en>Cydebot: Robot - Moving category Constraint satisfaction to :Category:Constraint programming per CFD at Wikipedia:Categories for discussion/Log/2011 June 15.

2011-06-30T16:24:44Z

Robot - Moving category Constraint satisfaction to Category:Constraint programming per CFD at Wikipedia:Categories for discussion/Log/2011 June 15.

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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

<math>\dot{x} = \epsilon f( t, x , \epsilon )</math>

where <math>f(t,x)</math> is [[Frequency|periodic]] in <math>t</math> with period <math>T</math>. The evolution of this system is said to occur in two timescales: a fast oscillatory one associated with the presence of <math>t</math> in <math>f</math> and a slow one associated with the presence of <math>\epsilon</math> in front of <math>f</math>. The corresponding (leading order in <math>\epsilon</math>) '''averaged''' system is

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

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 <math>x^{a}</math>. Standard methods for time-invariant (autonomous) systems may then be employed to analyze the [[equilibrium point|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

<math>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)</math>

where <math>a \sin \omega t</math> describes the motion of the suspension point and <math>\theta</math> is the angle made by the pendulum with the vertical.

The [[state space]] form of this equation is given as

[[Category:Dynamical systems]]

{{mathapplied-stub}}

Ordered graph - Revision history

en>Cydebot: Robot - Moving category Constraint satisfaction to :Category:Constraint programming per CFD at Wikipedia:Categories for discussion/Log/2011 June 15.