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


For citation:

Podlazov A. V. Self-organized criticality and risk analysis. Izvestiya VUZ. Applied Nonlinear Dynamics, 2001, vol. 9, iss. 1, pp. 49-88. DOI: 10.18500/0869-6632-2001-9-1-49-88

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 0)
Language: 
Russian
Article type: 
Article
UDC: 
519.216: 519.876

Self-organized criticality and risk analysis

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

We analyze the catastrophic behaviour of many complex systems from the point of view of the theory of self—organized criticality. This theory 18 now one of the most rapidly growing branches of the nonlinear science. Main attention is focused on presentation and generalization of conceptions of the theory of self-organized criticality related to the risk management. Among the phenomena considered there are flicker—noise, punctuated equilibrium, power probability distributions and property of integrity.

Key words: 
Acknowledgments: 
The work was supported by the RFBR - grant № 99-01-01091 and РГНФ - grant № 99-03-19696.
Reference: 
  1. Vorobyev YL, Malinetsky GG. Makhutov HA. Theory of risk and security technologies. Approach from the position of nonlinear dynamics. Part 1. Problems of safety in emergency situations. 1998.
  2. Vorobyev YL, Malinetsky GG, Makhutov NA. Risk theory and security technologies. Approach from the position of nonlinear dynamics. Part 1. Problems of safety in emergency situations. 1999.
  3. Vladimirov BA, Vorobyev YL, Salov SS. Risk Management: Risk. Sustainable Development. Synergetics. Moscow: Nauka, 2000. 431 p.
  4. Catastrophes and Society. Moscow: Contact-Culture, 2000. 332 p.
  5. Vorobyov YL, Malinetsky GG, Makhutov HA. Risk Management and Sustainable Development. Human dimension. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):1
  6. Nicolis G, Prigozhin I. Self-Organization in Nonequilibrium Systems. Moscow: Mir, 1979. 512 p.
  7. Zykov VS. Modeling of wave processes in excitable media. Moscow: Nauka, 1984. 166 p.
  8. Obukhov SP. Self-organized criticality: Goldstone modes and their interactions. Phys. Rev. Lett. 1990;65(12):1395–1398. DOI:https://doi.org/10.1103/PhysRevLett.65.1395.
  9. Gould SJ, Eldredge М. Punctuated equilibrium comes оf age. Nature. 1993;366:223–227. DOI: 10.1038/366223a0.
  10. Lowen SB, Teich MC. Fractal renewal processes generate 1/f noise. Phys. Кеу. Е. 1993;47(2):992–1001. DOI:https://doi.org/10.1103/PhysRevE.47.992.
  11. Maslov S, Paczuski M, Bak P. Avalanches and 1/f noise in evolution аnd growth models. Phys. Rev. Lett. 1994;73(16):2162–2165. DOI:10.1103/PhysRevLett.73.2162.
  12. Kanamori BH, Anderson DL. Theoretical basis of some empirical relations in seismology. Bull. Seism. Soc. Am. 1975;65(5):1073–1095.
  13. Golitsyn GS. Earthquakes from the point of view of similarity theory. Moskow: DAN USSR.. 1996;346(4):536–539.
  14. Reduction and predictability оf natural disaster. In: Rundle JB, Тurcotte DL, Klein W. Reduction and predictability of natural disasters. Oxfordshire: Routledge; 1996. 320 p.
  15. Rhodes CJ, Anderson RM. Power laws governing epidemics in isolated populations. Nature. 1996;381:600–602.
  16. Turcotte D. Fractals and Chaos in Geology and Geophysics. Cambridge Univ. Press. 1997;34(5):158.
  17. Mantegna RN, Stanley HE. Scaling behavior in the dynamics of an economic index. Nature. 1995;376:46–49.
  18. Bak P. How nature works: the science of self-organized criticality. New York: SpringerInc; 1996. 205 p.
