Skip to main content
SpringerLink
Log in

Dynamical process of complex systems and fractional differential equations

  • Research Article
  • Published:
Central European Journal of Physics

Abstract

Behavior of dynamical process of complex systems is investigated. Specifically we analyse two types of ideal complex systems. For analysing the ideal complex systems, we define the response functions describing the internal states to an external force. The internal states are obtained as a relaxation process showing a “power law” distribution, such as scale free behaviors observed in actual measurements. By introducing a hybrid system, the logarithmic time, and double logarithmic time, we show how the “slow relaxation” (SR) process and “super slow relaxation” (SSR) process occur. Regarding the irregular variations of the internal states as an activation process, we calculate the response function to the external force. The behaviors are classified into “power”, “exponential”, and “stretched exponential” type. Finally we construct a fractional differential equation (FDE) describing the time evolution of these complex systems. In our theory, the exponent of the FDE or that of the power law distribution is expressed in terms of the parameters characterizing the structure of the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Ide, A. Baltay, C. C. Beroza, Sciencexpress, 19 May 2011, science 1207020 (2011)

    Google Scholar 

  2. J. C. Bohorquez, S. Gourley, A. R. Dixon, M. Spagat, N. F. Johnson, Nature 462, 911

  3. M. Kamaya M. Itakura, Eng. Fract. Mech. 75, 386 (2009)

    Google Scholar 

  4. H. Hara, S. Okayama, Phys. Rev. B37, 9504 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  5. H. Hara, S. S. Lee, T. Obata, Y. Tamura, Proc. Inst. Stat. Math. 46, 477 (1998)

    Google Scholar 

  6. J. Koyama, The Complex Faulting Process of Earthquakes (Kluwer Academic Pub., Dordrecht, 1997)

    Book  Google Scholar 

  7. J. Koyama, H. Hara, Phys. A46, 1844 (1992)

    ADS  Google Scholar 

  8. J. Koyama, H. Hara, Chaos Soliton. Fract. 3 (1993)

    Google Scholar 

  9. Y. Kawada, H. Nagahama, H. Hara, Tectonophysics 427, 255 (2006)

    Article  ADS  Google Scholar 

  10. Y. Kawada, Ph.D. thesis, Tohoku University (Sendai, JPN, 2006)

  11. H. Hara, Phys. Rev. 31, 4612 (1985)

    Article  ADS  Google Scholar 

  12. H. Hara, Research of Physical Properties 57 (in Japanese), 58 (1991)

    Google Scholar 

  13. M. Koganezawa, H. Hara, Y. Hayakawa, I. Shimada, J. Theor. Biol. 260, 353 (2006)

    Article  Google Scholar 

  14. C.-C. Lo, L. A. Nunes Amaral, S. Havlin, P. Ch. Ivanov, T. Penzel, J. -H. Peter, H. E. Stanley, Europhys. Lett. 57(5), 625 (2002)

    Article  ADS  Google Scholar 

  15. H. Haken, Brain Dynamics An Introduction to Models and Simulations (Springer-Verlag, Berlin, 2008)

    MATH  Google Scholar 

  16. H. Hara, Physics in Neural Networks Model of Memory, In: A. Isihara and M. Wadati (Eds.) New Physical Properties (in Japanese, Kyoritu, Tokyo, 1990) 135

    Google Scholar 

  17. N. Takahashi, K. Kiatamura, N. Matsuno, M. Mayford, M. Kano, N. Matsuki, Y. Ikegaya, Science 335, 353 (2012)

    Article  ADS  Google Scholar 

  18. N. Ikeda, Mod. Phys. Lett. B23, 2073 (2009)

    Article  ADS  Google Scholar 

  19. N. Ikeda, Physica A 389, 3336 (2010)

    Article  ADS  Google Scholar 

  20. W. J. West, M. Bologna, P. Girigolini, Physics of Fractal Operators (Springer-Verlag, New York, 2003)

    Book  Google Scholar 

  21. M. D. Ortigueira, Physics of Fractal Operators (Springer Science+ Business Media R.V. Heidelberg 2011)

    Google Scholar 

  22. D. Baleanu, K. Diethelm, E. Scalas, FRACTIONAL CALCULUS Models and Numerical Methods (World Scientific, Singapore, 2012)

    MATH  Google Scholar 

  23. N. Nasuno, Ph.D. thesis, Iwaki Meisei University (in Japanese, Iwaki, JPN, 2009)

  24. N. Nasuno, N. Shimizu, M. Fukunaga, Trans. Jpn. Soc. Mech. Eng. 76, 116 (2010) (in Japanese)

    Google Scholar 

  25. J. H. He, Therm. Sci. 15, S145 (2011)

    Google Scholar 

  26. J. H. He, Phys. Lett. A 376, 257 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. C. M. Ionescu, R. D. Keyser, IEEE Transact. Biomed. Eng. 56(4), 978 (2009)

    Article  Google Scholar 

  28. N. Heymans, I. Podlubny, Rheol. Acta 45, 765 (2006)

    Article  Google Scholar 

  29. R. R. Nigmatullin, A. Le Méthauté, J. Non-Cryst. Solids 351, 2888 (2005)

    Article  ADS  Google Scholar 

  30. R. R. Nigmatullin, Physica A 363, 282 (2006)

    Article  ADS  Google Scholar 

  31. H. Hara, O. K. Chan, J. Koyama, Phys. Rev. B46, 838 (1992)

    Article  ADS  Google Scholar 

  32. H. Hara, N. Ikeda, M. Furukawa, In: Proc. of 2nd. IFAC on Fractional differentiation and its applications, 304 (Port, Portugal, July 19–21, 2006)

    Google Scholar 

  33. H. Hara, Phys. Rev. B20, 4062 (1979)

    Article  ADS  Google Scholar 

  34. H. Hara, T. Obata, Phys. Rev. B28, 4403 (1983)

    Article  ADS  Google Scholar 

  35. H. Hara, T. Obata, S. J. Lee, Phys. Rev. B37, 476 (1986)

    ADS  MathSciNet  Google Scholar 

  36. I. Suyari, Fundamental Mathematics for Complex Systems Power Law and Tsallis entropy (in Japanese, Makino Shoten, Tokyo, 2010)

    Google Scholar 

  37. C. Tsallis, Introduction to Non extensive Statistical Mechanics (Springer Science+Business Media. LLC, New York, 2009)

    Google Scholar 

  38. I. Podlubny, In: A. Le Méthauté et al.(Eds.), Fractional differentiation and its applications, 3 (2005)

Download references

Author information

Authors and Affiliations

  1. Tsutumidori-Amamiya, 10-21-406, Aoba-ku, Sendai, 981-0914, Japan

    Hiroaki Hara

  2. The Institute of Statistical Mathematics, 10-3, Midorimachi, Tachikawa, Tokyo, 190-8562, Japan

    Yoshiyasu Tamura

Authors
  1. Hiroaki Hara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshiyasu Tamura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroaki Hara.

About this article

Cite this article

Hara, H., Tamura, Y. Dynamical process of complex systems and fractional differential equations. centr.eur.j.phys. 11, 1238–1245 (2013). https://doi.org/10.2478/s11534-013-0224-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11534-013-0224-2

Keywords

Search

Navigation