Generalized q analysis of log-periodicity: applications to critical ruptures.

We introduce a generalization of the q analysis, which provides a nonparametric tool for the description and detection of log-periodic structures associated with discrete scale invariance. We use this generalized q analysis to construct a signature called the (H,q) derivative of discrete scale invariance, which we use to detect the log-periodicity in the cumulative energy release preceding the rupture of five pressure tanks made of composite carbon-matrix material. We investigate the significance level of the spectral Lomb periodogram of the optimal (H,q) derivative. We confirm and strengthen previous parametric results that the cumulative energy release exhibits log-periodicity before rupture. However, our tests to use this method as a scheme for the prediction of the critical value of the stress at rupture are not encouraging.

[1]  A. Johansen,et al.  Evidence of Discrete Scale Invariance in DLA and Time-to-Failure by Canonical Averaging , 1998 .

[2]  D. Sornette.  Discrete scale invariance and complex dimensions , 1997, cond-mat/9707012.

[3]  S. Baliunas,et al.  A Prescription for period analysis of unevenly sampled time series , 1986 .

[4]  Didier Sornette,et al.  Predicting Financial Crashes Using Discrete Scale Invariance , 1999 .

[5]  Modeling the stock market prior to large crashes , 1998, cond-mat/9811066.

[6]  D. Sornette,et al.  The Nasdaq crash of April 2000: Yet another example of log-periodicity in a speculative bubble ending in a crash , 2000 .

[7]  Failure time and microcrack nucleation , 1999, cond-mat/9905161.

[8]  D. Sornette,et al.  SPONTANEOUS GENERATION OF DISCRETE SCALE INVARIANCE IN GROWTH MODELS , 1997 .

[9]  D. Sornette,et al.  Sornette, Leung, and Andersen Reply: , 1998 .

[10]  D. Sornette,et al.  Large financial crashes , 1997, cond-mat/9704127.

[11]  D. Sornette,et al.  Tricritical Behavior in Rupture Induced by Disorder , 1997 .

[12]  Jean-Pierre Eckmann,et al.  Q-ANALYSIS OF FRACTAL SETS , 1997 .

[13]  Sornette,et al.  Dynamics and memory effects in rupture of thermal fuse networks. , 1992, Physical review letters.

[14]  Novel mechanism for discrete scale invariance in sandpile models , 1999, cond-mat/9903402.

[15]  Didier Sornette,et al.  Discrete scale invariance, complex fractal dimensions, and log‐periodic fluctuations in seismicity , 1996 .

[16]  Angel Garcimartín,et al.  Statistical Properties of Fracture Precursors , 1997 .

[17]  S. Ciliberto,et al.  An experimental test of the critical behaviour of fracture precursors , 1998 .

[18]  A. Johansen,et al.  Critical ruptures , 2000 .

[19]  Olivier Ledoit,et al.  CRASHES AS CRITICAL POINTS , 1998, cond-mat/9810071.

[20]  A. Johansen,et al.  Artifactual log‐periodicity in finite size data: Relevance for earthquake aftershocks , 1999, cond-mat/9911421.

[21]  Didier Sornette,et al.  Statistical Significance of Periodicity and Log-Periodicity with Heavy-Tailed Correlated Noise , 2002 .

[22]  D. Sornette,et al.  Significance of log-periodic precursors to financial crashes , 2001, cond-mat/0106520.

[23]  New evidence of earthquake precursory phenomena in the 17 January 1995 Kobe earthquake, Japan , 1999, cond-mat/9911444.

[24]  Ayşe Erzan,et al.  Finite q-differences and the discrete renormalization group , 1997 .

[25]  D. Sornette,et al.  The concept of ‘critical earthquakes’ applied to mine rockbursts with time‐to‐failure analysis , 2000 .

[26]  D. Sornette,et al.  Reexamination of log periodicity observed in the seismic precursors of the 1989 Loma Prieta earthquake , 2000 .

[27]  Sornette,et al.  Statistical model of earthquake foreshocks. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[28]  M. Sahimi,et al.  Scaling Laws for Fracture of Heterogeneous Materials and Rock. , 1996, Physical review letters.