TY - JOUR

T1 - Fractal dimensions and ƒ(α) spectrum of the Hénon attractor

AU - Arneodo, A.

AU - Grasseau, G.

AU - Kostelich, Eric J.

PY - 1987/10/12

Y1 - 1987/10/12

N2 - We measure the generalized fractal dimensions Dq(q≥0) of the Hénon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle the extremely large numbers of iterates needed for the convergence of the algorithms. We study quantitatively the oscillations which appear in the usual linear regressions of the log-log plot and which are inherent in lacunar fractal sets. These oscillations are the cause of previous underestimates of the Renyi dimensions and in fact make accurate dimension estimates an elusive goal. The Legendre transform of the Dq yields the f{hook}(α) spectrum which characterizes the multifractal structure of the attractor. We point out that this spectrum of singularities can be extracted directly from the computed invariant measure, avoiding the log-log regression procedure.

AB - We measure the generalized fractal dimensions Dq(q≥0) of the Hénon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle the extremely large numbers of iterates needed for the convergence of the algorithms. We study quantitatively the oscillations which appear in the usual linear regressions of the log-log plot and which are inherent in lacunar fractal sets. These oscillations are the cause of previous underestimates of the Renyi dimensions and in fact make accurate dimension estimates an elusive goal. The Legendre transform of the Dq yields the f{hook}(α) spectrum which characterizes the multifractal structure of the attractor. We point out that this spectrum of singularities can be extracted directly from the computed invariant measure, avoiding the log-log regression procedure.

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U2 - 10.1016/0375-9601(87)90546-9

DO - 10.1016/0375-9601(87)90546-9

M3 - Article

AN - SCOPUS:0000133395

VL - 124

SP - 426

EP - 432

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

IS - 8

ER -