Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applicationsReport as inadecuate

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Mathematical Problems in Engineering - Volume 2004 2004, Issue 2, Pages 145-168

Istituto di Scienza e Tecnica delle Costruzioni, Università Politecnica delle Marche, via Brecce Bianche, Monte D'Ago, Ancona 60131, Italy

Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza,” via A. Gramsci 53, Roma 00197, Italy

Received 30 October 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

Author: Stefano Lenci and Giuseppe Rega

Source: https://www.hindawi.com/


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