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Pavlova O. N., Pavlov A. N. Blow­up with complex exponents. Log­periodic oscillations in the democratic fiber bundle model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 2, pp. 3-14. DOI:


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

Pavlova Olga Nikolaevna, Saratov State University
Pavlov Aleksej Nikolaevich, Saratov State University

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.

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1. Mandelbrot B.B. The fractal geometry of nature. San Francisco: W.H. Freeman and company, 1982. 2. Halsey T.C., Jensen M.H., Kadanoff L.P., Procaccia I., Shraiman B.I. Fractal measures and their singularities: the characterization of strange sets // Phys. Rev. A. 1986. Vol. 33. P. 1141. 3. Tel T. Fractals, multifractals, and thermodynamics // Z. Naturforsh. 1988. Vol. 43a. P. 1154. 4. Шредер М. Фракталы, хаос, степенные законы. М., Ижевск: Регулярная и хаотическая динамика, 2001. 5. Mandelbrot B.B. Fractals and multifractals: noise, turbulence and galaxies. New York: Springer-Verlag, 1989. 6. Eisenberg E., Bunde A., Havlin S., Roman H.E. Range of multifractality for random walks on random fractals // Phys. Rev. E. 1993. Vol. 47. P. 2333. 7. Drager J.  ? Multifractal features of random walks and localized vibrational excitations on random fractals: dependence on the averaging procedures // Phys. Rev. E. 1996. Vol. 54. P. 4596. 8. Arneodo A., Decoster N., Roux S.G. Intermittency, log-normal statistics, and multifractal cascade process in high-resolution satellite images of cloud structure // Phys. Rev. Lett. 1999. Vol. 83. P. 1255. 9. Chabra A., Meneveau C., Jensen R.V. Direct determination of the f(?) singularity spectrum and its application to fully developed turbulence // Phys. Rev. A. 1989. Vol. 40. P. 5284. 10. Benzi R., Paladin G., Parisi G., Vulpiani A. On the multifractal nature of fully developed turbulence and chaotic systems // J. Phys. A. 1984. Vol. 17. P. 3521. 11. Strait B.J., Dewey T.G. Multifractals and decoded walks: applications to protein sequence correlations // Phys. Rev. E. 1995. Vol. 52. P. 6588. 12. Muzy J.F., Bacry E., Arneodo A. Wavelets and multifractal formalism for singular signals: Application to turbulence data // Phys. Rev. Lett. 1991. Vol. 67. P. 3515. 13. Muzy J.F., Bacry E., Arneodo A. Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method // Phys. Rev. E. 1993. Vol. 47. P. 875. 14. Frish U., Parisi G. Fully developed turbulence and intermittency // Turbulence and predictability in geophysical fluid dynamics and climate dynamics / Ed. by Ghil M., Benzi R., Parisi G. 1985. P. 71. 15. Gagne Y., Hopfinger E., Frisch U. A new universal scaling for fully developed turbulence: the distribution of velocity increments // New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium / Ed. by Coullet P., Huerre P. 1989. P. 315. 16. Ivanov P.Ch., Nunes Amaral L.A., Goldberger A.L., Havlin S., Rosenblum M.G., Struzik Z.R., Stanley H.E. Multifractality in human heartbeat dynamics // Nature. 1999. Vol. 399. P. 461. 17. Arneodo A., Aubenton-Carafa Y.D., Audit B., Bacry E., Muzy J.F., Thermes C. What can we learn with wavelets about DNA sequences? // Physica A. 1998. Vol. 249. P. 439. 18. Stanley H.E., Nunes Amaral L.A., Goldberger A.L., Havlin S., Ivanov P.Ch., Peng C.-K. Statistical physics and physiology: monofractal and multifractal approaches // Physica A. 1999. Vol. 270. P. 309. 19. Nunes Amaral L.A., Ivanov P.Ch., Aoyagi N., Hidaka I., Tomono S., Goldberger A.L., Stanley H.E., Yamamoto Y. Behavioral-independent features of complex heartbeat dynamics // Phys. Rev. Lett. 2001. Vol. 86. P. 6026. 20. Ivanov P.Ch., Nunes Amaral L.A., Goldberger A.L., Havlin S., Rosenblum M.G., Stanley H.E., Struzik Z.R. From 1/f noise to multifractal cascades in heartbeat dynamics // Chaos. 2001. Vol. 11. P. 641. 21. Marrone A., Polosa A.D., Scioscia G., Stramaglia S., Zenzola A. Multiscale analysis of blood pressure signals // Phys. Rev. E. 1999. Vol. 60. P. 1088. 