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

Cite this article as:

Podlazov A. V. Blow­up with complex exponents. Log­periodic oscillations in the democratic fiber bundle model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 2, pp. 15-30. DOI:


Blow­up with complex exponents. Log­periodic oscillations in the democratic fiber bundle model

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

The main trend of some blow­up systems is disturbed by log­periodic oscillations infinitely accelerating when approaching the blow­up point. Explanation of such behavior typical e.g. for seismic and economic phenomena could give an insight into the nature of blow­up 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. Log­periodic 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 self­similar 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 log­periodic 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 log­periodic output. Periodic events are quite worldwide, so one can assume that log­periodicity in other systems originates from a similar transformation.

Key words: 

1. Johansen A., Sornette D., Wakita H., Tsunogai U., Newman W.I., Saleur H. Discrete scaling in earthquake pre-cursory phenomena: Evidence in the Kobe earthquake, Japan // J. Phys. I (France). 1996. Vol. 6. P. 1391. 2. Sornette D., Johansen A. Large financial crashes // Physica A. 1997. Vol. 245, No 3–4. P. 411. 3. Сорнетте Д. Как предсказывать крахи финансовых рынков: критические события в комплексных финансовых системах. М.: Интернет-трейдинг, 2003. 400 с. 4. Sornette D., Sammis C.G. Complex critical exponents from renormalization group theory of earthquakes: Implications for earthquake predictions // J. Phys. I (France). 1995. Vol. 5, No 5. P. 607. 5. Johansen A., Sornette D. Critical crashes // Risk. 1999. Vol. 12, No 1. P. 91. 6. Johansen A., Sornette D., Ledoit O. Predicting financial crashes using discrete scale invariance // Journal of Risk. 1999. Vol. 1, No 4. P. 5. 7. Sornette D., Johansen A. Significance of log-periodic precursors to financial crashes // Quantitative Finance. 2001. Vol. 1, No 4. P. 452. 8. Saleur H., Sammis C.G., Sornette D. Discrete scale invariance, complex fractal dimensions and log-periodic fluctuations in seismicity // J. Geophys. Res. 1996. Vol. 101. P. 17661. 9. Ide K., Sornette D. Oscillatory finite-time singularities in finance, population and rupture // Physica A. 2002. Vol. 307, No 1–2. P. 63. mat/0106047 10. Sornette D., Ide K. Theory of self-similar oscillatory finite-time singularities in finance, population and rupture // Int. J. Mod. Phys. C. 2002. Vol. 14, No 3. P. 267. 11. Basin M.A. Differential equations determining the function that describes precatastrophic behavior of a system // Technical Physics Letters. 2006. Vol. 32, No 4. P. 338. 12. Самарский А.А., Галактионов В.А., Курдюмов С.П., Михайлов А.П. Режимы с обострением в задачах для квазилинейных параболических уравнений. М.: Наука, 1987. 480 с. 13. Режимы с обострением. Эволюция идеи: Законы коэволюции сложных структур // Сб.: «Кибернетика: неограниченные возможности и возможные ограничения» / Ред. Г.Г. Малинецкий. М.: Наука, 1998. 255 с. 14. Режимы с обострением: эволюция идеи // Сборник статей. 2-е изд. испр. и доп. / Под ред. Г.Г. Малинецкого. М.: Физматлит, 2006. 312 с. 15. Andersen J.V., Sornette D., Leung K.-T. Tri-critical behavior in rupture induced by disorder // Phys. Rev. Lett. 1997. Vol. 78. P. 2140. 16. Zhang S., Fan Q, Ding E. Critical processes, Langevin equation and universality // Physics Letters A. 1995. Vol. 203. P. 83. 17. Ма Ш. Современная теория критических явлений. М.: Мир, 1980. 298 с. 18. Федер Е. Фракталы. М.: Мир, 1991. 254 с. 19. 20. Stauffer D. New simulations on old biased diffusion// Physica A. 1999. Vol. 266, No 1–4. P. 35. 21. Sornette D., Johansen A. A hierarchical model of financial crashes // Physica A. 1998. Vol. 261, No 3–4. P. 351.

Short text (in English):
(downloads: 12)
Full text:
(downloads: 25)
Умер Дмитрий Иванович Трубецков. Прощание состоится 14 августа в 11.30 в Актовом зале 10 корпуса СГУ.