Singular Hopf bifurcation in singularly perturbed system with delay

Xin Ai; Yue Zhang; Xin Ai; Yue Zhang

doi:10.3934/era.2025318

Electronic Research Archive

2025, Volume 33, Issue 11: 7198-7215. doi: 10.3934/era.2025318

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Research article

Singular Hopf bifurcation in singularly perturbed system with delay

Xin Ai ,
Yue Zhang ^,

College of Sciences, Northeastern University, Shenyang 110004, China

Received: 27 July 2025 Revised: 08 October 2025 Accepted: 31 October 2025 Published: 27 November 2025

In this paper, we study the singular Hopf bifurcation in a class of singularly perturbed systems with delay. The key condition for the generation of a singular Hopf bifurcation involves the existence of eigenvalues of the singular pure imaginary part. The necessary and sufficient conditions for the generation of a singular Hopf bifurcation are discussed in a class of general nonlinear $ (n, 1) $-fast-slow systems with delay on slow variables $ (n\ge1) $.
- singularly perturbed system,
- delay,
- singular Hopf bifurcation
Citation: Xin Ai, Yue Zhang. Singular Hopf bifurcation in singularly perturbed system with delay[J]. Electronic Research Archive, 2025, 33(11): 7198-7215. doi: 10.3934/era.2025318

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Abstract

In this paper, we study the singular Hopf bifurcation in a class of singularly perturbed systems with delay. The key condition for the generation of a singular Hopf bifurcation involves the existence of eigenvalues of the singular pure imaginary part. The necessary and sufficient conditions for the generation of a singular Hopf bifurcation are discussed in a class of general nonlinear $ (n, 1) $-fast-slow systems with delay on slow variables $ (n\ge1) $.

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