An Extended Method of Multiple Scales

Perturbation techniques have long proven useful for analytically describing the dynamics of weakly nonlinear systems. The Method of Multiple Scales (MMS) is one such technique, popular for its versatility and structured approach to handling systems with multiple underlying time scales. However, in assuming an asymptotic series solution, classical MMS often has a restricted domain of applicability. For example, the generated solution for a nonlinear oscillator ultimately ends up as a perturbation about the homogeneous solution of the corresponding linear oscillator. While useful for capturing nonlinear phenomena such as amplitude dependent resonance, the solution will in general diverge from the direct numerical solution as the strength of the nonlinearity and/or the amplitude of the response increases. For example, in the lowest-order approximation the maximum amplitude of the response in resonance is independent of the strength of the nonlinearity, which does not match the response developed from the exact solution obtained from numerical integration. Higher-order solutions can of course be sought but immediately confront the practitioner with an abundance of choices to resolve under-determined equations. This is further complicated by the fact that classical asymptotic expansions lack uniform convergence properties by definition (therefore the name ``singular perturbation''). This work first examines how higher-order MMS has been historically addressed, and then proceeds to offer a novel take of its own. Inspiration is drawn from the exact solution for a harmonically forced and damped linear oscillator which can be expressed in the form of a rational polynomial fraction that is impossible to recover with traditional MMS (which uses a power law expansion). This motivates expansion of both the solution and the amplitude of the excitation in an ordering parameter. Based on this insight, the eXtended Method of Multiple Scales (XMMS) is introduced and applied to a damped-driven nonlinear (Duffing) oscillator. The XMMS approach is shown to consistently outperform traditional MMS and alternative higher-order approaches.