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NCTS Webinar on Nonlinear Evolutionary Dynamics
13:30 - 15:00, May 15, 2024 (Wednesday)
Cisco Webex, Online seminar
(線上演講 Cisco Webex)
Andronov-Hopf Bifurcations in Mass-Action Systems
Balázs Boros (University of Vienna)

Mass-action dynamical systems are probably the most common mathematical models in biochemistry, cell biology, and population dynamics. Further, they represent a large class of polynomial dynamical systems that are very important both theoretically and from the point of view of applications.
Since oscillatory phenomena are ubiquitous in nature, it is desirable to develop a theory that ensures the existence of periodic solutions in mass-action differential equations. Probably the most powerful tool for showing the existence of an oscillation in a parametrized family of differential equations is the concept of bifurcations of equilibria, and in particular, Andronov-Hopf bifurcation.
This talk presents the list of the smallest bimolecular mass-action systems that admit an Andronov-Hopf bifurcation. The analysis of these networks allows us to fully classify which networks admit a supercritical, a subcritical, or a degenerate Andronov-Hopf bifurcation, and thereby we get information about the stability of the emerging limit cycles. A deeper study of the degenerate cases led us to a classification of the smallest bimolecular mass-action systems that admit a Bautin bifurcation, a codimension-two bifurcation which produces two small limit cycles. In case the outer limit cycle is asymptotically stable, we obtain a bistable system, a property that is important in applications. We also comment on the computational aspects.
Finally, we discuss how the investigation of small, simplified (and therefore unrealistic) models of reaction networks can contribute to the understanding of the dynamical behaviour of larger, more realistic models. Namely, we give a brief introduction to the theory of inheritance of equilibria, periodic orbits, and bifurcations in mass-action systems.
Meeting number (access code): 2516 342 2216
Meeting password: nQh6qPVJm36 
Organizers: Chueh-Hsin Chang (CCU), Jia-Yuan Dai (NCHU), Bo-Chih Huang (CCU), Chih-Chiang Huang (CCU), Alejandro López-Nieto (NTU), Chang-Hong Wu (NYCU)


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