How do rapidly shaking forces reshape what systems do?

When a system gets hit by a rapidly oscillating force, its true dynamics emerge only at larger scales. This work builds an effective description order by order, like peeling away layers of noise, valid up to 1/ω⁶ precision. A surprising finding: many nonlinear systems have the same stability boundaries as the simple linear Mathieu equation, allowing exact predictions even in extreme conditions. Practical payoff: explaining magnetic traps for charged particles.