STUDIES OF SCALE INVARIANT CHANGE-OVER DYNAMICS IN THE HIERARCHICAL MODEL OF DEFECTS DEVELOPMENT
Cite this article as:
Podlazov А. V. STUDIES OF SCALE INVARIANT CHANGE-OVER DYNAMICS IN THE HIERARCHICAL MODEL OF DEFECTS DEVELOPMENT. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, iss. 3, pp. 3-16. DOI: https://doi.org/10.18500/0869-6632-2012-20-3-3-16
Hierarchical model of defect development makes possible the consideration of both ordinary and self-organized criticality from the common viewpoint. Scale invariant critical state in this model is presented by fixed points of a renormalization transformation, connected with lifting to the next level of hierarchy. So stable fixed points of the transformation correspond to the self-organized criticality and unstable points correspond to the ordinary one. We supplement the renormalizational approach to the critical state with the dynamical one, which is more usual to the theory of self-organized criticality. We show that individual disturbances at the lowest level of hierarchical system result in the power-law distributed response. We investigate the dependence of distribution indices on the model parameters.
1. Ма Ш. Современная теория критических явлений. М.: Мир, 1980. 298 с.
2. Bak P., Tang C., Wiesenfeld K. Self-organized criticality// Phys. Rev. A. 1988. Vol. 38, No 1. P. 364.
3. Bak P. How nature works: The science of self-organized criticality. Springer-Verlag, New York, Inc. 1996.
4. Dhar D., Ramaswamy R. Exactly solved model of self-organized critical phenomena// Phys. Rev. Lett. 1989. Vol. 63, No 16. P. 1659.
5. Paczuski M, Maslov S., Bak P. Avalanche dynamics in evolution, growth, and depinning models// Phys. Rev. E. 1996. Vol. 53, No 1. P. 414.
6. Подлазов А.В., Осокин А.Р. Самоорганизованная критичность эруптивных процессов в солнечной плазме// Математическое моделирование. 2002. Т. 14, No 2. C. 118.
7. Подлазов А.В. Теория самоорганизованной критичности – наука о сложности// Будущее прикладной математики. Лекции для молодых исследователей/ Под. ред. Г.Г. Малинецкого. М.: Эдиториал УРСС, 2005. С. 404.
8. Наркунская Г.С., Шнирман М.Г. Иерархическая модель дефектообразования и сейсмичность// Теория и алгоритмы интерпретации геофизических данных. М.: Наука, 1989; Выч. сейсмология: Вып. 22. С. 56.
9. Narkunskaya G.S., Shnirman M.G. Hierarchical model of defect development and seismicity// Phys. Earth Planet. Inter. 1990. Vol. 61. P. 29.
10. Shnirman M.G., Blanter E.M. Mixed hierarchical model of seismicity: Scaling and prediction// Phys. Earth Planet. Inter. 1999. Vol. 111. P. 295.
11. Blanter E.M., Shnirman M.G. Self-organized in a hierarchical model of defects development// Phys. Rev. E. 1996. Vol. 53, No 4. P. 3408.
12. Blanter E.M., Shnirman M.G. Simple hierarchical systems: Stability, self-organized criticality, and catastrophic behavior // Phys. Rev. E. 1997. Vol. 55, No 6. P. 6397.
13. Shnirman M.G., Blanter E.M. Scale invariance and invariant scaling in a mixed hierarchical system// Phys. Rev. E. 1999. Vol. 60, No 5. P. 5111.
14. Владимиров В.А., Воробьев Ю.Л. и др. Управление риском. Риск, устойчивое развитие, синергетика. М.: Наука, 2000. 432 с.
BibTeX
author = {А. V Podlazov},
title = {STUDIES OF SCALE INVARIANT CHANGE-OVER DYNAMICS IN THE HIERARCHICAL MODEL OF DEFECTS DEVELOPMENT},
year = {2012},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {20},number = {3},
url = {https://old-andjournal.sgu.ru/en/articles/studies-of-scale-invariant-change-over-dynamics-in-the-hierarchical-model-of-defects},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2012-20-3-3-16},pages = {3--16},issn = {0869-6632},
keywords = {Self-organized criticality,scale invariance,renormalization,power laws.},
abstract = {Hierarchical model of defect development makes possible the consideration of both ordinary and self-organized criticality from the common viewpoint. Scale invariant critical state in this model is presented by fixed points of a renormalization transformation, connected with lifting to the next level of hierarchy. So stable fixed points of the transformation correspond to the self-organized criticality and unstable points correspond to the ordinary one. We supplement the renormalizational approach to the critical state with the dynamical one, which is more usual to the theory of self-organized criticality. We show that individual disturbances at the lowest level of hierarchical system result in the power-law distributed response. We investigate the dependence of distribution indices on the model parameters. }}