22. Thurner S., Feurstein M.C., Teich M.C. Multiresolution wavelet analysis of heartbeat intervals discriminates healthy patients from those with cardiac pathology // Phys. Rev. Lett. 1998. Vol. 80. P. 1544. 23. Павлов А.Н., Анищенко В.С. Мультифрактальный анализ сложных сигналов // Успехи физических наук. 2007. Т. 177, вып. 8. C. 859. 24. Pavlov A.N., Ziganshin A.R., Klimova O.A. Multifractal characterization of blood pressure dynamics: stress-induced phenomena // Chaos, Solitons and Fractals. 2005. Vol. 24. P. 57. 25. Daubechies I. Ten lectures on wavelets. Philadelphia: S.I.A.M., 1992. 26. Mallat S.G. A wavelet tour of signal processing. New York: Academic Press, 1998. 27. Addison P.S. The illustrated wavelet transform handbook: Applications in science, engineering, medicine and finance. Bristol, Philadelphia: IOP Publishing, 2002. 28. Kaiser G. A friendly guide to wavelets. Boston: Birkhauser, 1994.  ? 29. Астафьева Н.М. Вейвлет-анализ: основы теории и примеры применения // Успехи физических наук. 1996. T. 166, No 11. С. 1145. 30. Короновский А.А., Храмов А.Е. Непрерывный вейвлетный анализ и его приложения. М.: Физматлит, 2003. 31. Marsh D.J., Sosnovtseva O.V., Pavlov A.N., Yip K.-P., Holstein-Rathlou N.-H. Frequency encoding in renal blood flow regulation // American Journal of Physiology. Regul. Integr. Comp. Physiol. 2005. Vol. 288. P. R1160. 32. Sosnovtseva O.V., Pavlov A.N., Mosekilde E., Holstein-Rathlou N.-H., Marsh D.J. Double-wavelet approach to studying the modulation properties of nonstationary multimode dynamics // Physiological Measurement. 2005. Vol. 26. P. 351. 33. Pavlov A.N., Makarov V.A., Mosekilde E., Sosnovtseva O.V. Application of wavelet-based tools to study the dynamics of biological processes // Briefings in Bioinformatics. 2006. Vol. 7. P. 375. 34. Sosnovtseva O.V., Pavlov A.N., Mosekilde E., Yip K.-P., Holstein-Rathlou N.-H., Marsh D.J. Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats // American Journal of Physiology. Renal Physiology. 2007. Vol. 293. P. F1545. 35. Pavlov A.N., Sosnovtseva O.V., Pavlova O.N., Mosekilde E., Holstein-Rathlou N.- H. Characterizing multimode interaction in renal autoregulation // Physiological Measurement. 2008. Vol. 29. P. 945. 36. Sosnovtseva O.V., Pavlov A.N., Pavlova O.N., Mosekilde E., Holstein-Rathlou N.-H. Characterizing the effect of L-name on intra- and inter-nephron synchronization // European Journal of Pharmaceutical Sciences. 2009. Vol. 36. P. 39. 37. Holstein-Rathlou N.-H., Marsh D.J. A dynamic model of renal blood flow autoregulation // Bull. Math. Biol. 1994. Vol. 56. P. 411. 38. Barfred M., Mosekilde E., Holstein-Rathlou N.-H. Bifurcation analysis of nephron pressure and flow regulation // Chaos. 1996. Vol. 6. P. 280. 39. Mosekilde E. Topics in nonlinear dynamics: applications to physics, biology and economic systems. World Scientific: Singapore, 1996. 40. Holstein-Rathlou N.-H., Leyssac P.P. TGF-mediated oscillations in the proximal intratubular pressure: differences between spontaneously hypertensive rats and Wistar–Kyoto rats // Acta Physiol. Scand. 1986. Vol. 126. P. 333. 41. Leyssac P.P., Holstein-Rathlou N.-H. Tubulo-glomerular feedback response: Enhan-cement in adult spontaneously hypertensive rats and effects of anaesthetics // Pflugers  ? Arch. 1989. Vol. 413. P. 267. 42. Павлов А.Н., Павлова О.Н. Анализ корреляционных свойств случайных процессов по сигналам малой длительности // Письма в ЖТФ. 2008. Т. 34, No 7. С. 71. 43. Pavlov A.N., Sosnovtseva O.V., Mosekilde E., Anishchenko V.S. Extracting dynamics from threshold-crossing interspike intervals: possibilities and limitations // Phys. Rev. E. 2000. Vol. 61, No 5. P. 5033. 44. Postnov D.E., Sosnovtseva O.V., Mosekilde E., Holstein-Rathlou N.-H. Cooperative phase dynamics in coupled nephrons // Int. J. Modern Physics B. 2001. Vol. 15. P. 3079. 45. Павлова О.Н., Павлов А.Н., Анисимов А.А., Назимов А.И., Сосновцева О.В. Синхронизация колебаний в динамике ансамблей корковых нефронов // Известия вузов. Прикладная нелинейная динамика. 2011. Т. 19, No 1. C. 14.

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