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Complexity and the Fractional Calculus

Author

Listed:
  • Pensri Pramukkul
  • Adam Svenkeson
  • Paolo Grigolini
  • Mauro Bologna
  • Bruce West

Abstract

We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.

Suggested Citation

  • Pensri Pramukkul & Adam Svenkeson & Paolo Grigolini & Mauro Bologna & Bruce West, 2013. "Complexity and the Fractional Calculus," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-7, April.
    • Handle: RePEc:hin:jnlamp:498789
    DOI: 10.1155/2013/498789
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    Cited by:

    1. Zare, Marzieh & Grigolini, Paolo, 2013. "Criticality and avalanches in neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 55(C), pages 80-94.
    2. Fabio Vanni & David Lambert, 2024. "Aging Renewal Point Processes and Exchangeability of Event Times," Mathematics, MDPI, vol. 12(10), pages 1-27, May.
    3. Grigolini, Paolo, 2015. "Emergence of biological complexity: Criticality, renewal and memory," Chaos, Solitons & Fractals, Elsevier, vol. 81(PB), pages 575-588.
    4. Yu, Shuhong & Zhou, Yunxiu & Du, Tingsong, 2022. "Certain midpoint-type integral inequalities involving twice differentiable generalized convex mappings and applications in fractal domain," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

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