Averaging - davidar/scholarpedia GitHub Wiki

Solution to perturbed logistic equation \dot{x}=\varepsilon\left( x(1-x) +\sin t \right) (blue) and the averaged equation \dot{z}=\varepsilon z(1-z) (red) with \varepsilon=0.05\ .

Averaging is the procedure of replacing a vector field by its average (over time or an angular variable) with the goal to obtain asymptotic approximations to the original system and to obtain periodic solutions.

Table of Contents Basic definitions, the periodic case Averaging: the periodic case Remark on the method of proof Example: perturbed logistic growth Example: the van der Pol oscillator The non-periodic case Averaging: the general case Remarks Averaging over angles Higher order approximations Related Theories References Recommended reading External links Other kinds of averaging See also

Consider an ordinary differential equation of the type

<math>\label{eq1}

\dot{x}=\varepsilon f(x,t,\varepsilon), \quad x(0)=x_0,\quad x,\ x_0\in D\subset\mathbb{R}^n, </math>

where <math>D</math> is an open set with compact (that is, closed and bounded) closure, on which <math>f</math> is defined. The parameter <math>\varepsilon</math> is assumed to be small. The equation often arises by expansion in the neighborhood of an equilibrium. The vector field <math>f</math> is assumed to be differentiable with respect to all variables, but this can be relaxed.

Since <math>f</math> depends explicitly on time <math>t\ ,</math> equation \eqref{eq1} is a nonautonomous differential equation. This type of equation is usually very difficult to analyze, so one is interested in finding an autonomous system, the solutions of which approximate the original system, where the accuracy of the approximation is a function of <math>\varepsilon\ .</math>

Putting <math>\varepsilon=0</math> is not going to do much good: it will give us an approximation that is valid on the interval <math>0\leq t\leq L</math> for some constant <math>L\ ,</math> that is on time scale <math> 1 \ .</math> On a longer time scale, for instance <math>1/\varepsilon\ ,</math> this is a singular perturbation problem, that is to say, the solution of the unperturbed problem (with <math>\varepsilon=0</math>) is not an approximation of the solution of the full problem \eqref{eq1}.

On this longer time <math>1/\varepsilon</math> another natural idea works better: average the right hand side over the time <math>t\ .</math> Assume, for simplicity, that <math>f</math> is periodic in <math>t</math> with period <math>T\ .</math> Then define the average

<math>\label{eq2}

\bar{f}(x)=\frac{1}{T}\int_{0}^T f(x,s,0)\, d s. </math>

One now considers the first order averaged equation

<math>\label{eq3}

\dot{z}=\varepsilon \bar{f}(z), \quad z(0)=x_0. </math>

Let <math>z(t)</math> be the solution of \eqref{eq3} and let <math>L\ ,</math> independent of <math>\varepsilon</math> be such that <math>z(t)\in D</math> for <math>0 \leq \epsilon t \leq L\ .</math> Then there exists an <math>\varepsilon</math>-independent constant <math>C</math> such that

<math>\label{eq4}

||x(t)-z(t)||\leq C \varepsilon </math>

 for <math>0 \leq \epsilon t \leq L\ ,</math>

where <math>x(t)</math> is the solution of \eqref{eq1}. We say that <math>x(t)=z(t)+O(\varepsilon)</math> on <math> 1/\varepsilon\ .</math>

Observe that we only require <math>z(t)\in D\ .</math> Since <math>x(t)</math> is <math> \varepsilon</math>-close to <math>z(t)</math> and <math>D</math> is an open set, we can always choose <math> \varepsilon</math> small enough.

There are two methods of proof: a direct method and a formal transformation method.

• The direct method needs less differentiability assumptions and can be generalized to more complicated situations, for instance delay equations, and
• The transformation method, which can be used to obtain higher order transformations.

Consider the equation

<math>

\dot{x}=\varepsilon\left( x(1-x) +\sin t \right),\quad x(0)=x_0, </math> which models slow logistic growth (the <math> x(1-x)</math> term) with a seasonal influence (the <math> \sin t</math> term). The averaged equation is

<math>

\dot{z}=\varepsilon z(1-z), </math> and this can be integrated by separation of variables.

To apply the periodic averaging theorem one needs to fix the domains <math>D\subset K \subset \mathbb{R}\ .</math> A good choice would be <math>D=(0,1), K=[0,1] \ .</math> In general, this choice will determine <math>L</math> and <math>C\ .</math> It then follows that in this model the seasonal influence on the solutions is <math>O(\varepsilon)\ ,</math> as one can see in <figref>Sanders_averaging.gif</figref>.

The van der Pol oscillator equation is (see van der Pol, 1926)

<math>\label{eq5}

\ddot{V} +V=\varepsilon (1-V^2)\dot{V}, </math>

where <math>V</math> is the current in a circuit. This equation is not of the form \eqref{eq1}, but one can put it in this form by the method of variation of constants. First we write \eqref{eq5} as a system:

<math>\label{eq6}

\begin{matrix} \dot{V}&=&I&& \\ \dot{I}&=&-V &+&\varepsilon (1-V^2) I. \end{matrix} </math>

<math>I</math> is the current in the circuit. The change of variables

<math>\label{eq9}

\left(\begin{matrix} X\\ Y\end{matrix}\right)= \left(\begin{matrix} \cos t & -\sin t\\ \sin t &\cos t \end{matrix}\right) \left(\begin{matrix} V \\ I \end{matrix}\right) </math>

transforms the system to the form \eqref{eq1}, with

<math>\label{eq11}

x=\left(\begin{matrix} X \\ Y \end{matrix}\right) </math>

and

<math>\label{eq12}

f(X,Y,t,\varepsilon)=\left(\begin{matrix} - (1-(X \cos t + Y \sin t)^2) (-X \sin t +Y \cos t) \sin t\\ (1-(X \cos t + Y \sin t)^2) (-X \sin t +Y \cos t) \cos t\end{matrix}\right). </math>

One can now compute the average equation using \eqref{eq2} with <math>T=2\pi</math> to obtain

<math>\label{eq13}

\bar{f}(X,Y)=1/8 \left( 4-({X}^{2}+{Y}^{2}) \right) \left(\begin{matrix}X \\ Y \end{matrix}\right). </math>

One sees that <math>\bar{f}(X,Y)=0</math> if <math> V^2+I^2=X^2+Y^2=4\ .</math> This corresponds to the famous limit cycle of the van der Pol oscillator. The expression <math> \tau=X^2+Y^2</math> is an invariant of the flow of the linear part of the equation. Dropping all information on the phase, one can reduce the averaged equation to

<math>\label{eq14}

\dot{\tau}=(1-\frac{\tau}{4})\tau. </math>

This equation has two equilibria, one unstable at <math>\tau=0\ ,</math> one stable at <math>\tau=4\ .</math> Equation \eqref{eq14} can be integrated explicitly. The fact that the averaged equation is simpler is an important aspect. Even if the system is high dimensional, and the averaged system is still difficult to analyze, there is a gain: the time-scale of the original equation is <math>1\ ,</math> and of the averaged equation it is <math>1/\varepsilon\ ,</math> which makes numerical methods much more efficient, since it increases the step size with a factor <math>1/\varepsilon\ .</math>

When one drops the assumption of periodicity, one can still try and define the averaged equation by

<math>

\bar{f}(x) = \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T f(x, s,0) \, ds. </math> If this expression exists, and the limit is uniform in <math>x</math> on compact sets <math> E\subset D\ ,</math> one has to compute a suitable order function, defined by

<math></math>