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What is Andronov-Hopf Bifurcation

Chapter 3
Handbook of Research on Systems Biology Applications in Medicine
From a mathematical point of view, the onset of sustained oscillations generally correspond to the passage through an Andronov-Hopf bifurcation point. Obviously, for a critical value of a control parameter (named bifurcation), the system displays damped oscillations and eventually reaches the steady state- stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [Golbeter, A., Nature, 420:238-245, 2002]. This bifurcation is very typical for biological systems.
Published in Chapter:
Mathematical Description of Time Delays in Pathways Cross Talk
S. Nikolov (Institute of Mechanics and Biomechanics, Bulgaria)
Copyright: © 2009 |Pages: 47
DOI: 10.4018/978-1-60566-076-9.ch003
Abstract
In this chapter we investigate how the inclusion of time delay alters the dynamic properties of (a) delayed protein cross talk model, (b) time delay model of RNA silencing (also known as RNA interference), and (c) time delay in ERK and STAT interaction. The consequences of a time delay on the dynamics of those systems are analysed using Hopf’s theorem and Lyapunov-Andronov theory. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations.
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