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Y. Zarmi
8. Theory of the method of averaging
8.1 Basic Theorem
8.1.1 Definitions
Consider the equation
dx
  f t,x; 
dt
x 0  x0
x, f  R n
(8. 1)
If the function f is well behaved (say, bounded in some domain D in Rn) then the variation in
time of x is expected to be slow (it is proportional to ). If, in addition, the explicit time
dependence of f is fast compared to the time dependence of x, then replacing f(t,x;) by its
average over appropriate time scales (for fixed x) should not make a sizable difference. The
solution of the "averaged" equation should be close to the solution of the exact equation over an
extended period of time. This is the intuition behind the method of averaging. This chapter
follows Ref. [17].
We define a local time average
1
fT t, x;  
T
t T
 f .x; d
(8. 2)
t
where T is an arbitrary constant. We shall be particularly interested in functions that have an
average over an infinite time span
T
1
0
f x;   lim  f .x; d
T  T
0
(8. 3)
0
f x;   
A function f that satisfies the Lipschitz condition, and for which f0 exists, is called a KBM field
(named after Krylov, Bogoliubov and Mitropolsky, who were instrumental in the development of
the method).
Of particular interest is the case of a well behaved f(t,x;) for which the explicit time dependence
(with x fixed) is periodic, with period T0:
f t  T0 ,x;   f t,x; 
It is easy to show that the local time average [Eq. (8.2)] with T chosen to be the period, T0, is
independent of time, and equal to the average over one cycle:
Nonlinear Dynamics
1
fT0 t, x; 
T0
-2-
t  T0
T0
1
1
 f , x; d  T0  f ,x; d  T0
t
t
T0
t  T0
 f ,x; d 
T0
t
1
1
f , x; d   f   T0 ,x; d 

T0 t
T0 0
T0
(8.4)
T0
t
1
1
1
f , x; d   f ,x; d   f , x; d

T0 t
T0 0
T0 0
We will now show that, for a periodic f, f0 exists and is also equal to the average over one cycle.
To this end, for large T in Eq. (8.3), we define
T 
N   
T0 
T  N T0   N
0   N  T0
(The square bracket stands for the "integer part".) The integral in Eq. (8.3) can be re-written as
2 T0
T0
 f .x; d   f .x; d 
T
0
T0
1
1
f

.x;

d


T 0
N T0   N 



f

.x;


d




 N  1 T0

N T0   N

 f .x; d 
N T0

N T0

Due the periodic time dependence of f, each of the first N integrals is equal to
T0
 f ,x; d
0
Therefore,
T
1
1
f .x; d 

T0
N T0   N
T
t  N T0   N

 0
N  f ,x; d 
 f.x; d

 0
t  N T0





If f is bounded in the domain D, then, the second integral in the last result is bounded.
Consequently, as N (T), we have
T
T0
1
1
f x;   lim  f .x; d   f ,x; d
T  T
T0 0
0
0
(8.5)
Thus, bounded periodic functions that satisfy the Lipschitz condition are KBM fields.
8.1.2 The basic theorem
We now leave periodic functions for a while. Our goal is to show that if we replace Eq. (8.1) by
the "averaged" equation
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Y. Zarmi
dy
0
  f y; 
dt
for a KBM field then y(t) will approximate x(t) with a known error over t≤O(1/).
Theorem Given the two differential equations
dx
  f t,x; 
dt
dy
0
  f y; 
dt
having the following properties:
x0  x0
(8. 6)
y0   x 0
(8. 7)
1. f is a KBM field with average f0
2. y(t) exists in a domain D in which x(t) exists for t≤O(1/),
then

xt   yt   O  
12

  0, t  O1  
(8. 8)
where
   sup 
xD
0t  L  
t
 f ,x;   f x; d
0
(8. 9)
0
Proof
Define
M  sup f t,x; 
(8.10)
xD
0 t  L  
and add to Eqs. (8.6&7) an intermediate dynamical system for a variable u, so that we now have
dx
  f t,x; 
dt
du
  fT t, u; 
dt
dy
0
  f y; 
dt
x0  x 0
u0  x 0
y0  x0
(8.11)
We want to estimate
xt   yt   xt   ut   ut   yt 
(8.12)
The proof will find the averaging time, T, that provides the bound of Eq. (8.8). We shall see that
only if T«L/ with L constant, will the result of the theorem be useful. Since we are studying the
Nonlinear Dynamics
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first order corrections only, we ignore from now on the possibility that, f may depend on , as
Eq. (8.1) has an explicit  in front of f. We start by developing an estimate for ||x(t)-u(t)||.
t
x t   ut     f ,x    fT ,u d 
0
t
T


1
 f ,x  
f

s,u




ds
d 
T 


0
t
 T
T

 f ,xds   f s, x ds 
 0

d
 T
 T



T
 f s,x ds  f s,u ds
 


 


