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

• Published in: Computational & applied mathematics Vol. 37; no. Suppl 1; pp. 7 - 15
• Main Authors: de Almeida, A. K., de Almeida Prado, A. F. B., de Moraes, R. Vilhena, Lara, M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Astrodynamics; Computational mathematics; Computational Mathematics and Numerical Analysis; Duffing oscillators; Integrals; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Trajectory analysis; 70F99; Perturbation expansion method; 41-02; Astrodynamics; Nonlinear systems; Perturbed dynamical analysis; 85-02
• ISSN: 0101-8205; 2238-3603; 1807-0302
• DOI: 10.1007/s40314-017-0471-8

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.