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Volumn 57, Issue 3, 1998, Pages R2511-R2514

Spatially localized unstable periodic orbits of a high-dimensional chaotic system

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CALCULATIONS; CONVERGENCE OF NUMERICAL METHODS; DAMPING; ERRORS; FRACTALS; NUMERICAL ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0032028875     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.57.R2511     Document Type: Article
Times cited : (62)

References (23)
  • 13
    • 85035199561 scopus 로고    scopus 로고
    • 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 [
    • 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 [
  • 15
    • 34047228827 scopus 로고
    • 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
    • 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.
    • (1990) Physica D , vol.44 , pp. 38
    • Jolly, J.1
  • 18
    • 0002099058 scopus 로고    scopus 로고
    • 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
    • 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.
    • (1997) Nonlinearity , vol.10 , pp. 55
    • Christiansen, F.1    Cvitanović, P.2    Putkaradze, V.3
  • 20
    • 85035218647 scopus 로고    scopus 로고
    • 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]
    • 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.
  • 21
    • 85035227370 scopus 로고    scopus 로고
    • 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
    • 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.
  • 22
    • 85035196077 scopus 로고    scopus 로고
    • 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
    • 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.


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.