Computational and Applied Mathematics

, Volume 37, Supplement 1, pp 7–15 | Cite as

Analyzing “integral indices” to quantify the effects of a perturbing force in the harmonic and Duffing oscillators

  • A. K. de AlmeidaJr.Email author
  • A. F. B. de Almeida Prado
  • R. Vilhena de Moraes
  • M. Lara


The present paper has the goal of studying the use of “integral indices” to quantify the effects of a perturbing force in the driven harmonic and Duffing oscillators. The main idea is to define a scalar index that can represent the cumulative effects over time that a perturbing force causes in a dynamical system. An index of this type can help to prepare “perturbation maps”, which can identify situations of larger or smaller effects. This idea appeared in the astrodynamics literature with the goal of finding less perturbed orbits for a spacecraft, but it is applied here to the driven harmonic and Duffing oscillators. The reason for those applications is that those problems have analytical solutions, which allows a better comparison of the indices. In particular, the effects of calculating this index using a perturbed and a non-perturbed trajectory are evaluated with the goal of better understanding these effects. The results show that the difference between both indices depends on the frequency and amplitude of the perturbing force.


Astrodynamics Perturbation expansion method Perturbed dynamical analysis Nonlinear systems 

Mathematics Subject Classification

70F99 41-02 85-02 


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  • A. K. de AlmeidaJr.
    • 1
    Email author
  • A. F. B. de Almeida Prado
    • 1
  • R. Vilhena de Moraes
    • 2
  • M. Lara
    • 2
  1. 1.Instituto Nacional de Pesquisas Espaciais (INPE)São José dos CamposBrazil
  2. 2.Instituto de Ciência e Tecnologia (ICT-UNIFESP)São José dos CamposBrazil

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