  19. Lu ET, Hamilton RJ. Avalanches and thе distribution оf solar flares. Astro- physical Journal. 1991;380:1–89.
  20. Lu ET, Hamilton RJ, McTiernan JM, Bromund KR. Solar flares аnd avalanches in driven dissipative systems. Astrophysical Journal. 1993;412:841– 852. DOI:10.1086/172966.
  21. Podlazov AB, Osokin AP. Self-organized critical model of solar flares. In: Math. Computer. Education. 2000, Russia, Moskow. Moskow: Progress-Gradition 2000;7(2):384—392.
  22. Yablonsky AI. Mathematical Models in the Study of Science. Moscow: Nauka, 1986. 352 p.
  23. Feller W. Introduction to Probability Theory and Its Applications. Moscow: Mir, 1967. 752 p.
  24. Harris T. Theory of Winding Processes. Moscow: Mir, 1966. 355 p.
  25. Malinetsky GG, Podlazov AB. The paradigm of self-organized criticality. Hierarchy of models and limits of predictability. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(5):89.
  26. Zolotarev VM. One-dimensional Stable Distributions. Theory of Probabilities and Mathematical Statistics. Moscow: Nauka, 1983. 304 p.
  27. Zolotarev VM. Stable laws and their applications. New in life, science, technology. Mathematics, cybernetics. Moscow: Znanie, 1984;11:64.
  28. Ваk Р, Tang C, Wiesenfeld K. Self-organized criticality. Phys. Кеу. А. 1988;38(1):364–374. DOI: 10.1103/PhysRevA.38.364.
  29. Zhang У-С. Scaling theory оf self-organized criticality. Phys. Rev. Lett. 1989;63(5):470–473. DOI: 10.1103/PhysRevLett.63.470.
  30. Majumdar SN, Dhar О. Height correlations in the Abelian sandpile mode. J. Phys. А: Math. Gen. 1991;24:357. DOI: 10.1088/0305-4470/24/7/008.
  31. Ma Sh. Modern Theory of Critical Phenomena. Moscow: Mir, 1980. 298 p.
  32. Sornette D, Johansen А, Dornic I. Mapping self-organized criticality onto criticality. J. Phys. I (France). 1995;5:325–335.
  33. Clar S, Drossel B, Schwabl Е. Forest fires and other examples оf self-organized criticality. J. Phys. Cond. Mat. 1996;8:6803. DOI: 10.1088/0953-8984/8/37/004.
  34. Dhar, Ramaswamy R. Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett. 1989;63(16):1659–1662.  DOI: 10.1103/PhysRevLett.63.1659.
  35. Bak Р, Chen K. Aggregate fluctuations from independent sectoral shocks: self-organized criticality in а model of production and inventory dynamics. Ricerche Economiche. 1993;47:3–30. DOI: 10.1016/0035-5054(93)90023-V.
  36. Feder HJS, Feder J. Self-organized criticality in а stick-slip process. Phys. Rev. Lett. 1991;66(20):2669–2672. DOI: 10.1103/PhysRevLett.66.2669.
  37. Manna SS. Critical exponents оf the sand pile models in two dimensions. Physica A. 1991;179(2):249–268. DOI: 10.1016/0378-4371(91)90063-I.
  38. Olami Z, Feder HJS, Christensen K. Self-organized criticality in а continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett. 1992;68(8):1244–1247.
  39. Christensen K, Olami Z. Scaling, phase transition, and nonuniversality in a self-organized critical cellularautomaton model. Phys. Rev. А. 1992;46(4):1829. DOI: 10.1103/PhysRevA.46.1829.
  40. Grassberger P. Efficient large-scale simulations of a uniformly driven system. Phys. Rev. Е. 1994:49(3):2436–2444. DOI: 10.1103/PhysRevE.49.2436.
  41. Bottani S, Delamorte В. Self-organized criticality and synchronization in pulse coupled relaxation oscillator systems: the Olami, Feder and Christensen model and the Feder and Feder model. Physica D. 1997;103(1-4):430–441. DOI: 10.1016/S0167-2789(96)00275-8.
