Stability and Bifurcation in a Delayed Reaction–Diffusion Equation with Dirichlet Boundary Condition

In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov–Schmidt reduction. The existence of Hopf bifurcation at the s...

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Bibliographic Details
Published inJournal of nonlinear science Vol. 26; no. 2; pp. 545 - 580
Main Authors Guo, Shangjiang, Ma, Li
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2016
Subjects
Analysis
Classical Mechanics
Economic Theory/Quantitative Economics/Mathematical Methods
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Stability
Center manifold
Lyapunov–Schmidt reduction
Normal form
34K15
Hopf bifurcation
92B20
Reaction–diffusion
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Summary:In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov–Schmidt reduction. The existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution is derived by analyzing the distribution of the eigenvalues. The direction of Hopf bifurcation and stability of the bifurcating periodic solution are also investigated by means of normal form theory and center manifold reduction. Moreover, we illustrate our general results by applications to the Nicholson’s blowflies models with one- dimensional spatial domain.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-016-9285-x