Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales

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. ...

Resonant Bifurcations

We consider dynamical systems depending on one or more real parameters, and assuming that, for some " critical " value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence – under suitable hypotheses – of a general class of bifurcating solutions in correspondence to this resonance. These bifurcating solutions include, as particular cases, the usual statio...

Toroidal normal forms for bifurcations in retarded functional differential equations I: Multiple Hopf and transcritical/multiple Hopf interaction

For finite-dimensional bifurcation problems, it is well-known that it is possible to compute normal forms which possess nice symmetry properties. Oftentimes, these symmetries may allow for a partial decoupling of the normal form into a so-called “radial” part and an “angular” part. Analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its...

Discretizing Dynamical Systems with Hopf-Hopf Bifurcations

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that Hopf-Hopf bifurcations are O(h)-shifted and turned into double Neimark-Sacker points by general one-step methods of order p. Then we discuss the effect of discretization methods on the emanating Hopf curves. The numerical approximation of the critical eigenvalues is analyzed too. The resul...

Double Hopf Bifurcations