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.

Definition

Consider a general, nonlinear dynamical system

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

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

$\dot{x}^{a} = \epsilon \frac{1}{T}\int_{0}^{T}f(\tau,x,0) d\tau = \tilde{f}(x^{a}).$

Averaging mods out the fast oscillatory dynamics by averaging their effect (through time integration - see the formula above). In this way, the mean (or long-term) behavior of the system is retained in the form of the dynamical equation for the evolution for x^a. Standard methods for time-invariant (autonomous) systems may then be employed to analyze the equilibria (and their stability) as well as other dynamical objects of interest present in the phase space of the averaged system.

Example

Consider a simple pendulum whose point of suspension is vibrated vertically by a small amplitude, high frequency signal (this is usually known as dithering). The equation of motion for such a pendulum is given by

$m(l\ddot{\theta} - ak\omega^2 \sin \omega t \sin \theta) = -mg \sin \theta - k(l\dot{\theta} + a\omega \cos \omega t \sin \theta)$

where asin ωt describes the motion of the suspension point and θ is the angle made by the pendulum with the vertical.

The state space form of this equation is given as

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