Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales | Journal of Mathematical Neuroscience 2011 1 9 DOI 2190-8567-1-9 0 The Journal of Mathematical Neuroscience a SpringerOpen Journal RESEARCH Open Access Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales Wenjun Zhang Vivien Kirk James Sneyd Martin Wechselberger Received 31 May 2011 Accepted 23 September 2011 Published online 23 September 2011 2011 Zhang et al. licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models with one common technique being to eliminate certain fast variables using a quasi-steady-state assumption. In this article we show when such a physiological model reduction technique in a slow-fast system is mathematically justified. We provide counterexamples showing that this technique can give erroneous results near the onset of oscillatory behaviour which is practically the region of most importance in a model. In addition we show that the singular limit of the first Lyapunov coefficient of a Hopf bifurcation in a slow-fast system is in general not equal to the first Lyapunov coefficient of the Hopf bifurcation in the corresponding layer problem a seemingly counterintuitive result. Consequently one cannot deduce in general the criticality of a Hopf bifurcation in a slow-fast system from the lower-dimensional layer problem. Keywords Physiological model reduction geometric singular perturbation theory Hopf bifurcation first Lyapunov coefficient quasi-steady-state reduction W Zhang S V Kirk J Sneyd Department of Mathematics University of Auckland Auckland 1142 New Zealand e-mail V Kirk e-mail J Sneyd e-mail M Wechselberger School of Mathematics and Statistics University of
