# Mathematical Biology Seminar

### Brian Ryals

Courant Institute of Mathematical Sciences, NYU

#### Dynamics of some models of forced oscillators

#### Wednesday, February 6, 2013, at 3:05pm

LCB 219

Most of the talk will be about chaotic behavior of the periodically kicked van der Pol
system. The aim is to determine the extent to which chaotic behavior occurs as well as
the nature of the chaos. Unlike previous studies, which used continuous forcing, I will
work with instantaneous kicks, for which the geometry is simpler. This study covers a
range of parameters describing nonlinearity, kick sizes, and kick periods. I will show
that horseshoes are abundant whenever the limit cycle is kicked to a specific region of
the phase space, and offer a geometric explanation for the stretch-and-fold behavior
which ensues. I would also like to report on some ongoing work in the last 5-10 minutes
on a system of coupled oscillators similar to the Kuramoto model. Synchronization and
the classification of steady states will be briefly discussed.