  42. Klein W, Rundel J. Comment оn «Self-organized criticality in а continuous, nonconservative cellular automaton modeling earthquakes». Phys. Rev. Lett. 1993;71(8):1288. DOI: 10.1103/PhysRevLett.71.1288.
  43. Carlson JM, Langer JS. Properties of earthquake generated by fault dynamics. Phys. Rev. Lett. 1989;62(22):2632–2635. DOI: 10.1103/PhysRevLett.62.2632.
  44. Carlson JM, Langer JS. Mechanical model оf аn earthquake fault. Phys. Кеу. А. 1989;40(11):6470–6484. DOI: 10.1103/PhysRevA.40.6470.
  45. dе Sousa Vieira M. Self-organized criticality ш а deterministic mechanical model. Phys. Rev. А. 1992;46(10):6288–6293. DOI: 10.1103/PhysRevA.46.6288.
  46. Liu WS, Lu YN, Ding EJ. Dynamical phase transition and self-organized criticality in а theoretical spring-block model. Phys. Rev. Е. 1995;51(3):1916–1928. DOI: 10.1103/physreve.51.1916.
  47. Held GA, Solina DHII, Keane DT, Haag WJ, Horn P.M., Grinstein С. Experimental study оf critical-mass fluctuations in ап evolving sandpile. Phys. Rev. Lett. 1990;65(9):1120. DOI: 10.1103/PhysRevLett.65.2066.
  48. Jaeger НМ, Liu C, Nagel SR. Relaxation аt the angle оf repose. Phys. Rev. Lett. 1989;62(1):40–43. DOI: 10.1103/PhysRevLett.62.40.
  49. Paczuski M, Boettcher S. Universality in sandpiles, interface depinning, and earthquake models. Phys. Rev. Lett. 1996;77(1):111–114. DOI: 10.1103/PhysRevLett.77.111.
  50. Andersen JV, Sornette Р, Leung KT. Tri-critical behavior in rupture induced by disorder. Phys. Rev. Lett. 1997;78:2140. DOI: /10.1103/PhysRevLett.78.2140.
  51. Zhang S, Fan Q, Ding Е. Critical processes, Langevin equation аnd universality. Phys. Lett. А. 1995;203:83–87. DOI: 10.1016/0375-9601(95)00397-L.
  52. Bak P, Chen K, Tang С. A forest-fire model аnd some thoughts оп turbulence. Phys. Lett. А. 1990;147(5-6):297–300. DOI:10.1016/0375-9601(90)90451-S.
  53. Drossel В, Schwabl Е. Self organization in а forest-fire model. Fractals. 1993;1(4):1022–1029. DOI: 10.1142/S0218348X93001118.
  54. Drossel B, Clar S, Schwabl Е. Crossover from percolation to self-organized criticality. Phys. Rev. Е. 1994;50(4):2399–2402. DOI: 10.1103/physreve.50.r2399.
  55. Grassberger Р, Kantz H. On а forest fire model with supposed self-organized criticality. J. оf Stat. Phys. 1991;63(3-4):685–700. DOI: 10.1007/BF01029205.
  56. Drossel В. Self-organized criticality and synchronization in the foresti-fire model. Phys. Rev. Lett. 1996;76(6):936–939. DOI: 10.1103/PhysRevLett.76.936.
  57. Newman MEJ, Sibani P. Extinction, diversity and survivorship of taxa in the fossil record. Proceedings: Biological Sciences. 1999;266(1428):1593–1599.
  58. Newman MEJ. A model оf mass extinction. J. Theor. Biol. 1997;189(3):235– 252. DOI: 10.1006/jtbi.1997.0508.
  59. Roberts BW, Newman MEJ. A model for evolution and extinction. J. Theor. Biol. 1996;180:39–54.
  60. Sole RV, Manrubia SC. Criticality and unpredictability in macroevolution. Phys. Rev. Е. 1997;55(4):4500. DOI: 10.1103/PhysRevE.55.4500.
  61. Sneppen K, Bak Р, Flyvbjerg H, Jensen MH. Evolution аs а self-organized critical phenomena. USA: Proc. Natl. Acad. Sci; 1995;92(11):5209–5213. DOI: 10.1073/pnas.92.11.5209.
  62. Sole RV, Bascompte J. Are critical phenomena relevant 10 large-scale evolution? Proc. R. Soc. London B. 1996;263:161–168.
  63. Sole RV, Alonso D, McKane А. Scaling in а multispecies network model ecosystem. Physica A. 2000;286(1):337–344. DOI: 10.1016/S0378-4371(00)00304-6.
  64. Keitt TH, Stanley HE. Dynamics of North American breeding bird populations. Nature. 1998;393:257–260.
