Dynamic bifurcations in smooth and non-smooth systems with forcing and delays

Monday, June 4, 2018 - 9:00am - 9:50am
Lind 305
Rachel Kuske (Georgia Institute of Technology)
We contrast the behavior of dynamic bifurcations near smooth and non-smooth fold bifurcations.  Dynamic bifurcation refers to the state transition or tipping that takes place when a parameter slowly varies through a value corresponding to a bifurcation in the static parameter case. Note that these models correspond to non-autonomous systems with multiple time scales.  As is well known for a smooth fold bifurcation, the dynamic bifurcation is delayed relative to the static value, with the delay as a function of the rate that the parameter is changing.  Historically dynamic bifurcations have been studied for unforced systems and white noise forcing, but have received relatively little attention for systems with oscillatory forcing and coefficients, delayed feedback, and non-smooth bifurcations.

We give results for high and low frequency forcing, and contrast the behavior for smooth and non-smooth dynamics near saddle node bifurcations. We also develop new reduction methods for dynamic Hopf bifurcations with forcing, that lead to efficient semi-analytical approaches. These results capture conditions under which the slow time scale, the type of forcing, or a combination of factors determine the tipping behavior. We illustrate these results in simple 1D examples, and then apply to a variety of applications including box models, energy models for arctic sea ice, Morris Lecar-type models and machine tool dynamics.