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A fractional approach to the Fermi-Pasta-Ulam problem
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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.
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Springer