On higher-order averaging of time-periodic systems: reconciliation of two averaging techniques

In this paper we show how higher-order averaging can be used to remedy serious technical issues with the direct application of the averaging theorem. While doing so, we reconcile two higher-order averaging methodologies that were developed independently using different tools and within different communities: (i) perturbation theory using a near-identity transformation and (ii) chronological calculus using Lie algebraic tools. We provide the underpinning concepts behind each averaging approach and provide a mathematical proof for their equivalence up to the fourth order. Moreover, we provide a higher-order averaging study and analysis for two applications: the classical problem of the Kapitza pendulum and the modern application of flapping flight dynamics of micro-air-vehicles and/or insects.