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Sensitivity to switching rates in stochastically switched ODEs
Abstract
We consider a stochastic process driven by a linear ordinary differential equation
whose right-hand side switches at exponential times between a collection of different
matrices. We construct planar examples that switch between two matrices where the
individual matrices and the average of the two matrices are all Hurwitz (all eigenvalues
have strictly negative real part), but nonetheless the process goes to infinity at
large time for certain values of the switching rate. We further construct examples
in higher dimensions where again the two individual matrices and their averages are
all Hurwitz, but the process has arbitrarily many transitions between going to zero
and going to infinity at large time as the switching rate varies. In order to construct
these examples, we first prove in general that if each of the individual matrices
is Hurwitz, then the process goes to zero at large time for sufficiently slow switching
rate and if the average matrix is Hurwitz, then the process goes to zero at large
time for sufficiently fast switching rate. We also give simple conditions that ensure
the process goes to zero at large time for all switching rates. © 2014 International
Press.
Type
Journal articlePermalink
https://hdl.handle.net/10161/9515Published Version (Please cite this version)
10.4310/CMS.2014.v12.n7.a9Publication Info
Lawley, Sean D; Mattingly, Jonathan C; & Reed, Michael C (2014). Sensitivity to switching rates in stochastically switched ODEs. Communications in Mathematical Sciences, 12(7). pp. 1343-1352. 10.4310/CMS.2014.v12.n7.a9. Retrieved from https://hdl.handle.net/10161/9515.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
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Show full item recordScholars@Duke
Jonathan Christopher Mattingly
Kimberly J. Jenkins Distinguished University Professor of New Technologies
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave
elementary and Charlotte Country Day. He graduated from the NC School of Science
and Mathematics and received a BS is Applied Mathematics with a concentration in physics
from Yale University. After two years abroad with a year spent at ENS Lyon studying
nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to
attend Princeton University where he obtained a PhD in Applied and
Michael C. Reed
Arts & Sciences Distinguished Professor of Mathematics
Professor Reed is engaged in a large number of research projects that involve the
application of mathematics to questions in physiology and medicine. He also works
on questions in analysis that are stimulated by biological questions. For recent work
on cell metabolism and public health, go to sites@duke.edu/metabolism.
Since 2003, Professor Reed has worked with Professor Fred Nijhout (Duke Biology) to
use mathematical methods to understan
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