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This article also contains an extensive list of references
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A. T. Winfree, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 487 (1997). This article also contains an extensive list of references.
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Winfree, A.T.1
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For other experiments exhibiting phenomena in vertically vibrated granular layers, see T. Metcalf, J. B. Knight, and H. M. Jaeger, Physica A 236, 202 (1997);
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Physica A
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C. K. K. Lun, S. B. Savage, D. J. Jeffery, and N. Chepurniy, J. Fluid Mech. 140, 223 (1983);
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J. R. de Bruyn, C. Bizon, M. D. Shattuck, D. Goldman, J. B. Swift, and H. L. Swinney, Phys. Rev. Lett. 81, 1421 (1998).
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de Bruyn, J.R.1
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C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Phys. Rev. E 60, 4340 (1999);
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M. D. Shattuck, C. Bizon, J. B. Swift, and H. L. Swinney, Physica A 274, 158 (1999);
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54
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0000159571
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The authors modified the Swift-Hohenberg equation by introducing a second length scale, and this led to patterns besides stripes
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R. Lifshitz and D. M. Petrich, Phys. Rev. Lett. 79, 1261 (1997). The authors modified the Swift-Hohenberg equation by introducing a second length scale, and this led to patterns besides stripes.
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Lifshitz, R.1
Petrich, D.M.2
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M. Golubitsky, I. N. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory (Springer-Verlag, New York, 1988), Vol. II
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M. Golubitsky, I. N. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory (Springer-Verlag, New York, 1988), Vol. II.
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57
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85035266333
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E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991); R. E. O’Malley Jr., Introduction to Singular Perturbations (Academic Press, New York, 1974)
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E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991); R. E. O’Malley Jr., Introduction to Singular Perturbations (Academic Press, New York, 1974).
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58
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85035302093
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We do not have to restrict (Formula presented) to be time periodic. In particular, we can let (Formula presented) be a chaotic sequence or a random sequence. In certain parameter regimes varying (Formula presented) irregularly gives states that intermittently switch between stripe patterns and hexagon patterns [Markus Löcher (private communication)]
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We do not have to restrict (Formula presented) to be time periodic. In particular, we can let (Formula presented) be a chaotic sequence or a random sequence. In certain parameter regimes varying (Formula presented) irregularly gives states that intermittently switch between stripe patterns and hexagon patterns [Markus Löcher (private communication)].
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59
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85035260561
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Since the time index n is a discrete variable, it is incorrect to say that (Formula presented) continuously. However, (Formula presented) varies on a scale (Formula presented) and by appropriate coarse graining on a scale T we can define a function (Formula presented) for a continuous variable (Formula presented) such that (Formula presented) By the statement (Formula presented) in a continuous fashion, we mean that the coarse graining yields a continuous, slowly varying, function (Formula presented) such that (Formula presented) as (Formula presented)
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Since the time index n is a discrete variable, it is incorrect to say that (Formula presented) continuously. However, (Formula presented) varies on a scale (Formula presented) and by appropriate coarse graining on a scale T we can define a function (Formula presented) for a continuous variable (Formula presented) such that (Formula presented) By the statement (Formula presented) in a continuous fashion, we mean that the coarse graining yields a continuous, slowly varying, function (Formula presented) such that (Formula presented) as (Formula presented)
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60
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34848902004
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This issue of Chaos focuses on the subject of coupled map lattices
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See, e.g., K. Kaneko, Chaos 2, 279 (1992). This issue of Chaos focuses on the subject of coupled map lattices.
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(1992)
Chaos
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, pp. 279
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Kaneko, K.1
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61
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0000420222
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Another in between case is provided by coupled ordinary differential equations on a spatial lattice; see, e.g., D. K. Umberger, C. Grebogi, E. Ott, and B. Afeyan, Phys. Rev. A 39, 4835 (1989).
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Phys. Rev. A
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Umberger, D.K.1
Grebogi, C.2
Ott, E.3
Afeyan, B.4
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62
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85035258260
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our numerical solutions of the model [Eqs. (12345678)], a spatial grid is used, but the spacing between the grid points is small compared to the characteristic scale (Formula presented)
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In our numerical solutions of the model [Eqs. (12345678)], a spatial grid is used, but the spacing between the grid points is small compared to the characteristic scale (Formula presented)
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65
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0001548087
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O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, Phys. Rev. Lett. 83, 3190 (1999).
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Lioubashevski, O.1
Hamiel, Y.2
Agnon, A.3
Reches, Z.4
Fineberg, J.5
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