We consider systems characterized by the presence of a rapidly oscillating force. A general method is presented for the construction of the effective action governing the large-scale nonlinear dynamics of such systems order by order in inverse powers of the oscillation frequency $\omega$. The explicit expression for the effective Lagrangian is derived up to $\mathcal{O}(1/{\omega }^6)$ next-to-next-to-leading approximation in the strong-drive regime. The general structure of the high-frequency expansion reveals a broad class of nonlinear systems whose transition curves are identical to those of the linear Mathieu equation, which enables a fully nonperturbative stability analysis in the case of strong driving and nonlinearity. The method is generalized to velocity-dependent forces and configuration space with curvature, characteristic to systems with constraints. Several applications are discussed in detail, including the dynamical magnetic trapping of electric charges.

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• Bifurcations
• Classical mechanics
• Floquet systems
• Multiple time scale dynamics
• Chaos & nonlinear dynamics