Bogdanov-Takens bifurcation of a two species chemical reaction system and its stability using parameter identification

Xue Zhang; Jinzhao Wang; Feng Li; Hongwei Li; Muhammad Marwan; Xue Zhang; Jinzhao Wang; Feng Li; Hongwei Li; Muhammad Marwan

doi:10.3934/nhm.2026016

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 349-367. doi: 10.3934/nhm.2026016

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Bogdanov-Takens bifurcation of a two species chemical reaction system and its stability using parameter identification

School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, China

Received: 26 November 2025 Revised: 05 March 2026 Accepted: 06 March 2026 Published: 19 March 2026

In this paper, we mathematically analyzed a chemical reaction-based dynamical model to investigate codimension-two and codimension-three Bogdanov–Takens (BT) bifurcations and their impact on system dynamics. Generalized eigenvectors were first computed using normalization techniques to derive unfolding parameters for codim-two BT bifurcation. A codim-three BT bifurcation case was then obtained, with its unfolding parameters determined with the aid of the seven-step method. A control technique based on anonymous parameters and updated law was used to control oscillations in our considered dynamical system. Analytical results were validated through numerical simulations, highlighting the role of autocatalytic reactions $ 2 D_1 \xrightarrow{\alpha_4} D_2 $ and $ 2 D_1+D_2 \xrightarrow{\alpha_5} 3 D_1 $ in system behavior. These bifurcations generated oscillations, bi-stability, and excitability, behaviors essential for understanding and controlling complex chemical and biological systems.
- chemical reaction system,
- codim-two BT bifurcation,
- codim-three BT bifurcation,
- normalization,
- stability,
- updated law
Citation: Xue Zhang, Jinzhao Wang, Feng Li, Hongwei Li, Muhammad Marwan. Bogdanov-Takens bifurcation of a two species chemical reaction system and its stability using parameter identification[J]. Networks and Heterogeneous Media, 2026, 21(2): 349-367. doi: 10.3934/nhm.2026016

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Abstract

In this paper, we mathematically analyzed a chemical reaction-based dynamical model to investigate codimension-two and codimension-three Bogdanov–Takens (BT) bifurcations and their impact on system dynamics. Generalized eigenvectors were first computed using normalization techniques to derive unfolding parameters for codim-two BT bifurcation. A codim-three BT bifurcation case was then obtained, with its unfolding parameters determined with the aid of the seven-step method. A control technique based on anonymous parameters and updated law was used to control oscillations in our considered dynamical system. Analytical results were validated through numerical simulations, highlighting the role of autocatalytic reactions $ 2 D_1 \xrightarrow{\alpha_4} D_2 $ and $ 2 D_1+D_2 \xrightarrow{\alpha_5} 3 D_1 $ in system behavior. These bifurcations generated oscillations, bi-stability, and excitability, behaviors essential for understanding and controlling complex chemical and biological systems.

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