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Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications

Author

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  • Stefano Lenci
  • Giuseppe Rega

Abstract

A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincaré sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds (one-half Melnikov functions) and the distance between perturbed stable and unstable manifolds (full Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations.

Suggested Citation

  • Stefano Lenci & Giuseppe Rega, 2004. "Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications," Mathematical Problems in Engineering, Hindawi, vol. 2004, pages 1-24, January.
    • Handle: RePEc:hin:jnlmpe:294307
    DOI: 10.1155/S1024123X04310045
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