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 $2D_1\stackrel{{\alpha }_4}{\to }{D}_2$ and $2D_1+D_2\stackrel{{\alpha }_5}{\to }3D_1$ in system behavior. These bifurcations generated oscillations, bi-stability, and excitability, behaviors essential for understanding and controlling complex chemical and biological systems.