Abstract
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.
Similar content being viewed by others
References
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
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)
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)
Oustaloup, A.: La Commande CRONE: Commande Robuste D’ordre non Entier. Hermes, Oslo (1991)
Anastasio, T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72(1), 69–79 (1994)
Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)
Machado, J.T.: Analysis and design of fractional-order digital control systems. J. Syst. Anal., Model. Simul. 27, 107–122 (1997)
Nigmatullin, R.: The statistics of the fractional moments: Is there any chance to “read quantitatively” any randomness? Signal Process. 86(10), 2529–2547 (2006)
Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of systems with long-range interaction. Commun. Nonlinear Sci. Numer. Simul. 11(8), 885–898 (2006)
Baleanu, D.: About fractional quantization and fractional variational principles. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2520–2523 (2009)
Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999)
Tenreiro Machado, J.: Discrete-time fractional-order controllers. J. Fract. Calc. Appl. Anal. 4, 47–66 (2001)
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)
Tseng, C.C.: Design of fractional order digital fir differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001)
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)
Tenreiro Machado, J., Galhano, A.M.S.: Statistical fractional dynamics. ASME J. Comput. Nonlinear Dyn. 3(2), 021201 (2008)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 623–656 (1948)
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957)
Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover, New York (1957)
Plastino, A., Plastino, A.R.: Tsallis entropy and Jaynes’ information theory formalism. Braz. J. Phys. 29(1), 50–60 (1999)
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)
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)
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)
Carter, T.: An introduction to information theory and entropy. Complex Systems Summer School, Santa Fe (June 2007)
Rathie, P., Da Silva, S.: Shannon, Levy, and Tsallis: a note. Appl. Math. Sci. 2(28), 1359–1363 (2008)
Beck, C.: Generalised information and entropy measures in physics. Contemp. Phys. 50(4), 495–510 (2009)
Gray, R.M.: Entropy and Information Theory. Springer, Berlin (2009)
Al-Alaoui, M.A.: Novel digital integrator and differentiator. Electron. Lett. 29(4), 376–378 (1993)
Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373(30), 2516–2519 (2009)
Tenreiro Machado, J., Galhano, A.M.: Statistical fractional dynamics. ASME J. Comput. Nonlinear Dyn. 3(2), 021201 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9724-4