Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise

Chi Phan

doi:10.3934/dcdsb.2020060

Article Contents

2020, Volume 25, Issue 8: 3233-3256. Doi: 10.3934/dcdsb.2020060

This issue Previous Article Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay Next Article On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures

Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise

Chi Phan

University of South Florida, Department of Mathematics and Statistics, Tampa, FL 33620, USA

Received: July 31, 2019

Revised: September 30, 2019

Early access: February 2020

Published: August 2020

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Abstract

The longtime and global pullback dynamics of stochastic Hind-marsh-Rose equations with multiplicative noise on a three-dimensional bounded domain in neurodynamics is investigated in this work. The existence of a random attractor for this random dynamical system is proved through the exponential transformation and uniform estimates showing the pullback absorbing property and the pullback asymptotically compactness of this cocycle in the $ L^2 $ Hilbert space.

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Mathematics Subject Classification: Primary: 35K55, 35Q80, 37L30, 37L55, 37N25; Secondary: 35B40, 60H15, 92B20.

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Figure 1. Time responses of the membrane potential for various value of the stimulated current: (a) resting state when J = 0, (b) tonic spiking when J = 1.2, (c) regular bursting when J = 2.2, (d) chaotic bursting when J = 3.1, (e) the x-z phase portrait when J = 2.2, (f) the x-z phase portrait when J = 3.1. Source: [25]

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