(8.13)
0
The first pair of integrals in square brackets in Eq. (8.13) is equal to
t
 T
d  dsf ,x  f s  ,x 
 0
0
t
 T

  ds f ,x  f s  ,xs   


 0
d


T




   ds f s  ,xs     f s  ,x 
  0





0
(8.14)
T
ts
 t

ds  d f , x    df ,x
 0

s
0
t
 T
d  ds f s  , xs    fs  ,x  
 0

0
Consequently,

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Y. Zarmi
T
ts
s
ds  d f ,x   d f ,x  
       
 



0
t
0
t
 T
 
xt   ut   d  ds f s  , xs    f s  ,x
T  0
0
t
 T
d  ds f s,x  f s, u 
 

 
 






0















(8.15)
where the second pair of integrals in Eq. (8.13) has been included, and the Cauchy-Schwarz
inequality has been used. In Eq. (8.15) we now replace ||f|| in the first two integrals by its bound
over the whole domain, M. For the second and third terms we use the Lipschitz condition with
Lipschitz constant . The result after some trivial integration's is
t


T
 T

 
2
xt   ut   M T   d  ds x s    x    T  d x    u  
T 
 0

0


0
(8.16)
A bound for the middle term in Eq. (8.16) can be found as follows. The solution of Eq. (8.1) can
be formally written as
t
x t   x 0    f ,xd
(8.17)
0
Hence
xs     x  
s
s 


 ds f s, xs    ds f s,x s   M s
(8.18)
Inserting this result in Eq. (8.16), we finally find
t
xt   ut    M T  12  M  t T     d x   u 
2
(8.19)
0
Now use the Gronwall lemma to obtain
xt   ut   12  M T 3exp t 1 12 M 3exp  L 1 T  Const.T
for t≤(L/).
Now we need to find a bound for ||u(t)-y(t)|| of Eq. (8.12).
(8.20)
Nonlinear Dynamics
-6t
ut   yt     d fT ,u   f
0
y
0
t
 1   T

 d   ds f s,u   f 0 u 
 T 

0
t
  d f
0
u  f 0 y 
0
t
T
 
d ds f s,u   f
T  
0
u 
(8.21)
0
t
T1

1
 d lim  ds f s,u   f s,y
 T1  T1 0
0
Using Eq. (8.9) for estimating the first term in Eq. (8.21), and the Lipschitz condition for the
second one, we obtain
ut   yt  
 
t     d u   y
T
0
t
(8.22)
Again, making use of Gronwall's lemma, we find
ut   yt  
 

exp  t   1  Const.

 T
T
(8.23)
for t≤L/. As a result, Eq. (8.12) yields

(8.24)
,
t  L  
T
This inequality is true for any T. The smallest r.h.s. of Eq. (8.24) attains its smallest value when
ut   yt   Const. T  Const.
 T  Const.  
12
(8.25)
Hence, under the conditions of the theorem

xt   yt   O  
12

t  OL  
(8.26)
which proves the theorem.
However, for the theorem to be of any use, and the bound to be useful, we must have
 
 
0 0
In many examples, this is true. For instance, if f(t,x;) is periodic in t, with period T0, assign the
value T0 for the averaging time T. With this value, u(t)=y(t) in Eq. (8.12)). Eq. (8.20) leads to:
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Y. Zarmi
xt   yt   xt   ut   Const. T0
t  L  
(8.27)
Thus, in the periodic case the error bound is rather stringent. It is O() for t≤O(1/). This is not
true in aperiodic functions. There, usually, the error bound is much weaker (e.g., 1/2 etc.).
Example Consider the equation
xÝ
Ý  2  F t   x  0
F0  1
x0  r0 xÝ0  0
(8.28)
Ft  
0
t
In polar coordinates,
x  r cost   
y  rsint  
Eq. (8.28) has the form
2
rÝ  r sin t  Ft   2
r0   r0
Ý  sint   cost   F t   2 

0  0
(8.29)
Here the right hand side vector field is
 r sin 2 t  Ft   2 
f t,r,   1

2 sin 2 t   F t   2
(8.30)
Although f is not periodic in t, the fact that F vanishes when t implies
T
r
1
f  lim  f t,r, dt   
T T
 0 
0
0
(8.31)
from which the averaged equation follows:
r
r  r0 exp  t 
d r 
0
    f r,      
  

dt 
 0 
 
0

In order to evaluate () (see Eq. (8.9)), we compute the integral
t

2
t
 r sin t   F t   cos 2 t  dt 
0

  f  f dt   0
t


1
0

2 sin2 t   F t   2dt




0
(8.32)
(8.33)
The result for Eq. (8.33) depends on the properties of F(t). The two integrals that do not involve
F(t) are integrals of trigonometric functions, which are O(0) for 0. Now consider
Nonlinear Dynamics
-8t
t
1
 sin t   F t dt  2  F t dt 
2
0
t
1
2
0
 cos 2 t    Ft dt 
0
(8.34)
t
1
2
 F t dt  O 
0
0
The result depends on the integral of F(t), over times t≤L/. The following are a few examples:

I.

t  L  

F

t

dt

II.


