UNFOLDING A HOPF BIFURCATION IN A LINEAR REACTION-DIFFUSION EQUATION WITH STRONGLY LOCALIZED IMPURITY

Ji Li; Qing Yu; Qian Zhang

doi:10.11948/20240379

2025 Volume 15 Issue 3

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Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(3): 1742-1769

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Ji Li, Qing Yu, Qian Zhang. UNFOLDING A HOPF BIFURCATION IN A LINEAR REACTION-DIFFUSION EQUATION WITH STRONGLY LOCALIZED IMPURITY[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1742-1769. doi: 10.11948/20240379

Citation:

Ji Li, Qing Yu, Qian Zhang. UNFOLDING A HOPF BIFURCATION IN A LINEAR REACTION-DIFFUSION EQUATION WITH STRONGLY LOCALIZED IMPURITY[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1742-1769. doi: 10.11948/20240379

UNFOLDING A HOPF BIFURCATION IN A LINEAR REACTION-DIFFUSION EQUATION WITH STRONGLY LOCALIZED IMPURITY

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Author Bio: Email: liji@hust.edu.cn(J. Li); Email: yuqing@hust.edu.cn(Q. Yu)
Corresponding author: Email: qian_z@hust.edu.cn(Q. Zhang)
Fund Project: The authors were supported by National Natural Science Foundation of China (12171174)

Abstract

Abstract

This paper presents a general framework to derive the weakly nonlinear stability near a Hopf bifurcation in a special class of multi-scale reaction-diffusion equations. The main focus is on how the linearity and nonlinearity of the fast variables in system influence the emergence of the breathing pulses when the slow variables are linear and the bifurcation parameter is around the Hopf bifurcation point. By applying the matching principle to the fast and slow changing quantities and using the singular perturbation theory, we obtain explicit expressions for the stationary pulses. Then, the normal form theory and the center manifold theory are applied to give Hopf normal form expressions. Finally, one of these expressions is verified by the numerical simulation.
- Pinned solution /
- Hopf bifurcation /
- breathing pulse /
- center manifold expansion /
- normal form
MSC: 37L10, 35K57, 35B25, 35B36

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References

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