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Taghvafard, H., Jardón-Kojakhmetov, H., Szmolyan, P., & Cao, M. (2021). Geometric analysis of oscillations in the Frzilator model. Journal of Mathematical Analysis and Applications, 495(1), Article 124725. https://doi.org/10.1016/j.jmaa.2020.124725
A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a ...
A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a strongly attracting limit cycle. This cycle corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model. In addition, we give a detailed description of the structure of the limit cycle, and the timescales along it.