Synchronization between two Hele-Shaw Cells

2014

Phys. Rev. E

We investigate common-noise-induced phase synchronization between uncoupled identical Hele-Shaw cells exhibiting oscillatory convection. Using the phase description method for oscillatory convection, we demonstrate that the uncoupled systems of oscillatory Hele-Shaw convection can exhibit in-phase synchronization when driven by weak common noise. We derive the Lyapunov exponent determining the relaxation time for the synchronization, and develop a method for obtaining the optimal spatial pattern of the common noise to achieve synchronization. The theoretical results are confirmed by direct numerical simulations.

“…(10) is also well established (see Ref. [32], and also references therein). In general, a stable limit-cycle solution to Eq.…”

Section: B Limit-cycle Solution and Its Floquet Zero Eigenfunctionsmentioning

confidence: 80%

“…The dynamics of the temperature field T (x, y, t) in the Hele-Shaw cell is described by the following dimensionless form (see Ref. [32] and also references therein):…”

Section: A Dimensionless Form Of the Governing Equationsmentioning

confidence: 99%

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Noise-induced synchronization of oscillatory convection and its optimization

2014

Phys. Rev. E

“…For example, experimental investigations on the synchronization of convection flows have been performed in both periodic and chaotic regimes in a pair of thermally coupled rotating baroclinic annulus systems [12,13]. Numerical studies on the synchronization of spatiotemporal chaos have also been conducted in a pair of quasitwo-dimensional channel models [14] and in a pair of Hele-Shaw cells [15].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, as the first step toward theoretical understanding of the synchronization phenomena in fluid systems, we analyze oscillatory thermal convection in the Hele-Shaw cell [15]. The Hele-Shaw cell is a rectangular cavity in which the gap between two vertical walls is much smaller than the other two spatial dimensions.…”

Section: Introductionmentioning

confidence: 99%

Collective phase description of oscillatory convection

Chaos: An Interdisciplinary Journal of Nonlinear Science

2013

We formulate a theory for the collective phase description of oscillatory convection in Hele-Shaw cells. It enables us to describe the dynamics of the oscillatory convection by a single degree of freedom which we call the collective phase. The theory can be considered as a phase reduction method for limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable time-periodic solutions to partial differential equations, representing the oscillatory convection. We derive the phase sensitivity function, which quantifies the phase response of the oscillatory convection to weak perturbations applied at each spatial point, and analyze the phase synchronization between two weakly coupled Hele-Shaw cells exhibiting oscillatory convection on the basis of the derived phase equations.

Phase description of oscillatory convection with a spatially translational mode

Physica D: Nonlinear Phenomena

2015

h i g h l i g h t s• We develop a phase reduction theory for oscillatory convection with a spatial mode. • The theory can be considered as a phase description method for limit-torus solutions.• We derive phase sensitivity functions for spatial and temporal phases of convection.• We can quantify spatiotemporal phase responses of convection to weak perturbations. • We can analyze spatiotemporal phase synchronization between weakly coupled systems. a b s t r a c tWe formulate a theory for the phase description of oscillatory convection in a cylindrical Hele-Shaw cell that is laterally periodic. This system possesses spatial translational symmetry in the lateral direction owing to the cylindrical shape as well as temporal translational symmetry. Oscillatory convection in this system is described by a limit-torus solution that possesses two phase modes; one is a spatial phase and the other is a temporal phase. The spatial and temporal phases indicate the ''position'' and ''oscillation'' of the convection, respectively. The theory developed in this paper can be considered as a phase reduction method for limit-torus solutions in infinite-dimensional dynamical systems, namely, limit-torus solutions to partial differential equations representing oscillatory convection with a spatially translational mode. We derive the phase sensitivity functions for spatial and temporal phases; these functions quantify the phase responses of the oscillatory convection to weak perturbations applied at each spatial point. Using the phase sensitivity functions, we characterize the spatiotemporal phase responses of oscillatory convection to weak spatial stimuli and analyze the spatiotemporal phase synchronization between weakly coupled systems of oscillatory convection.