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in a cylinder
Constrained stability and instability of polynomial difference
equations with state-dependent noise
We examine the stability and instability of solutions of a
polynomial difference equation with state-dependent Gaussian
perturbations, and describe a phenomenon that can only occur in
discrete time. For a particular set of initial values, we find
that solutions approach equilibrium asymptotically in a highly
regulated fashion: monotonically and bounded above by a
deterministic sequence. We observe this behaviour with a probability
that can be made arbitrarily high by choosing the initial value
sufficiently small.
However, for any fixed initial value, the probability of
instability is nonzero, and in fact we can show that as the
magnitude of the initial value increases, the probability of
instability approaches $1$.