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9
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0002623916
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E. J. Kostelich, I. Kan, C. Grebogi, E. Ott, and J. Yorke, Physica D 109, 81 (1997).
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(1997)
Physica D
, vol.109
, pp. 81
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Kostelich, E.J.1
Kan, I.2
Grebogi, C.3
Ott, E.4
Yorke, J.5
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13
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85035199561
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In previous work, Politi et al., 7 have studied UPOs of a one-dimensional extensively chaotic system with discrete time, consisting of up to six weakly coupled Hénon maps. In contrast, our analysis concerns a continuous-time strongly-coupled system with an emphasis on the spatial structure of the UPOs. Also, Kevrekidis and collaborators [
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In previous work, Politi et al. 7 have studied UPOs of a one-dimensional extensively chaotic system with discrete time, consisting of up to six weakly coupled Hénon maps. In contrast, our analysis concerns a continuous-time strongly-coupled system with an emphasis on the spatial structure of the UPOs. Also, Kevrekidis and collaborators [
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15
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34047228827
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have computed periodic orbits from partial differential equations in a low-fractal-dimension or stable regime and studied bifurcations of these solutions. Our emphasis is again different; we start in a high-fractal-dimension regime where we do not have weakly unstable solutions to apply continuation
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Jolly Physica D 44, 38 (1990)] have computed periodic orbits from partial differential equations in a low-fractal-dimension or stable regime and studied bifurcations of these solutions. Our emphasis is again different; we start in a high-fractal-dimension regime where we do not have weakly unstable solutions to apply continuation.
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(1990)
Physica D
, vol.44
, pp. 38
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Jolly, J.1
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17
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0000871381
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edited by O. Pironneau, Springer-Verlag, New York
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P. Manneville, in Macroscopic Modeling of Turbulent Flows, Vol. 230 of Lecture Notes in Physics, edited by O. Pironneau (Springer-Verlag, New York, 1985), pp. 319–326.
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(1985)
Macroscopic Modeling of Turbulent Flows, Vol. 230 of Lecture Notes in Physics
, pp. 319-326
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Manneville, P.1
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18
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0002099058
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Recently, also studied the KS equation to investigate how a chaotic attractor may be related to the UPOs whose closure defines that attractor. Their emphasis and results differ from ours in that these authors considered a smaller periodic domain for which the chaotic dynamics is low-dimensional (we estimate [formula presented] from their data) and for which the set of UPOs has an explicit symbolic dynamics
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Recently, F. Christiansen, P. Cvitanović and V. Putkaradze Nonlinearity 10, 55 (1997) also studied the KS equation to investigate how a chaotic attractor may be related to the UPOs whose closure defines that attractor. Their emphasis and results differ from ours in that these authors considered a smaller periodic domain for which the chaotic dynamics is low-dimensional (we estimate D<2.1 from their data) and for which the set of UPOs has an explicit symbolic dynamics.
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(1997)
Nonlinearity
, vol.10
, pp. 55
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Christiansen, F.1
Cvitanović, P.2
Putkaradze, V.3
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20
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85035218647
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We used the convergence criterion [formula presented], [formula presented] for the residual and [formula presented] for the correction, where the infinity norm [formula presented] and where the vector [formula presented] is the numerical trajectory evaluated at time [formula presented] that passes through the initial value [formula presented] at time [formula presented]
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We used the convergence criterion ‖U(T)-U0‖∞ <10-3‖U0‖∞ for the residual and ‖δX‖∞<10-3‖X0‖∞ for the correction, where the infinity norm ‖X‖∞=maxi|Xi|and where the vector U(T) is the numerical trajectory evaluated at time t=T that passes through the initial value U0 at time t=0.
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21
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85035227370
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Our inability to find approximate recurrences from numerical output [formula presented] suggests that some of the widely used control methods 4 will not succeed in stabilizing UPOs since they depend on knowing an approximate recurrence as input to the algorithm
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Our inability to find approximate recurrences from numerical output U(t) suggests that some of the widely used control methods 4 will not succeed in stabilizing UPOs since they depend on knowing an approximate recurrence as input to the algorithm.
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22
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85035196077
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UPOs were considered distinct if their time-averaged pattern [formula presented] and variance [formula presented], [formula presented] averaged over one period differed by at least 0.01 and if their periods differed by at least 0.01
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UPOs were considered distinct if their time-averaged pattern m(x)=〈u(t, x)〉 and variance v(x)=〈(u(t, x)-m(x))2〉 averaged over one period differed by at least 0.01 and if their periods differed by at least 0.01.
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