Method of averaging

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 formformula_1of a phase space variable formula_2The "fast oscillation" is given by formula_3 versus a "slow drift" of formula_4. The averaging method yields an autonomous dynamical systemformula_5which approximates the solution curves of formula_4 inside a connected and compact region of the phase space and over time of formula_7.
Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for formula_8. 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 formula_9, with the corresponding averaged system formula_10, in order to use the averaged system to make a prediction and then test the prediction against the results of a physical experiment.
The averaging method has a long history, which is deeply rooted in perturbation problems that arose in celestial mechanics (see, for example in ).
First example.
Consider a perturbed logistic growthformula_11and the averaged equationformula_12The purpose of the method of averaging is to tell us the qualitative behavior of the vector field when we average it over a period of time. It guarantees that the solution formula_13 approximates formula_14 for times formula_15 Exceptionally: in this example the approximation is even better, it is valid for all times. We present it in a section below.
Definitions.
We assume the vector field formula_16 to be of differentiability class formula_17 with formula_18 (or even we will only say smooth), which we will denote formula_19. We expand this time-dependent vector field in a Taylor series (in powers of formula_20) with remainder formula_21. We introduce the following notation:formula_22
where formula_23is the formula_24-th derivative with formula_25. As we are concerned with averaging problems, in general formula_26is zero, so it turns out that we will be interested to vector fields given byformula_27Besides, we define the following initial value problem to be in the standard form:formula_28
Theorem: averaging in the periodic case.
Consider for every formula_29 connected and bounded and every formula_30 there exist formula_31 and formula_32 such that the original system (a non-autonomous dynamical system) given byformula_33has solution formula_34, where formula_35 is periodic with period formula_36 and formula_37 both with formula_18 bounded on bounded sets. Then there exists a constant formula_39 such that the solution formula_40 of the "averaged" "system" (autonomous dynamical system) isformula_41isformula_42for formula_43 and formula_44.
Strategy of the proof.
Krylov-Bogoliubov realized that the slow dynamics of the system determines the leading order of the asymptotic solution.
In order to proof it, they proposed a "near-identity transformation," which turned out to be a change of coordinates with its own time-scale transforming the original system to the averaged one.
Non-autonomous class of systems: more examples.
Along the history of the averaging technique, there is class of system extensively studied which give us meaningful examples we will discuss below. The class of system is given by:
formula_59
where formula_60 is smooth. This system is similar to a linear system with a small nonlinear perturbation given by 
formula_61:
formula_62
differing from the standard form. Hence there is a necessity to perform a transformation to make it in the standard form explicitly. We are able to change coordinates using variation of constants method. We look at the unperturbed system, i.e. formula_63, given by
formula_64
which has the fundamental solution formula_65corresponding to a rotation. Then the time-dependent change of coordinates is formula_66 where formula_67 is the coordinates respective to the standard form.
If we take the time derivative in both sides and invert the fundamental matrix we obtain
formula_68
Remarks.
formula_73 If formula_74we average it as long as a neighborhood of the origin is excluded (since the polar coordinates fail) yields:formula_75 where the averaged system isformula_76
Example: Misleading averaging results.
The method contains some assumptions and restrictions. These limitations play important role when we average the original equation which is not into the standard form, and we can discuss counterexample of it. The following example in order to discourage this hurried averaging:formula_77where we put formula_78following the previous notation.
This systems corresponds to a damped harmonic oscillator where the damping term oscillates between formula_79 and formula_80. Averaging the friction term over one cycle of formula_81 yields the equation:formula_82The solution isformula_83which the convergence rate to the origin is formula_20. The averaged system obtained from the standard form yields:formula_85which in the rectangular coordinate shows explicitly that indeed the rate of convergence to the origin is formula_86differing from the previous crude averaged system:formula_87
Example: Van der Pol Equation.
Van der Pol was concerned with obtaining approximate solution for equations of the typeformula_88where formula_89following the previous notation. This system is named Van der Pol oscillator. If we apply periodic averaging to this nonlinear oscillator, this give us qualitative knowledge of the phase space without solving explicitly the system.
The averaged system isformula_90and we can analyze the fixed points and their stability. There is an unstable fixed point at the origin and a stable limit cycle represented by formula_91.
The existence of such stable limit-cycle can be stated as a theorem.
Theorem (Existence of a periodic orbit): If formula_92is a hyperbolic fixed point of formula_93Then there exists formula_30such that for all formula_95,
formula_96has a unique hyperbolic periodic orbit formula_97of the same stability type as formula_92.
The proof can be found at Guckenheimer and Holmes, Sanders "et al." and for the angle case in Chicone.
Example: Restricting the time interval.
The average theorem assumes existence of a connected and bounded region formula_29which affects the time interval formula_100 of the result validity. The following example points it out. Consider theformula_101where formula_102. The averaged system consists offormula_103which under this initial condition indicates that the original solution behaves likeformula_104where it holds on a bounded region over formula_105.
Damped Pendulum.
Consider a damped 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
formula_106
where formula_107 describes the motion of the suspension point, formula_108 describes the damping of the pendulum, and formula_109 is the angle made by the pendulum with the vertical.
The phase space form of this equation is given by
formula_110
where we have introduced the variable formula_111 and written the system as an "autonomous", first-order system in formula_112-space.
Suppose that the angular frequency of the vertical vibrations, formula_113, is much greater than the natural frequency of the pendulum, formula_114. Suppose also that the amplitude of the vertical vibrations, formula_115, is much less than the length formula_116 of the pendulum. The pendulum's trajectory in phase space will trace out a spiral around a curve formula_117, moving along formula_117 at the slow rate formula_114 but moving around it at the fast rate formula_113. The radius of the spiral around formula_117 will be small and proportional to formula_115. The average behaviour of the trajectory, over a timescale much larger than formula_123, will be to follow the curve formula_117.
Extension error estimates.
Average technique for initial value problems has been treated up to now with an validity error estimates of order formula_7. However, there are circumstances where the estimates can be extended for further times, even the case for all times. Below we deal with a system containing an asymptotically stable fixed point. Such situation recapitulates what is illustrated in Figure 1.
Theorem (Eckhaus /Sanchez-Palencia ) Consider the initial value problemformula_126Suppose
formula_127exists and contains an asymptotically stable fixed point formula_128 in the linear approximation. Moreover, formula_129is continuously differentiable with respect to formula_8 in formula_131 and has a domain of attraction formula_132. For any compact formula_133 and for all formula_134
formula_135
with formula_136 in the general case and formula_137 in the periodic case.