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Modeling Odd Nonlinear Oscillators with Fifth-Order Truncated Chebyshev Series

Author

Listed:
  • Daniele Ritelli

    (Department of Statistical Sciences, University of Bologna, 40126 Bologna, Italy
    These authors contributed equally to this work.)

  • Giulia Spaletta

    (Department of Statistical Sciences, University of Bologna, 40126 Bologna, Italy
    INdAM–GNCS, 00185 Roma, Italy
    These authors contributed equally to this work.)

Abstract

The aim of this work is to model the nonlinear dynamics of conservative oscillators, with restoring force originating from even-order potentials. In particular, we extend our previous findings on inverting the time-integral equation that arises in the solution of such dynamical systems, a task that is almost always intractable in exact form. This is faced and solved by approximating the restoring force with its Chebyshev series truncated to order five; such a quintication approach yields a quinticate oscillator, whose associated time-integral can be inverted in closed form. Our solution procedure is based on the quinticate oscillator coefficients, upon which a second-order polynomial is constructed, which appears in the time-integrand of the quinticate problem, and whose roots determine the expression of the closed-form solution, as well as that of its period. The presented algorithm is implemented in the Mathematica software and validated on some conservative nonlinear oscillators taken from the relevant literature.

Suggested Citation

  • Daniele Ritelli & Giulia Spaletta, 2025. "Modeling Odd Nonlinear Oscillators with Fifth-Order Truncated Chebyshev Series," Mathematics, MDPI, vol. 13(7), pages 1-13, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1125-:d:1623543
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