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Stability of equilibria of randomly perturbed maps

  • * Corresponding author: Georgi S. Medvedev

    * Corresponding author: Georgi S. Medvedev
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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  • 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.

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

    Citation:

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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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