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Entropy analysis of integer and fractional dynamical systems

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This paper investigates the adoption of entropy for analyzing the dynamics of a multiple independent particles system. Several entropy definitions and types of particle dynamics with integer and fractional behavior are studied. The results reveal the adequacy of the entropy concept in the analysis of complex dynamical systems.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    Google Scholar 

  2. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993)

    MATH  Google Scholar 

  3. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1998)

    Google Scholar 

  5. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  6. Oustaloup, A.: La Commande CRONE: Commande Robuste D’ordre non Entier. Hermes, Oslo (1991)

    MATH  Google Scholar 

  7. Anastasio, T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72(1), 69–79 (1994)

    Article  Google Scholar 

  8. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Machado, J.T.: Analysis and design of fractional-order digital control systems. J. Syst. Anal., Model. Simul. 27, 107–122 (1997)

    MATH  Google Scholar 

  10. Nigmatullin, R.: The statistics of the fractional moments: Is there any chance to “read quantitatively” any randomness? Signal Process. 86(10), 2529–2547 (2006)

    Article  MATH  Google Scholar 

  11. Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of systems with long-range interaction. Commun. Nonlinear Sci. Numer. Simul. 11(8), 885–898 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Baleanu, D.: About fractional quantization and fractional variational principles. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2520–2523 (2009)

    Article  Google Scholar 

  13. Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Tenreiro Machado, J.: Discrete-time fractional-order controllers. J. Fract. Calc. Appl. Anal. 4, 47–66 (2001)

    MATH  MathSciNet  Google Scholar 

  15. Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(3), 363–367 (2002)

    Article  MathSciNet  Google Scholar 

  16. Tseng, C.C.: Design of fractional order digital fir differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001)

    Article  Google Scholar 

  17. Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Tenreiro Machado, J., Galhano, A.M.S.: Statistical fractional dynamics. ASME J. Comput. Nonlinear Dyn. 3(2), 021201 (2008)

    Article  Google Scholar 

  19. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)

    MATH  MathSciNet  Google Scholar 

  20. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 623–656 (1948)

    MathSciNet  Google Scholar 

  21. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957)

    Article  MathSciNet  Google Scholar 

  22. Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover, New York (1957)

    MATH  Google Scholar 

  23. Plastino, A., Plastino, A.R.: Tsallis entropy and Jaynes’ information theory formalism. Braz. J. Phys. 29(1), 50–60 (1999)

    MathSciNet  Google Scholar 

  24. Li, X., Essex, C., Davison, M., Hoffmann, K.H., Schulzky, C.: Fractional diffusion, irreversibility and entropy. J. Non-Equilib. Thermodyn. 28(3), 279–291 (2003)

    Article  Google Scholar 

  25. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Boltzmann–Gibbs entropy versus Tsallis entropy: recent contributions to resolving the argument of Einstein concerning “Neither Herr Boltzmann nor Herr Planck has given a definition of W”? Astrophys. Space Sci. 290(3–4), 241–245 (2004)

    Article  MATH  Google Scholar 

  26. Mathai, A.M., Haubold, H.J.: Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy. Phys. A, Stat. Mech. Appl. 375(1), 110–122 (2007)

    Article  MathSciNet  Google Scholar 

  27. Carter, T.: An introduction to information theory and entropy. Complex Systems Summer School, Santa Fe (June 2007)

  28. Rathie, P., Da Silva, S.: Shannon, Levy, and Tsallis: a note. Appl. Math. Sci. 2(28), 1359–1363 (2008)

    MATH  Google Scholar 

  29. Beck, C.: Generalised information and entropy measures in physics. Contemp. Phys. 50(4), 495–510 (2009)

    Article  Google Scholar 

  30. Gray, R.M.: Entropy and Information Theory. Springer, Berlin (2009)

    Google Scholar 

  31. Al-Alaoui, M.A.: Novel digital integrator and differentiator. Electron. Lett. 29(4), 376–378 (1993)

    Article  Google Scholar 

  32. Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373(30), 2516–2519 (2009)

    Article  MathSciNet  Google Scholar 

  33. Tenreiro Machado, J., Galhano, A.M.: Statistical fractional dynamics. ASME J. Comput. Nonlinear Dyn. 3(2), 021201 (2008)

    Article  Google Scholar 

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Tenreiro Machado, J.A. Entropy analysis of integer and fractional dynamical systems. Nonlinear Dyn 62, 371–378 (2010). https://doi.org/10.1007/s11071-010-9724-4

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