Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise

2019

J Dyn Diff Equat

Self Cite

For stochastic Hindmarsh-Rose equations with additive noises in the study of neurodynamics, the longtime and global pullback dynamics on a twodimensional bounded domain is explored in this work. Using the additive transformation and by the sharp uniform estimates, we proved the pullback absorbing and the pullback asymptotically compact characteristics of the Hindmarsh-Rose random dynamical system in the L 2 Hilbert space. It shows the existence of a random attractor for this random dynamical system.2000 Mathematics Subject Classification. Primary: 35K55, 35Q80, 37L30, 37L55, 37N25 ; Secondary: 35B40, 60H15, 92B20.

Section: Introductionmentioning

confidence: 77%

“…Very recently, we have proved the existence of a random attractor for the stochastic Hindmarsh-Rose equations with multiplicative noise in [23].…”

Section: Introductionmentioning

confidence: 99%

Random Attractor for Stochastic Hindmarsh–Rose Equations with Additive Noise

Phan

2019

J Dyn Diff Equat

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“…Another model is the three-dimensional Hindmarsh-Rose equations [14] (1984) and the diffusive or partly diffusive Hindmarsh-Rose equations recently proposed and studied [9,10,23,24] on topics of regular and chaotic bursting dynamics, global attractors, and random attractors.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and Synchronization of Boundary Coupled FitzHugh-Nagumo Neural Networks

Skrzypek

2020

Preprint

In this work a new mathematical model for complex neural networks is presented by the partly diffusive FitzHugh-Nagumo equations with ensemble boundary coupling. We analyze the dissipative dynamics and boundary coupling dynamics of the solution semiflow with sharp estimates. The exponential synchronization of this kind complex neural networks is proved under the condition that synaptic stimulation signal strength reaches a threshold quantitatively expressed.

“…Very recently, the authors in [30] and [29] proved the existence of global attractors for the diffusive and partly diffusive Hindmarsh-Rose equations as well as the existence of a random attractor for the stochastic Hindmarsh-Rose equations with multiplicative noise.…”

Section: Introductionmentioning

confidence: 99%

Global dynamics of nonautonomous Hindmarsh–Rose equations

Phan

Nonlinear Analysis: Real World Applications

2020

Self Cite

Global dynamics of nonautonomous diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process.