0

III.

1  expa L  
F t   exp at 
a
1  L    O  s  1

1 s

F t   1  t 
log1  L    Olog  
F t  
s
0s
(8.35)
1
1 t
This yields
I.

   II.

II.
O 
O
s


 
12
O log  
O  1 2
  
 s 2
 O 

12

O  log  

(8.36)

For F=Const., the problem becomes periodic, and ()=O(). In similarity to the error estimate
theorem of Section 4, the present theorem is weak. It estimates the error as O((())1/2), while
intuitively we expect the maximum deviation between the two solutions to be O(()).
8.2 Crude averaging
The basic theorem of averaging requires that x is a slow variable, so that it may be "frozen", as
one averages over the fast variable, t. To conform with this requirement, we have converted Eq.
(8.28) into first order equations for the slow variables r and . Can't we simply average over the
original Eq. (8.28)? We call this "crude" averaging. We will now show by the example of:
Ý 2  1  cos2 t xÝ x  0
xÝ
x0 1
xÝ0   0
(8.37)
how crude averaging leads to failure. If we simply average over that equation, we obtain
Ý
xÝ 2 xÝ x  0
(8.38)
x  exp  t cos 1   2 t
(8.39)
which is solved by
We now apply the correct method of averaging to Eq. (8.38). Defining y=dx/dt, we obtain
-9-
Y. Zarmi
xÝ y
x0  1
yÝ  x  2 1  cos 2t  y
y0   0
(8.40)
We need to transform (x,y) into a pair of slow variables. We do this by the transformation:
x
z1 
  exp Jt  
y
z2 
 0 1
J  

1 0
(8.41)
After some calculation, the equation for (z1, z2) is found to be
zÝ1 
0 0
z1 
expJt   
  2 1  cos 2t exp J t 
zÝ2 
0 1
z2 
(8.42)
 sin 2 t
 sint cos t z1 
2 1  cos 2t 
 
cos 2 t z2 
 sint cos t
(z1, z2) are, indeed, slow variables. Their averaged equation is
12 z1 
d z1 


 
3 
dt z2 
2 z2 
(8.43)
which is solved by (with a and b constants of integration)
z1  exp 12  t a
  

z2  exp  32  t b
(8.44)
As a result, after imposing the conditions, we obtain the following approximate solution:
x 
 cos t 
  exp  12 t 

y 
 sint 
(8.45)
The r.h.s. of Eq. (8.42) is periodic in t. Hence, averaging ensures that the deviation of the
approximation from the exact solution is
x  x 
     O 
y  y 
t  O1  
(8.46)
Over long times, the solution derived by crude averaging [Eq. (8.39)] deviates from the solution
of the correct averaged equation [Eq. (8.45)], by a large error. We find:
 0

2
x crude  xcorrect  exp t cos 1   t  exp  12 t cos t  O 
averaging
averaging

O1
t0
0
t  O 
t  O1  
The same conclusions are reached through an analysis in polar coordinates. Eq. (8.40) leads to
Nonlinear Dynamics
x  r cost   
-10-
y  r sint  
rÝ  r 1  cos 2t 1  cos 2 t  
r0   1
Ý  1  cos 2t sin 2 t   

0  0
(8.47)
The averaged equations are
r 1  12 cos 2 
d r 
     1

dt 
 2 sin2  
(8.48)
With the (0,1) initial conditions, Eq. (8.48) is solved by
r  exp  12 t 
 0
(8.49)
which, yields the same solution as Eq. (8.45).
In Fig. 8.1, we show the error
x t   x t 
for =0.05, for crude and correct averaging. The solution for x(t) was obtained numerically.
Fig. 8.1 Errors in correct and crude averaging for Eq. (8.37)
For t=O(1) (e.g., t up to about 2) the errors are comparable and of O(). For longer times,
the error generated by crude averaging is much greater than the one generated by correct
averaging. If it were not for the exponential fall off of both the exact solution and the
approximations, the "performance" of crude averaging would have been much worse. (Clearly,
for very long times, the approximations, the exact solution and the errors all decay to zero.)
-11-
Y. Zarmi
Close examination of the procedure that leads to the correct averaged equation [Eq. (8.43)]
shows that, in crude averaging, we miss the delicate interference between the leading time
behavior, exp(±it), and the perturbation. This interference produces the resonant terms which
are retained by the method of averaging, while they are lost by crude averaging.
Exercises
8.1 Obtain Eqs. (8.47) & (8.48).