arXiv 2201.012002022-01math.AP

The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system

Yehu Lv

备注：: arXiv admin note: substantial text overlap with arXiv:2112.14105; text overlap with arXiv:2104.00330 by other authors
DOI：: 10.1016/j.chaos.2022.111826

英文摘要：

In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., digestion, gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in the proposed system. The developed algorithm for calculating the normal form of Hopf bifurcation can be used to investigate the direction and stability of Hopf bifurcation. As a real application, we consider a diffusive predator-prey model with ratio-dependent Holling type-3 functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. By using the developed algorithm for calculating the normal form of Hopf bifurcation, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Singular limit in Hopf bifurcation for doubly diffusive convection equations II: bifurcation and stability

A singular perturbation problem from the artificial compressible system to the incompressible system is considered for a doubly diffusive convection when a Hopf bifurcation from the motionless state occurs in the incompressible system. It is proved that the Hopf bifurcation also occurs in the artificial compressible system for small singular perturbation parameter, called the artificial Mach number. The time periodic solution branch of the artificial compressible system is shown to converge to the corresponding bifurcating branch of the incompressible system in the singular limit of vanishing …

2021-03-04math.AP

Singular limit in Hopf bifurcation for doubly diffusive convection equations I: linearized analysis at criticality

A singularly perturbed system for doubly diffusive convection equations, called the artificial compressible system, is considered on a two-dimensional infinite layer for a parameters range where the Hopf bifurcation occurs in the corresponding incompressible system. The spectrum of the linearized operator in a time periodic function space is investigated in detail near the bifurcation point when the singular perturbation parameter is small. The results of this paper are the basis of the study of the nonlinear Hopf bifurcation problem and the singular limit of the time periodic bifurcating solu…

2021-03-04math.AP

Hopf Bifurcation in Structural Population Models

We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely counter-migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration.

2018-07-24math.AP

Hopf bifurcation of a free boundary problem modeling tumor growth with angiogenesis and two time delays

This paper concerns a free boundary problem modeling tumor growth with angiogenesis and two time delays. The two delays represent the time taken for cells to undergo mitosis and modify the rate of cell loss because of apoptosis, respectively. We study the stability of stationary solutions and find that Hopf bifurcation occurs under some conditions, which extends the results of Xu. Furthermore, numerical simulations are performed to investigate the relationship among the rate of angiogenesis, two time delays and Hopf bifurcation.

2021-09-11math.AP

Hopf bifurcations in the full SKT model and where to find them

In this paper, we consider the Shigesada-Kawasaki-Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator-inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies increasing the interspecific competition pressure. The major changes in the bifurcation structure, the type of pitchfork bifurcatio…

2022-02-08math.AP

The Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations

We prove a Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations, which improves a classical result by Crandall and Rabinowitz in the case where basic spaces are Hilbert spaces. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear equations in unbounded domains of $\mathbb{R}^n$.

2020-12-13math.AP

更多相关论文收起更多论文

Hopf bifurcation from fronts in the Cahn-Hilliard equation

We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard equation. The primary front is induced by a moving source term. Models of this form have been used to study a variety of physical phenomena, including pattern formation in chemical deposition and precipitation processes. Technically, we study bifurcation in the presence of essential spectrum. We contribute a simple and direct functional analytic method and determine bifurcation coefficients explicitly. Our approach uses exponential weights to recover Fredholm properties and spectral flow ideas to compute Fredholm ind…

2014-07-09math.AP

Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions

We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic "galloping", or "pulsating", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also he…

2010-10-05math.AP

抛物-椭圆型约化抑制具有级联趋化性的觅食者-剥削者系统中的霍普夫分岔

Parabolic-elliptic reduction suppresses Hopf bifurcation in a forager-exploiter system with cascade taxis

