability

Considering the fact that results for static neural networks are much more scare than results for local field neural networks and our purpose letting the problem researched be more general in many aspects, in this paper, a generalized neural networks model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stability and bifurcation problems for it are investigated under Neumann boundary conditions. First, by discussing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcations is shown. By using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae which determine the direction and stability of bifurcating periodic solutions are acquired. Finally, numerical simulations show the results.

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