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

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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

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

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

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. 

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