Turing instability and Hopf bifurcation of a spatially discretized diffusive Brusselator model with zero-flux boundary conditions

In the present paper, a spatially discretized diffusive Brusselator model with zero-flux boundary conditions is considered. Firstly, the global existence and uniqueness of the positive solution are proved. Then the local stability of the unique spatially homogeneous steady state is considered by ana...

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Bibliographic Details
Published in:Nonlinear dynamics 2023, Vol.111 (1), p.713-731
Main Authors: Li, Zunxian, Song, Yongli, Wu, Chufen
Format: Article
Language:English
Subjects:
Automotive Engineering
Bifurcation theory
Boundary conditions
Boundary value problems
Classical Mechanics
Control
Decoupling method
Discretization
Dynamical Systems
Eigenvalues
Engineering
Hopf bifurcation
Mathematical models
Mathematics
Mechanical Engineering
Original Paper
Periodic structures
Simulation
Stability analysis
Steady state
Vibration
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Summary:In the present paper, a spatially discretized diffusive Brusselator model with zero-flux boundary conditions is considered. Firstly, the global existence and uniqueness of the positive solution are proved. Then the local stability of the unique spatially homogeneous steady state is considered by analyzing the relevant eigenvalue problem with the aid of decoupling method. Hence, the occurrence conditions of Turing bifurcation and Hopf bifurcation for the model at this steady state are obtained. Meanwhile, the comparative simulations on the stability regions of the steady state between the spatially discretized diffusive Brusselator model and its counterpart in continuous space are given. Furthermore, the approximate expressions of the bifurcating periodic solutions are derived according to Hopf bifurcation theorem. The bifurcating spatially nonhomogeneous periodic solutions show the formation of a special kind of periodic structures for this model. Finally, numerical simulations are given to demonstrate the theoretical results.
ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-022-07863-z