On the multiple time-scales perturbation method for differential-delay equations

• Published in: Nonlinear dynamics Vol. 112; no. 10; pp. 8431 - 8451
• Main Authors: Binatari, N., van Horssen, W. T., Verstraten, P., Adi-Kusumo, F., Aryati, L.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.05.2024; Springer Nature B.V
• Subjects: Applied mathematics; Approximation; Automotive Engineering; Classical Mechanics; Control; Delay; Differential equations; Dynamical Systems; Eigenvalues; Eigenvectors; Engineering; Finite differences; Initial conditions; Mathematical analysis; Mechanical Engineering; Operators (mathematics); Ordinary differential equations; Original Paper; Perturbation methods; Vibration; Asymptotic validity; 41A60; Multiple time-scales; Perturbation methods; Delay differential equations; 74H10
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-024-09485-z

In this paper, we present a new approach on how the multiple time-scales perturbation method can be applied to differential-delay equations such that approximations of the solutions can be obtained which are accurate on long time-scales. It will be shown how approximations can be constructed which branch off from solutions of differential-delay equations at the unperturbed level (and not from solutions of ordinary differential equations at the unperturbed level as in the classical approach in the literature). This implies that infinitely many roots of the characteristic equation for the unperturbed differential-delay equation are taken into account and that the approximations satisfy initial conditions which are given on a time-interval (determined by the delay). Simple and more advanced examples are treated in detail to show how the method based on differential and difference operators can be applied.