A fractional approach to the Fermi-Pasta-Ulam problem
Institute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, Rua Dr. António Bernardino de Almeida, post431, 4200-072 Porto, Portugal
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Received: 3 June 2013
Revised: 6 August 2013
Published online: 7 October 2013
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.
© EDP Sciences, Springer-Verlag, 2013