4th annual Applied Analysis Day
October
2nd, 2020
University of
Ottawa and
Carleton University
This event is held on zoom.
To register, please send an email with subject line "Applied Analysis Day" to
frithjof.lutscher@uottawa.ca. You will be sent the zoom link on October 1st.
Key note speakers:
Mary Pugh (Toronto)
Bard Ermentrout (Pittsburgh)
Program:
12:30 pm,
Mary Pugh: Using Adaptive Time-Steppers to Explore
Stability Domains
Abstract: We've all looked at
stability domains for ODE time-steppers. At the most
basic level, these are found by studying how the
time-stepper handles the ODE x' = sigma x where sigma is
a complex number with negative real part. This leads to
a stability domain that has a continuous boundary. The
underlying analysis generalizes to systems of ODEs if
the linearized system is diagonalizable. In this talk,
I'll discuss an implicit-explicit time-stepping scheme
for which the linearized system is not diagonalizable
and so standard stability theory doesn't apply. I'll
demonstrate that an adaptive time-stepper can be used to
explore the stability domain and I'll give an example of
a system for which the stability domain can have a
discontinuous boundary; a small change in a parameter
can lead to a jump in the stability threshold of the
time-step size. This is joint work with Francis Dawson
(University of Toronto) and Dave Yan.
Abstract: Animals use stereo sampling of odor concentration to localize sources and follow odor trails. We analyze the dynamics of a bilateral model that depends on the simultaneous comparison between odor concentrations detected by left and right sensors. The general model consists of three differential equations for the positions in the plane and the heading. When the odor landscape is an infinite trail, then we reduce the dynamics to a planar system whose dynamics have just two fixed points. Using an integrable approximation (for short sensors) we estimate the basin of attraction. In the case of a radially symmetric landscape, we again can reduce the dynamics to a planar system, but the behavior is considerably richer with multi-stability, isolas, and limit cycles. As in the linear trail case, there is also an underlying integrable system when the sensors are short. In odor landscapes that consist of multiple spots and trail segments, we find periodic and chaotic dynamics and characterize the behavior on trails with gaps and that turn corners.
For questions, please email: frithjof.lutscher@uottawa.ca
