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

Pages: 913 - 933, Volume 11, Issue 4, June 2009

doi:10.3934/dcdsb.2009.11.913       Abstract        Full Text (235.1K)       Related Articles

Cónall Kelly - Department of Mathematics, University of the West Indies, Kingston, 7, Jamaica (email)
Alexandra Rodkina - Department of Mathematics, University of the West Indies, Kingston, 7, Jamaica (email)

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 $1$.

Keywords:  Stochastic difference equations, a.s. asymptotic stability, local stability, instability, state dependent perturbations.
Mathematics Subject Classification:  Primary: 34F05, 60H10; Secondary: 65C05, 93E15.

Received: August 2008;      Revised: February 2009;      Published: April 2009.