Sensitivity to switching rates in stochastically switched odes

By S. Lawley, Jonathan C. Mattingly, M. Reed. 2013

47 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6013dbb6b52808ffde1436784d99c613f5950ece

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Citations in the corpus

Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder” (2014, background): “The transitions from extinction of species y to extinction of species x when the jump rate parameter t increases is reminiscent of the transition occurring with linear systems analyzed in [6] and [26].”
Some simple but challenging Markov processes (2014, background): “The main probabilistic results of this section are established in [38].”