### Abstract

tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis-

cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region.

Language | English |
---|---|

Pages | 401-430 |

Number of pages | 30 |

Journal | Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis |

Volume | 17 |

Publication status | Published - 2010 |

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### Keywords

- nonlinear stochastic differencial equation
- stability
- instability

### Cite this

*Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis*,

*17*, 401-430.

}

*Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis*, vol. 17, pp. 401-430.

**On the local dynamics of polynomial difference equations with fading stochastic perturbations.** / Appleby, John A.D.; Kelly, C.; Mao, Xuerong; Rodkina, A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the local dynamics of polynomial difference equations with fading stochastic perturbations

AU - Appleby, John A.D.

AU - Kelly, C.

AU - Mao, Xuerong

AU - Rodkina, A.

PY - 2010

Y1 - 2010

N2 - We examine the stability-instability behaviour of a polynomial difference equa-tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis-cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region.

AB - We examine the stability-instability behaviour of a polynomial difference equa-tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis-cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region.

KW - nonlinear stochastic differencial equation

KW - stability

KW - instability

UR - http://dcdis001.watam.org/volumes/abstract_pdf/2010v17/v17n3a-pdf/7.pdf

M3 - Article

VL - 17

SP - 401

EP - 430

JO - Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis

T2 - Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis

JF - Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis

SN - 1201-3390

ER -