Stochastic effects on the angular velocity and translational displacements of spiral waves

We consider stochastic rotating spiral waves that are solutions to two-dimensional reactiondiffusion equations driven by a stochastic noise term. We are especially interested in the effects of this stochastic forcing on the angular velocity of the rotating spiral and the effects on possible translations of the core of the spiral wave. To study these effects, we introduce a reference stochastic rotating spiral wave with a stochastic angular velocity and expand these around their deterministic counterparts. Subsequently, we consider the fluctuations around this reference function. The main ingredients are finding a decomposition of the spiral wave in terms of this reference function and a perturbation, applying Itˆo’s lemma and calculating Taylor expansions. In addition, we introduce numerical methods to find the translations and rotation angle of a spiral wave. Using these methods, we can numerically find the angular velocities of simulations of stochastic rotating spiral waves and...Show moreWe consider stochastic rotating spiral waves that are solutions to two-dimensional reactiondiffusion equations driven by a stochastic noise term. We are especially interested in the effects of this stochastic forcing on the angular velocity of the rotating spiral and the effects on possible translations of the core of the spiral wave. To study these effects, we introduce a reference stochastic rotating spiral wave with a stochastic angular velocity and expand these around their deterministic counterparts. Subsequently, we consider the fluctuations around this reference function. The main ingredients are finding a decomposition of the spiral wave in terms of this reference function and a perturbation, applying Itˆo’s lemma and calculating Taylor expansions. In addition, we introduce numerical methods to find the translations and rotation angle of a spiral wave. Using these methods, we can numerically find the angular velocities of simulations of stochastic rotating spiral waves and compare these to the expansions found for the stochastic angular velocity. To be able to numerically compute the stochastic angular velocity and translations, we use noise that is uncorrelated in time but equal at every point in space and apply the theory to the Barkley model. We find that the difference between the stochastic spiral wave and the stochastic angular velocity and their deterministic counterparts are seen at order ? 2 , where ? ? 0 is the used noise strength. Furthermore, we find that the stochastic forcing does lead to orbital drift in the angular velocity, but not to a translational orbital drift.Show less

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Stochastic effects on the angular velocity and translational displacements of spiral waves