Cellular pattern formation in circular domains

Palacios, Antonio Lozano; Gunaratne, Gemunu H.; Gorman, M.; Robbins, Kay A.

doi:10.1063/1.166218

Cited by 61 publications

(39 citation statements)

References 25 publications

(31 reference statements)

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“…Using as an example the long period oscillations that are observed in systems with winnerless competition (WLC), we analyze below the ultrasubharmonic synchronization phenomenon. Competition without a winner based on heteroclinic cycles is a robust oscillatory behavior in diverse multiagent systems, such as biological communities (the simplest example is the competition between three species [2]), structured hydrodynamic flows (e.g., competition between different convective patterns of rotating layers of fluid heated from below [3]), Earth magnetic field dynamics [4,5], cellular flame dynamics in porous media [6,7], mode interaction in counterrotating von Karman swirling flow [8], multimode lasers [9,10], etc. In this Letter, we use as an example the competitive activity of neural circuits and their synchronization by weak high-frequency input.…”

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confidence: 99%

Heteroclinic Synchronization: Ultrasubharmonic Locking

2006

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According to the traditional view of synchronization, a weak periodic input is able to lock a nonlinear oscillator at a frequency close to that of the input (1:1 zone). If the forcing increases, it is possible to achieve synchronization at subharmonic bands also. Using a competitive dynamical system we show the inverse phenomenon: with a weak signal the 1:1 zone is narrow, but the synchronization of ultrasubharmonics is dominant. In the system's phase space, there exists a heteroclinic contour in the autonomous regime, which is the image of sequential dynamics. Under the action of a weak periodic forcing, in the vicinity of the contour a stable limit cycle with long period appears. This results in the locking of very low-frequency oscillations with the finite frequency of the forcing. We hypothesize that this phenomenon can be the origin for the synchronization of slow and fast brain rhythms.

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confidence: 99%

Heteroclinic Synchronization: Ultrasubharmonic Locking

2006

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“…In Table 2, we list all the control parameter values at which these ordered state patterns were attained. Experimentally, cellular patterns have been observed in other systems, such as convective fluids [16], combustion flames [21], and nonlinear optical media [7]. Nevertheless, due to the constraint of experimental conditions, only limited cellular pattern morphology was observed [7,16,21].…”

Section: Numerical Aspectsmentioning

confidence: 99%

“…First, circular domain is a natural and important geometry for many physical systems. Extensive experiments on pattern formation have been carried out in circular domains, including surface waves [16,17], Rayleigh-Bénard convections [18,19], Taylor-Couette flow [20], combustion flames [21] and vibrated granular materials [22,23]. The rich physics exhibited in these experiments leads us to believe that controlled pattern formation in binary systems in a circular domain deserves a systematic investigation.…”

Section: Introductionmentioning

confidence: 99%

Geometry and boundary control of pattern formation and competition

Guan

Lai

Wei

2003

Physica D: Nonlinear Phenomena

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This paper presents the effective control of the formation and competition of cellular patterns. Simulation and theoretical analyses are carried out for pattern formation in a confined circular domain. The Cahn-Hilliard equation is solved with the zero-flux boundary condition to describe the phase separation of binary mixtures. A wavelet-based discrete singular convolution algorithm is employed to provide high-precision numerical solutions. By extensive numerical experiments, a set of cellular ordered state patterns are generated. Theoretical analysis is carried out by using the Fourier-Bessel series. Modal decomposition shows that the pattern morphology of an ordered state pattern is dominated by a principal Fourier-Bessel mode, which has the largest Fourier-Bessel decomposition amplitude. Interesting modal competition is also observed. It is found that the formation and competition of cellular patterns are effectively controlled by the confined geometry and boundary condition.

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“…[72][73][74][75]99 These experiments were performed on a circular burner and provided an excellent example of pattern formation obtained from steady-state and Hopf bifurcation in the world of circular Oð2Þ symmetry. The pattern in the flame front, as shown in the figures, is seen by looking down on the flame from a point directly above the center of the burner.…”

Section: Laminar Premixed Flamesmentioning

confidence: 99%

Recent advances in symmetric and network dynamics

Golubitsky

Stewart

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts Rene Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network. Symmetry is a feature of many systems of interest in applied science. Mathematically, a symmetry is a transformation that preserves structure; for example, a square looks unchanged if it is rotated through any multiple of a right angle, or reflected in a diagonal or a line joining the midpoints of oppose edges. These symmetries also appear in mathematical models of real-world systems, and their effect is often extensive. The last thirty to forty years has seen considerable advances in the mathematical understanding of the effects of symmetry on dynamical systems-systems of ordinary differential equations. In general, symmetry leads to pattern formation, via a mechanism called symmetry-breaking. For example, if a system with circular symmetry has a time-periodic state, repeating the same behavior indefinitely, then typically this state is either a standing wave or a rotating wave. This observation applies to the movement of a flexible hosepipe as water passes through it: in the standing wave, the hosepipe moves to and fro like a pendulum; in the rotating wave, it goes round and round with its end describing a circle. This observation also applies to how flame fronts move on a circular burner. We survey some basic mathematical ideas that have ...

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Cellular pattern formation in circular domains

Cited by 61 publications

References 25 publications

Heteroclinic Synchronization: Ultrasubharmonic Locking

Heteroclinic Synchronization: Ultrasubharmonic Locking

Geometry and boundary control of pattern formation and competition

Recent advances in symmetric and network dynamics

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