Constrained stability and instability of polynomial differenceequations with state-dependent noise

Cónall Kelly; Alexandra Rodkina

doi:10.3934/dcdsb.2009.11.913

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2009, Volume 11, Issue 4: 913-933. Doi: 10.3934/dcdsb.2009.11.913

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Constrained stability and instability of polynomial difference equations with state-dependent noise

Cónall Kelly^1, and
Alexandra Rodkina^1,

1.
Department of Mathematics, University of the West Indies, Kingston, 7

Received: July 31, 2008

Revised: January 31, 2009

Published: October 2009

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Abstract

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 $1$.

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Mathematics Subject Classification: Primary: 34F05, 60H10; Secondary: 65C05, 93E15.

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