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The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1795–1803 | Cite as

A fractional approach to the Fermi-Pasta-Ulam problem

  • J. A. T. Machado
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Abstract

This paper studies the Fermi-Pasta-Ulam problem having in mind the generalization provided by Fractional Calculus (FC). The study starts by addressing the classical formulation, based on the standard integer order differential calculus and evaluates the time and frequency responses. A first generalization to be investigated consists in the direct replacement of the springs by fractional elements of the dissipative type. It is observed that the responses settle rapidly and no relevant phenomena occur. A second approach consists of replacing the springs by a blend of energy extracting and energy inserting elements of symmetrical fractional order with amplitude modulated by quadratic terms. The numerical results reveal a response close to chaotic behaviour.

Keywords

Fractional Order European Physical Journal Special Topic Fractional Derivative Fractional Calculus Circuit Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E. Fermi, J. Pasta, S. Ulam, Studies of Nonlinear Problems (Los Alamos report LA-1940 1955), published later in Collected Papers of Enrico Fermi, edited by E. Segré (University of Chicago Press, 1965)Google Scholar
  2. 2.
    J. Ford, Phys. Rep. 213, 271 (1992)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    T.P. Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem (Springer, New York, 1997)Google Scholar
  4. 4.
    M.A. Porter, N.J. Zabusky, B. Hu, D.K. Campbell, American Scientist 97, 214 (2009)CrossRefGoogle Scholar
  5. 5.
    G. Benettin, Chaos 15, 015108 (2004)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    B. Chirikov, F. Izrailev, V. Tayurskij, Comput. Phys. Commun. 5, 11 (1973)ADSCrossRefGoogle Scholar
  8. 8.
    D.J. Korteweg, G. de Vries, Philosophical Mag. 5th Series 36, 422 (1895)CrossRefGoogle Scholar
  9. 9.
    F.M. Izrailev, B.V. Chirikov, Soviet Phys. Dokl. 11, 30 (1966)ADSGoogle Scholar
  10. 10.
    N.J. Zabusky, G.S. Deem, J. Comp. Phys. 2, 126 (1967)ADSCrossRefGoogle Scholar
  11. 11.
    P. Bocchieri, A. Scotti, B. Bearzi, A. Loinger, Phys. Rev. A 2, 2013 (1970)ADSCrossRefGoogle Scholar
  12. 12.
    R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, A. Vulpiani, Phys. Rev. A 31, 1039 (1985)ADSCrossRefGoogle Scholar
  13. 13.
    M. Pettini, M. Landolfi, Phys. Rev. A 41, 768 (1990)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    J. De Luca, A.J. Lichtenberg, M.A. Lieberman, Chaos 5, 283 (1995)ADSCrossRefGoogle Scholar
  15. 15.
    D.L. Shepelyansky, Nonlinearity 10, 1331 (1997)MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. 16.
    L. Casetti, M. Cerruti-Sola, M. Pettini, E.G.D. Cohen, Phys. Rev. E 55, 6566 (1997)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    S. Flach, C.R. Willis, Phys. Rep. 295, 181 (1998)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    G. James, C.R. Acad. Sci. Paris Ser. I Math. 332, 581 (2001)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Dauxois, R. Khomeriki, F. Piazza, S. Ruffo, Chaos 15, 015110-1-11 (2005)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    S. Flach, M.V. Ivanchenko, O.I. Kanakov, Phys. Rev. Lett. 95, 064102-1-4 (2005)ADSGoogle Scholar
  21. 21.
    T. Penati, S. Flach, Chaos 17, 023102-1-16 (2007)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    D. Bambusi, A. Ponno, Comm. Math. Phys. 264, 539 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  23. 23.
    T. Dauxois, M. Peyrard, S. Ruffo, Eur. J. Phys. 26, S3 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York-London, 1974)Google Scholar
  25. 25.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 1993)Google Scholar
  26. 26.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley and Sons, New York, 1993)Google Scholar
  27. 27.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of the Fractional Differential Equations, Math. Studies, Vol. 204 (Elsevier (North-Holland), Amsterdam, 2006)Google Scholar
  28. 28.
    A. Gemant, Physics 7, 311 (1936)ADSCrossRefGoogle Scholar
  29. 29.
    R.L. Bagley, P.J. Torvik, AIAA J. 21, 741 (1983)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010)Google Scholar
  31. 31.
    J.A.T. Machado, J. Syst. Anal. Modell. Simul. 27, 107 (1997)zbMATHGoogle Scholar
  32. 32.
    I. Podlubny, Fractional Diferential Equations (Academic Press, San Diego, 1999)Google Scholar
  33. 33.
    J.A.T. Machado, J. Fract. Calculus Appl. Anal. 4, 47 (2001)zbMATHGoogle Scholar
  34. 34.
    V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011)Google Scholar
  35. 35.
    J.T. Machado, V. Kiryakova, F. Mainardi, Comm. Nonlinear Sci. Numer. Simul. 16, 1140 (2011)MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. 36.
    J.A.T. Machado, Some Notes About the Fermi-Pasta-Ulam Problem (Symposium on Fractional Signals and Systems, Coimbra, Portugal, 2011)Google Scholar
  37. 37.
    D. Raškovič, Teorija elastičnosti (Theory of Elasticity) (Nauna knjiga, 1985)Google Scholar
  38. 38.
    K. Hedrih, Signal Proc. 86, 2678 (2006)CrossRefzbMATHGoogle Scholar
  39. 39.
    L.T. Burton, IEEE Trans. Circuit Theory 16, 406 (1969)ADSCrossRefGoogle Scholar
  40. 40.
    A. Antoniou, IEEE Trans. Circuit Theory 17, 212 (1970)CrossRefGoogle Scholar
  41. 41.
    R. Senani, IEEE Trans. Circ. Syst. 33, 323 (1986)CrossRefGoogle Scholar
  42. 42.
    L.O. Chua, IEEE Trans. Circuit Theory 18, 507 (1971)CrossRefGoogle Scholar
  43. 43.
    L.O. Chua, Appl. Phys. A: Mater. Sci. Proc. 102, 765 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    D. Jeltsema, A. Dòria-Cerezo, Appl. Phys. A: Mater. Sci. Proc. 100, 1928 (2012)Google Scholar
  45. 45.
    J.A.T. Machado, Comm. Nonlinear Sci. Numer. Simul. 18, 264 (2013)CrossRefzbMATHGoogle Scholar
  46. 46.
    R.S. Barbosa, J.A.T. Machado, B.M. Vinagre, A.J. Calderón, J. Vibr. Control 13, 1291 (2007)CrossRefzbMATHGoogle Scholar
  47. 47.
    C.M. Pinto, J.A.T. Machado, Nonlinear Dyn. 65, 247 (2011)CrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Dept. of Electrical EngineeringInstitute of Engineering, Polytechnic of PortoPortoPortugal

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