ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Podlazov A. V. Blow-­up with complex exponents. Log-­periodic oscillations in the democratic fiber bundle model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 2, pp. 15-30. DOI: 10.18500/0869-6632-2011-19-2-15-30

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Article
UDC: 
519.6

Blow-­up with complex exponents. Log-­periodic oscillations in the democratic fiber bundle model

Autors: 
Podlazov Andrej Viktorovich, Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)
Abstract: 

The main trend of some blow-up systems is disturbed by log-periodic oscillations infinitely accelerating when approaching the blow-up point. Explanation of such behavior typical e.g. for seismic and economic phenomena could give an insight into the nature of blow-up point rising in this case as the condensation of constant phase points of oscillations. This viewpoint is a particular case of the more general approach that treats not oscillations as a disturbance of the growing trend, but the trend itself as a result of oscillatory process. Log-periodic oscillations indicate about the discrete scale invariance of described phenomenon. One can easily establish the connection of theirs with other its examples, such as considered here self-similar fractals or diffusion in anisotropic quenched random media. However these examples presuppose the presence of discrete levels of organization in the system nontrivial of themselves. We show that log-periodic oscillations arise in the classical democratic fiber bundle model with the strength of bundles generated by means of random number generator of limited depth. In this case possible strength values belong to a periodic set. And the nonlinear model just transforms this periodic input to the log-periodic output. Periodic events are quite worldwide, so one can assume that log-periodicity in other systems originates from a similar transformation. 

Reference: 
  1. Johansen A, Sornette D, Wakita H, Tsunogai U, Newman WI, Saleur H. Discrete scaling in earthquake pre-cursory phenomena: Evidence in the Kobe earthquake, Japan. J. Phys. I (France). 1996;6(10):1391–1402. DOI: 10.1051/jp1:1996143.
  2. Sornette D, Johansen A. Large financial crashes. Physica A. 1997;245(3–4):411–422. DOI: 10.1016/S0378-4371(97)00318-X.
  3. Sornette D. Why Stock Market Crash: Critical Events in Complex Financial Systems. Princeton University Press; 2009. 448 p.
  4. Sornette D, Sammis CG. Complex critical exponents from renormalization group theory of earthquakes: Implications for earthquake predictions. J. Phys. I (France). 1995;5(5):607–619. DOI: 10.1051/jp1:1995154.
  5. Johansen A, Sornette D. Critical crashes. Risk. 1999;12(1):91–94. Available from: https://arxiv.org/abs/cond-mat/9901035.
  6. Johansen A, Sornette D, Ledoit O. Predicting financial crashes using discrete scale invariance. Journal of Risk. 1999;1(4):5–32. Available from: https://arxiv.org/abs/cond-mat/9903321.
  7. Sornette D, Johansen A. Significance of log-periodic precursors to financial crashes. Quantitative Finance. 2001;1(4):452–471. Available from: http://arXiv.org/abs/cond-mat/0106520.
  8. Saleur H, Sammis CG, Sornette D. Discrete scale invariance, complex fractal dimensions and log-periodic fluctuations in seismicity. J. Geophys. Res. 1996;101(B8):17661–17677. DOI: 10.1029/96JB00876.
  9. Ide K, Sornette D. Oscillatory finite-time singularities in finance, population and rupture. Physica A. 2002;307(1–2):63–106. DOI: 10.1016/S0378-4371(01)00585-4.
  10. Sornette D, Ide K. Theory of self-similar oscillatory finite-time singularities in finance, population and rupture. Int. J. Mod. Phys. C. 2002;14(3):267–275. DOI: 10.1142/S0129183103004462.
  11. Basin MA. Differential equations determining the function that describes precatastrophic behavior of a system. Technical Physics Letters. 2006;32(4):338–339. DOI: 10.1134/S1063785006040195.
  12. Samarskiy AA, Galaktionov VA, Kurdyumov SP, Mikhailov AP. Regimes with Peaking in Problems for Quasilinear Parabolic Equations. Moscow: Nauka; 1987. 480 p. (in Russian).
  13. Malinetskii GG, editor. Exacerbation Modes. Evolution of an Idea: The Laws of Co-Evolution of Complex Structures. Collection: «Cybernetics: Unlimited Possibilities and Possible Limitations». Moscow: Nauka; 1998. 255 p. (in Russian).
  14. Malinetskii GG, editor. Regimes with Sharpening: The Evolution of an Idea. Collection of Articles. 2nd edition. Moscow: Fizmatlit; 2006. 312 p. (in Russian).
  15. Andersen JV, Sornette D, Leung KT. Tri-critical behavior in rupture induced by disorder. Phys. Rev. Lett. 1997;78(11):2140–2143. DOI: 10.1103/PhysRevLett.78.2140.
  16. Zhang S, Fan Q, Ding E. Critical processes, Langevin equation and universality. Physics Letters A. 1995;203(2–3):83–87. DOI: 10.1016/0375-9601(95)00397-L.
  17. Ma S. Modern Theory of Critical Phenomena. Moscow: Mir; 1980. 298 p. (in Russian).
  18. Feder E. Fractals. Boston: Springer; 1988. 284 p. DOI: 10.1007/978-1-4899-2124-6.
  19. Sierpinski Carpet [Electronic resource]. Available from: http://en.wikipedia.org/wiki/Sierpinski_carpet.
  20. Stauffer D. New simulations on old biased diffusion. Physica A. 1999;266(1–4):35–41. DOI: 10.1016/S0378-4371(98)00571-8.
  21. Sornette D, Johansen A. A hierarchical model of financial crashes. Physica A. 1998;261(3–4):581–598. DOI: 10.1016/S0378-4371(98)00433-6.
Received: 
08.07.2010
Accepted: 
11.04.2011
Published: 
31.05.2011
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