Computation of limit cycles and their isochrons: Applications to biology.

Mathematical Biology and Ecology Seminar
Wednesday, April 18, 2012 - 13:05
1 hour (actually 50 minutes)
Skiles 006
 In this talk we will present a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not reached yet the asymptotic state (a limit cycle). That is we extend the computation of the phase resetting curves (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRS). These are very useful tools for the study of synchronization of coupled oscillators. To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and the Lie symmetries approach), which allow to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we design a numerical scheme to compute both the isochrons and the PRSs of a given oscillator. Finally, we will show some examples of the computations we have carried out for some well-known biological models. This is joint work with Toni Guillamon and R. de la Llave