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.
Similar content being viewed by others
References
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)
J. Ford, Phys. Rep. 213, 271 (1992)
T.P. Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem (Springer, New York, 1997)
M.A. Porter, N.J. Zabusky, B. Hu, D.K. Campbell, American Scientist 97, 214 (2009)
G. Benettin, Chaos 15, 015108 (2004)
N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)
B. Chirikov, F. Izrailev, V. Tayurskij, Comput. Phys. Commun. 5, 11 (1973)
D.J. Korteweg, G. de Vries, Philosophical Mag. 5th Series 36, 422 (1895)
F.M. Izrailev, B.V. Chirikov, Soviet Phys. Dokl. 11, 30 (1966)
N.J. Zabusky, G.S. Deem, J. Comp. Phys. 2, 126 (1967)
P. Bocchieri, A. Scotti, B. Bearzi, A. Loinger, Phys. Rev. A 2, 2013 (1970)
R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, A. Vulpiani, Phys. Rev. A 31, 1039 (1985)
M. Pettini, M. Landolfi, Phys. Rev. A 41, 768 (1990)
J. De Luca, A.J. Lichtenberg, M.A. Lieberman, Chaos 5, 283 (1995)
D.L. Shepelyansky, Nonlinearity 10, 1331 (1997)
L. Casetti, M. Cerruti-Sola, M. Pettini, E.G.D. Cohen, Phys. Rev. E 55, 6566 (1997)
S. Flach, C.R. Willis, Phys. Rep. 295, 181 (1998)
G. James, C.R. Acad. Sci. Paris Ser. I Math. 332, 581 (2001)
T. Dauxois, R. Khomeriki, F. Piazza, S. Ruffo, Chaos 15, 015110-1-11 (2005)
S. Flach, M.V. Ivanchenko, O.I. Kanakov, Phys. Rev. Lett. 95, 064102-1-4 (2005)
T. Penati, S. Flach, Chaos 17, 023102-1-16 (2007)
D. Bambusi, A. Ponno, Comm. Math. Phys. 264, 539 (2006)
T. Dauxois, M. Peyrard, S. Ruffo, Eur. J. Phys. 26, S3 (2005)
K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York-London, 1974)
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 1993)
K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley and Sons, New York, 1993)
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)
A. Gemant, Physics 7, 311 (1936)
R.L. Bagley, P.J. Torvik, AIAA J. 21, 741 (1983)
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010)
J.A.T. Machado, J. Syst. Anal. Modell. Simul. 27, 107 (1997)
I. Podlubny, Fractional Diferential Equations (Academic Press, San Diego, 1999)
J.A.T. Machado, J. Fract. Calculus Appl. Anal. 4, 47 (2001)
V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011)
J.T. Machado, V. Kiryakova, F. Mainardi, Comm. Nonlinear Sci. Numer. Simul. 16, 1140 (2011)
J.A.T. Machado, Some Notes About the Fermi-Pasta-Ulam Problem (Symposium on Fractional Signals and Systems, Coimbra, Portugal, 2011)
D. Raškovič, Teorija elastičnosti (Theory of Elasticity) (Nauna knjiga, 1985)
K. Hedrih, Signal Proc. 86, 2678 (2006)
L.T. Burton, IEEE Trans. Circuit Theory 16, 406 (1969)
A. Antoniou, IEEE Trans. Circuit Theory 17, 212 (1970)
R. Senani, IEEE Trans. Circ. Syst. 33, 323 (1986)
L.O. Chua, IEEE Trans. Circuit Theory 18, 507 (1971)
L.O. Chua, Appl. Phys. A: Mater. Sci. Proc. 102, 765 (2011)
D. Jeltsema, A. Dòria-Cerezo, Appl. Phys. A: Mater. Sci. Proc. 100, 1928 (2012)
J.A.T. Machado, Comm. Nonlinear Sci. Numer. Simul. 18, 264 (2013)
R.S. Barbosa, J.A.T. Machado, B.M. Vinagre, A.J. Calderón, J. Vibr. Control 13, 1291 (2007)
C.M. Pinto, J.A.T. Machado, Nonlinear Dyn. 65, 247 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Machado, J.A.T. A fractional approach to the Fermi-Pasta-Ulam problem. Eur. Phys. J. Spec. Top. 222, 1795–1803 (2013). https://doi.org/10.1140/epjst/e2013-01964-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2013-01964-2