  65. Sole RV, Manrubia SC, Benton M, Bak P. Self-similarity оf extinction statistics in the fossil record. Nature. 1997;388:764–767.
  66. Burlando В. The fractal dimension of taxonomic systems. J. Theor. Biol. 1990;146:99–114.
  67. Burlando В. The fractal geometry оf evolution. J. Theor. Biol. 1993;163:161– 172. DOI: 10.1006/jtbi.1993.1114.
  68. Newman MEJ, Eble GJ. Decline in extinction rates and scale invariance in the fossil record. Cambridge University Press. 1999;25(4):434–439.
  69. Newman MEJ. Self-organized criticality, evolution and the fossil extinction record. Рrос. В. Soc. London В. 1996;263:1605–1610.
  70. Bak Р, Flyvbjerg H, Lautrup В. Coevolution in а rugged fitness landscape. Phys. Rev. А. 1992;46(10):6724–6730. DOI: 10.1103/PhysRevA.46.6724.
  71. Bak Р, Sneppen K. Punctuated equilibrium and criticality ш а simple model оf evolution. Phys. Rev. Lett. 1993;71(24):4083–4086. DOI: 10.1103/PhysRevLett.71.4083.
  72. Paczuski M, Maslov S, Bak P. Avalanche dynamics in evolution, growth, аnd depinning models. Phys. Rev. Е. 1996;53(1):414–443. DOI: 10.1103/PhysRevE.53.414.
  73. Grassberger P. The Bak - Sneppen model for punctuated evolution. Phys. Lett. A. 1995;200(3-4):277–282. DOI: 10.1016/0375-9601(95)00179-7.
  74. Maslov S. Time directed avalanches in invasion models. Phys. Rev. Lett. 1995;74(5):562–565. DOI: 10.1103/PhysRevLett.74.562.
  75. Klafter J, Shlesinger MF, Zumofen G. Beyond Brownian motion. Physics Today. 1996;49(2):33–39. DOI:10.1063/1.881487.
  76. Sole RV, Manrubia SC. Extinction аnd self-organized criticality in а model of large-scale evolution. Phys. Rev. Е. 1996;54(1):R42–R45. DOI:10.1103/PhysRevE.54.R42.
  77. Newman MEJ, Roberts BW. Mass-extinction: Evolution аnd the effects оf external influences оn unfit species. Proc. R. Soc. London В. 1995;260:31. DOI: 10.1098/rspb.1995.0055.
  78. Newman MEJ, Sneppen K. Avalanches, scaling аnd coherent noise. Phys. Rev. Е. 1996;54(6):6226–6231. DOI:https://doi.org/10.1103/PhysRevE.54.6226.
  79. Sneppen K, Newman MEJ. Coherent noise, scale invariance and intermittency in large systems. Physica D. 1997;110:209.
  80. Rubio MA, Edwards CA, Dougherty А, Gollub JP. Self-affine fractal  interface from immiscible displacement in porous media. Phys. Rev. Lett. 1989;63(16):1685–1688. DOI: 10.1103/PhysRevLett.63.1685.
  81. Leschhorn H, Tang LH. Avalanches and correlations in driven interface depinning. Phys. Rev. Е. 1994;49(2):1238. DOI: 10.1103/PhysRevE.49.1238.
  82. Sneppen K. Self-organized criticality and interface growth in а random medium. Phys. Rev. Lett. 1992;69(24):3539–3542. DOI: /10.1103/PhysRevLett.69.3539.
  83. Kim JM, Kosterlitz JM. Growth in restricted solid-on-solid model. Phys. Rev. Lett. 1989;62(19):2289–2292. DOI:10.1103/PhysRevLett.62.2289.
  84. Podlazov AV. Model of surface liberation hecatonheires and soft universality in the theory of self-organized criticality. Izvestiya VUZ. Applied Nonlinear Dynamics. 1999. V.7(6).
  85. Maslov S, Paczuski M. Scaling theory оf depinning in the Sneppen model. Phys. Rev. E. 1994;50(2):R643–R643. DOI: 10.1103/PhysRevE.50.R643.
  86. Sornette D. Linear stochastic dynamics with nonlinear fractal properties. Physica A. 1998;250(1-4):295–314. DOI: 10.1016/S0378-4371(97)00543-8
  87. Sornette D, Cont R. Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J. Phys. France 1. 1997;7:431–444.
  88. Sornerte D. Multiplicative processes and power laws. Phys. Rev. Е. 1998;57:4811. DOI: 10.1103/PhysRevE.57.4811.
Received: 
21.11.2000
Accepted: 
28.02.2001
Published: 
05.06.2001