Skip to main content
SpringerLink
Log in

Fractional Fractals

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

This paper introduces the notion of “fractional fractals”. The main idea is to establish a connection between the classical iterated function system and the first order truncation of the Grndwald-Letnikov fractional derivative. This allows us to consider higher order truncations, and also to study the limit sets for these higher order systems. We prove several results involving the existence and dimension of such limit sets, that will be called “fractional fractals”. Some numerical calculations and representations illustrate relevant examples.

MSC 2010: Primary 26A33, 28A80

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. D. Baleanu, A.M. Lopes, Handbook of Fractional Calculus with Applications. Vol.7: Applications in Engineering, Life and Social Sciences, Part A. De Gruyter, Berlin (2019).

    Book  Google Scholar 

  2. D. Baleanu, A.M. Lopes, Handbook of Fractional Calculus with Applications. Vol. 8: Applications in Engineering, Life and Social Sciences, Part B. De Gruyter, Berlin (2019).

    Book  Google Scholar 

  3. J. Clerc, A.M. Tremblay, G. Albinet, C. Mitescu, A.C. response of fractal networks. J. de Physique Lettres 45, No 19 (1984), 913–924.

    Article  Google Scholar 

  4. G. Devillanova, G.C. Marano, A free fractional viscous oscillator as a forced standard damped vibration. Fract. Calc. Appl. Anal. 19, No 2 (2016), 319–356; DOI: 10.1515/fca-2016-0018; https://www.degruyter.com/view/journals/fca/19/2/fca.19.issue-2.xml.

    Article  MathSciNet  Google Scholar 

  5. E. Deza, M.M. Deza, Encyclopedia of Distances. Springer-Verlag, Heidelberg (2009).

    Book  Google Scholar 

  6. K. Falconer, Fractal Geometry. John Wiley & Sons Limited, Chichester (1990).

    MATH  Google Scholar 

  7. T. Kaplan, L.J. Gray, S.H. Liu, Self-affine fractal model for a metal-electrolyte interface. Physical Review B 35, No 10 (1987), 5379–5381.

    Article  Google Scholar 

  8. G. Karniadakis, Handbook of Fractional Calculus with Applications. Vol. 3: Numerical Methods. De Gruyter, Berlin (2019).

    Google Scholar 

  9. A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Vol. 1: Basic Theory. De Gruyter, Berlin (2019).

    Google Scholar 

  10. A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications, Vol. 2: Fractional Differential Equations. De Gruyter, Berlin (2019).

    Google Scholar 

  11. S.H. Liu, Fractal model for the ac response of a rough interface. Physical Review Letters 55, No 5 (1985), 529–532.

    Article  Google Scholar 

  12. É. Lucas, Le calcul des nombres entiers. Le calcul des nombres rationnels. La divisibilité arithmétique. Gauthier-Villars, Paris (1891).

    MATH  Google Scholar 

  13. J.A.T. Machado, A.M. Lopes, Fractional-order modeling of a diode. Commun. Nonlin. Sci. and Numer. Simul. 70 (2019), 343–353.

    Article  MathSciNet  Google Scholar 

  14. J.T. Machado, Fractional generalization of memristor and higher order elements. Commun. Nonlin. Sci. and Numer. Simul. 18, No 12 (2013), 264–275.

    Article  MathSciNet  Google Scholar 

  15. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010).

    Book  Google Scholar 

  16. P.R. Massopust, A.I. Zayed, On the invalidity of Fourier series expansions of fractional order. Fract. Calc. Appl. Anal. 18, No 6 (2015), 1507–1517; DOI: 10.1515/fca-2015-0087; https://www.degruyter.com/view/journals/fca/18/6/fca.18.issue-6.xml.

    Article  MathSciNet  Google Scholar 

  17. A.L. Méhauté, Fractal Geometries: Theory and Applications. Penton Press, New York (1991).

    MATH  Google Scholar 

  18. R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi (b) 133, No 1 (1986), 425–430.

    Article  Google Scholar 

  19. A. Oustaloup, La Commande CRONE: Commande Robuste d’Ordre Non Entier. Hermes, Paris (1991).

    MATH  Google Scholar 

  20. I. Petráş, Handbook of Fractional Calculus with Applications. Vol. 6: Applications in Control. De Gruyter, Berlin (2019).

    Google Scholar 

  21. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).

    MATH  Google Scholar 

  22. L.E. Sigler, Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Springer Science & Business Media, Berlin (2003).

    Google Scholar 

  23. V. Tarasov, Handbook of Fractional Calculus with Applications. Vol. 4: Applications in Physics, Part A. De Gruyter, Berlin (2019).

    Google Scholar 

  24. V. Tarasov, Handbook of Fractional Calculus with Applications. Vol. 5: Applications in Physics, Part B. De Gruyter, Berlin (2019).

    Google Scholar 

  25. V. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer-Verlag, Heidelberg (2011).

    Google Scholar 

  26. P.J. Torvik, R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials. J. of Applied Mechanics 51, No 2 (1984), 294–298.

    Article  Google Scholar 

  27. V. Uchaikin, R. Sibatov, Fractional Kinetics in Space. World Scientific Publishing, Singapore (2017).

    MATH  Google Scholar 

  28. D. Valério, J. da Costa, An Introduction to Fractional Control. Institution of Engineering and Technology, London (2012).

    Book  Google Scholar 

  29. B.J. West, M. Bologna, P. Grigolini, Physics of Fractal Operators. Springer Science & Business Media, New York (2003).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Electrical Engineering Rua Dr. António Bernardino de Almeida, 431, Institute of Engineering, Polytechnic of Porto, 4249–015, Porto, Portugal

    J. A. Tenreiro Machado

  2. Departamento de Matemática Aplicada I Campus Universitario da Xunqueira, A Xunqueira, S/N, Escola de Enxeñarýa Forestal, Universidade de Vigo, Pontevedra–, 36005, Spain

    Daniel Cao Labora

Authors
  1. J. A. Tenreiro Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Cao Labora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. A. Tenreiro Machado.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tenreiro Machado, J.A., Cao Labora, D. Fractional Fractals. Fract Calc Appl Anal 23, 1329–1348 (2020). https://doi.org/10.1515/fca-2020-0066

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2020-0066

Keywords

Search

Navigation