Stability of equilibria of randomly perturbed maps

PaweŁ Hitczenko; Georgi S. Medvedev

doi:10.3934/dcdsb.2017017

Article Contents

2017, Volume 22, Issue 2: 369-381. Doi: 10.3934/dcdsb.2017017

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Stability of equilibria of randomly perturbed maps

PaweŁ Hitczenko and
Georgi S. Medvedev^,

Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

* Corresponding author: Georgi S. Medvedev
* Corresponding author: Georgi S. Medvedev

Received: March 2016

Revised: September 2016

Published: December 2017

PH was supported by a grant from Simons Foundation grant #208766. GSM was supported by the NSF grant DMS #1412066. GSM participated in a SQuaRe group 'Stochastic stabilisation of limit-cycle dynamics in ecology and neuroscience' sponsored by the American Institute of Mathematics.

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Abstract

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in $\mathbb{R}^d$. This condition can be used to stabilize unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed nonlinear maps in one-and two-dimensional spaces.

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Mathematics Subject Classification: Primary:37H10, 93E15;Secondary:39A30, 39A50, 34F05.

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Figure 1. a) Trajectories of the logistic map $x\mapsto f(x)$ (plotted in dashed line) and that of the randomly perturbed system (1). The former approaches the stable equilibrium of the deterministic system $\bar x_2$, while the latter returns to and remains in a small neighborhood of the origin. b) A trajectory of the two-dimesional system (46) stays near the orgin after brief transients. All trajectories of the underlying deterministic system $x\mapsto Ax+q(x)$ starting off the $x^{(2)}$-axis tend to infinity. In numerical simulations shown in a and b, the following parameter values were used: $\epsilon=0.05$ and $\rho=3$.

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