Method of averaging

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In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale. It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution.

More precisely, the system has the following form $${\displaystyle {\dot {x}}=\varepsilon f(x,t,\varepsilon ),\quad 0\leq \varepsilon \ll 1}$$of a phase space variable ${\displaystyle x.}$The fast oscillation is given by ${\displaystyle f}$ versus a slow drift of ${\displaystyle {\dot {x}}}$. The averaging method yields an autonomous dynamical system $${\displaystyle {\dot {y}}=\varepsilon {\frac {1}{T}}\int _{0}^{T}f(y,s,0)~ds=:\varepsilon {\bar {f}}(y)}$$ which approximates the solution curves of ${\displaystyle {\dot {x}}}$ inside a connected and compact region of the phase space and over time of ${\displaystyle 1/\varepsilon }$.

Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for ${\displaystyle y}$. In this way, qualitative methods for autonomous dynamical systems may be employed to analyze the equilibria and more complex structures, such as slow manifold and invariant manifolds, as well as their stability in the phase space of the averaged system.

In addition, in a physical application it might be reasonable or natural to replace a mathematical model, which is given in the form of the differential equation for ${\displaystyle {\dot {x}}}$, with the corresponding averaged system ${\displaystyle {\dot {y}}}$, in order to use the averaged system to make a prediction and then test the prediction against the results of a physical experiment.^[1]

The averaging method has a long history, which is deeply rooted in perturbation problems that arose in celestial mechanics (see, for example in ^[2]).