South China University of Technology（华南理工大学）

研究抛物-椭圆型觅食者-剥削者模型，通过建立经典解性质及研究长期行为，发现资源更新率小时解收敛到稳态，大时该系统无霍普夫分岔，与全抛物型情况不同。

2026-07-28math.AP

Bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction

We study the global bifurcation curves of a diffusive logistic equation, when harvesting is orthogonal to the first eigenfunction of the Laplacian, for values of the linear growth up to $λ_2+δ$, examining in detail their behavior as the linear growth rate crosses the first two eigenvalues. We observe some new behavior with regard to earlier works concerning this equation. Namely, the bifurcation curves suffer a transformation at $λ_1$, they are compact above $λ_1$, there are precisely two families of degenerate solutions with Morse index equal to zero, and the whole set of solutions below $λ_2…

2014-07-01math.AP

Hopf bifurcation of viscous shock wave solutions of compressible Navier-Stokes equations and MHD

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier--Stokes (cNS) and magnetohydrodynamics (MHD) equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in $H^s$ for data in $H^s$. We accomplish this by a novel energy estimate carried out in Lagrangian…

2006-12-16math.AP

Equivariant Hopf bifurcation with arbitrary pressure laws

The equivariant Hopf bifurcation dynamics of a class of system of partial differential equations is carefully studied. The connections between the current dynamics and fundamental concepts in hyperbolic conservation laws are explained. The unique approximation property of center manifold reduction function is used in the current work to determine certain parameter in the normal form. The current work generalizes the study of the second author ([J. Yao, $O(2)$-Hopf bifurcation for a model of cellular shock instability, Physica D, 269 (2014), 63-75.]) and supplies a class of examples of $O(2)$ H…

2014-06-30math.APmath.DS

A Hopf-bifurcation theorem for the critically dissipative quasi-geostrophic equation

This paper is devoted to the study of the dynamical behavior of the critically dissipative quasi-geostrophic equation in $\textbf{T}^2$. We prove that this system possesses time-dependent periodic solutions, bifurcating from a smooth steady solution, i.e. a Hopf-Bifurcation theorem.

2014-04-11math.APmath.DS

Computer assisted proof of branches of stationary and periodic solution, and Hopf bifurcations, for dissipative PDEs

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.

2021-05-10math.APmath.DS

A Hopf bifurcation in the planar Navier-Stokes equations

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

2020-09-27math.APmath.DS

A short proof for Hopf bifurcation in Gurtin-MacCamy's population dynamics model

In this paper, we provide a short proof for the Hopf bifurcation theorem in the Gurtin-MacCamy's population dynamics model. Here we use the Crandall and Rabinowitz's approach, based on the implicit function theorem. Compared with previous methods, here we require the age-specific birth rate to be slightly smoother (roughly of bounded variation), but we have a huge gain for the length of the proof.

2021-05-16math.APmath.DS

The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system

The memory-based diffusion systems have wide applications in practice. Hopf bifurcations are observed from such systems. To meet the demand for computing the normal forms of the Hopf bifurcations of such systems, we develop an effective new algorithm where the memory delay is treated as the perturbation parameter. To illustrate the effectiveness of the algorithm, we consider a diffusive predator-prey system with memory-based diffusion and Holling type-II functional response. By employing this newly developed procedure, we investigate the direction and stability of the delay-induced mode-1 and …

2021-04-01math.APmath.DS

具有闵可夫斯基曲率算子和恒产捕获的扩散广义逻辑问题的分支曲线图

Bifurcation Curve Diagrams for a Diffusive Generalized Logistic Problem with Minkowski Curvature Operator and Constant-Yield Harvesting

National Taipei University of Education（台北教育大学）

研究了一维扩散广义逻辑边值问题的正解分支图，证明了在(lambda, ||u||)-平面和(mu, ||u||)-平面上的分支曲线呈C形，并确定了正解的精确多重性。

2026-04-13math.APmath.CA

Banach空间中的霍普夫分支定理

The Hopf bifurcation theorem in Banach spaces

Faculty of Education, Saga University（佐贺大学教育学部）

本文在Banach空间中证明了改进的霍普夫分支定理，无需紧致条件，适用于非紧致领域中的半线性和拟线性偏微分方程。

2026-03-28math-phmath.APmath.MP

Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics

We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay. We also show that the distributed delay can destabilize the entire system. In particularly, it is shown that Hopf bifurcations can occur.

2009-04-16math.APq-bio.CBq-bio.TO