Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

Ying Su; Junjie Wei; Junping Shi

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Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

机译：时滞和时滞混合依赖的时滞扩散Logistic方程的Hopf分支。

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摘要

A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered. The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation. The direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are derived by the center manifold theory and normal form method. In particular, the global continuation of the Hopf bifurcation branches are investigated with a careful estimate of the bounds and periods of the periodic orbits, and the existence of multiple periodic orbits are shown.

机译：考虑具有时滞和瞬时密度的混合依赖和Dirichlet边界条件的扩散逻辑方程。通过对特征方程进行详细分析，可以获得唯一的正稳态解的稳定性以及该正稳态解的Hopf分支的发生。通过中心流形理论和正态形式方法推导了霍普夫分岔的方向和分岔周期轨道的稳定性。特别是，在仔细估计周期轨道的边界和周期的基础上，研究了Hopf分叉分支的整体连续性，并显示了多个周期轨道的存在。

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来源
《Journal of dynamics and differential equations》 |2012年第4期|897-925|共29页
作者
Ying Su; Junjie Wei; Junping Shi;
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作者单位

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001 Heilongjiang,People's Republic of China,Department of Applied Mathematics, University of Western Ontario, London. ON N6A 5B7, Canada;

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001 Heilongjiang,People's Republic of China;

Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA;

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正文语种 eng
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关键词
reaction-diffusion equation; logistic equation; delayed and instantaneous density dependence; stability; local hopf bifurcation; global hopf bifurcation;

机译：反应扩散方程逻辑方程延迟和瞬时密度依赖性;稳定性;局部Hopf分叉;全局Hopf分叉;

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3. HOPF BIFURCATION AND PATTERN FORMATION IN A DELAYED DIFFUSIVE LOGISTIC MODEL WITH SPATIAL HETEROGENEITY [J] . Shi Qingyan, Shi Junping, Song Yongli Discrete and continuous dynamical systems . 2019,第2期

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4. Imposing a Hopf bifurcation on a model estimated from noisy data from the delayed logistic equation [C] . Giovani G. Rodrigues, Luis A. Aguirre FAC Conference on Analysis and Control of Chaotic Systems . 2012

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5. Differential equations with state-dependent delay: Global Hopf bifurcation and smoothness dependence on parameters. [D] . Hu, Qingwen. 2008

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Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

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