Method of averaging

In the study of dynamical systems, the method of averaging is used to study certain time-varying systems by analyzing easier, time-invariant systems obtained by averaging the original system.

Consider a general, nonlinear dynamical system

${\displaystyle {\dot {x}}=\epsilon f(t,x,\epsilon )}$

where ${\displaystyle f(t,x)}$ is periodic in ${\displaystyle t}$ with period ${\displaystyle T}$. The evolution of this system is said to occur in two timescales: a fast oscillatory one associated with the presence of ${\displaystyle t}$ in ${\displaystyle f}$ and a slow one associated with the presence of ${\displaystyle \epsilon }$ in front of ${\displaystyle f}$. The corresponding (leading order in ${\displaystyle \epsilon }$) averaged system is

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