-1-
Y. Zarmi
9. Second order averaging
9.1 Introductory steps
We again follow closely Ref. [17]. At this stage, we study the case of functions f that do not
depend on , that is, f=f(t,x). The effect of dependence in f will be discussed later onTo find
a second-order approximation that is better than y(t), of first-order averaging (see Chapter 8), we
need to find a w(t) so that the error {x(t)w(t)} is O(1()), rather than O([1()]1/2). 1() is
defined in Chapter 8 as:
t
vd
(9. 1)
1  u t, v    f ,v  f v d
(9. 2)
1    sup 
vD
0t  L  
 f ,v  f
0
0
Eq. (9.1) implies that if we define
t
1
0
0
than u1 is O(1) for 0 for all v. Hence, if we find a w(t) such that
x t   wt   1 u 1t,wt 
u 10,w0   0
w0   x 0
u 1t,wt   O1
0  t  L 
(9. 3)
wt  D
then or goal of deriving a second order approximation to x(t) will be achieved. We now want to
find a differential equation for w(t). In the derivation, we shall repeatedly use the identities
du 1 t,wt  u 1
dwt 

 u 1 t,wt 
dt
t
dt
 1
u 1 i
u t,vij 
v j

n
u 1 i t,wt  dw j t 
 1
dwt 
u t, wt 
  


dt i j 1
w j
dt 

1  
u 1
0
  f t,v  f v
t
(9. 4a)
(9. 4b)
( is the gradient operator.) Eq. (9.4b) is based on Eq. (9.2). Eq. (9.4b) leads to:
t
1  u t, wt     f ,wt   f
1
0
wt d 
0
u , wt 
du , w
1  
d  1  
d

d
0
0
t
1
t
1
(9. 5)
Nonlinear Dynamics
-2-
The result on the second line of Eq. (9.5) is trivial. From the last expression, we derive the
following identity:
t
u 1 ,w
dw
1
1
1  u t, wt   1  
 u ,w
d

d 

(9. 6)
0
Using Eq. (9.3) we have
du 1 , w 
du 1 , w 
dw dx 

 1  
  f , x   1  

d
d
d
d
du 1 , w 
 f , w  1  u 1 ,w   1  
d
(9. 7)
Using Eqs. (9.4a&b), Eq. (9.7) becomes
dw
 f ,w  1 u 1 ,w   f , w   f 0 w
d
dw
 1  u 1 , w
d

dw 
1   u , w d   f , w    u ,w   f ,w   f w (9.8)
1
1
1
0
1
This is a differential equation that w has to satisfy, to ensure that Eqs. (9.2) and (9.3) are
compatible. To obtain useful information from the equation, we need to find an estimate of
dw/d. We exploit the assumption that derivatives of f are properly bounded in the domain to
perform two expansions, keeping the first order term only in each expansion.
First, we expand f(,w()+1()u1(w())) on the r.h.s. of Eq. (9.8) through first order around
w(), to find
dw 
1   u , w  d   f w     f ,wu , w  O   (9. 9)
1
1
0
1
1
2
1
Second, we invert the nn matrix, [I+1u1], expand it in powers of 1 and, again, retain only
the first order term. (This matrix is regular, because the 1 term is small compared to the unit
matrix. Thus, we can invert it, and solve for dw/d.) We obtain an approximate differential
equation for w(t):
dw
 f
d
0
w   1 f ,w u 1 ,w  u 1 ,wf 0w  O 1 2 
Defining
f 1,w   f , wu 1,w   u 1,w f 0 w 
(9.10)
-3-
Y. Zarmi
Eq. (9.10) yields the approximate differential equation for w(t) that will serve our purpose:
dw
 f
d
0
w    1 f 1, w  O 1 2
(9.11)
From Eq. (9.11), we obtain and approximate integral equation for w(t):
t
wt   x 0    f
t
0
wd  1 f 1,wd  O1 2 
0
t  OL  
(9.12)
0
The O(12) term in Eq. (9.11) is bounded after integration over time by O(12)t. As the time
interval may be as long as O(1/), the overall error in Eq. (9.12) may be O(12).
Eq. (9.11) provides a hint about a way of solving approximately for w. It is an extension of the
ideas developed in Chapter 8 for first order averaging. If f1(t,v) has an average, f10(v) (in the
sense defined in Chapter 8, that is f1 were a KBM field), then we can find an approximation for
w by solving an appropriate averaged equation:
zÝ  f 0z    1 f 10z 
z0  x 0
(9.13)
The difference between z and w is determined by the mismatch between f1(t,v) and f10(v). (A
reminder: In first-order averaging, the difference between the exact solution, x(t), and the
solution of the averaged equation, y(t), was determined by the mismatch between f(t,v) and
f0(v).) The error estimate will be
 2   sup 
vD
0t  L  
t
 f ,v  f
1
10
vd
(9.14)
0
In the following, we show how these ideas can be used to obtain a better approximation for x(t).
9.2 Second order averaging theorem
Theorem
If for all xD and 0≤t≤(L/:
f(t,x) satisfies the Lipschitz condition;
f(t,x) is a KBM field with average f0(x);
f1(t,x) is a KBM field with average f10(x).
Then z(t) of Eq. (9.13) is an approximation for x(t), satisfying
x t   zt   1 u 1t,zt   O1 2  1  2 1 2
Proof
The proof goes in parallel to first order averaging.
We first introduce an intermediate stage between w(t) and z(t):
(9.15)
Nonlinear Dynamics
-4xÝ  f t,x 
Ý  f 0 w   1 f 1t, w  O1 2 
w
Ý  f 0    f 1 t,

1
t  O1  
zÝ  f z   1 f z 
(9.16d)
f1T is the local time average of f1 [see Eq. (8.2)]. We will now show that for t=O(1/)
  
w    O1 2   T 1 
  z  O 1 2  
  T 
w  z  w      z  O1 2
(9.16b)
(9.16c)
T
10
0
(9.16a)
  
  T 1   O 1 2 
  T 
(9.17)
Clearly, since T is arbitrary, the error is largest when T=O(21/2), yielding
w  z  O1 2  1 2 1 2 
(9.18)
We use this conclusion to obtain an estimate for
xt   z t   1  u 1 t, zt  
wt     u t, wt  zt     u t,zt  
1
1
1
(9.19)
1
wt   zt   1   u 1 t,wt   u 1 t,zt 
Note that, since f satisfies the Lipschitz condition, so does u1. We denote its Lipschitz constant
by . Employing the Lipschitz condition for u1 and Eq. (9.18), Eq. (9.19) yields
xt   z t   1 u 1t, zt   1  1  wt   zt   O1 2  1 2 1 2 
(9.20)
for t=O(1/). This is the desired result, Eq. (9.15).
We now have to verify Eq. (9.17). Consider, first, ||w(t)-z(t)||. From Eq. (9.16c), we have
t
t
t   x 0    f d   1   fT ,  d
1
0
0
0
Consider the second integral. By the definition, Eq. (8.2),
t
1 
1
 fT , d  d
T 
0
t
0
t
 T
t
1 
1
1
d  f s  , ds 
 f s,ds  T 
 0

T
0
1 
1
1
1
d  f s  ,  f s  ,s    f s  ,s    ds 
T  0
0
T


(9.20)
-5-
Y. Zarmi
T
t
T
s t
1 
1 
1
1
1
d f s  ,  f s  ,s   ds  ds  f , d 
T  0
T  s


0
(9.21)
0
t
1 
1
1
d  f s  ,  f s  ,s   ds 
T  0
T


0
T
s t
s

1   1
1
1
f
,



d

ds

f
,



d

f
,



d










T   t

0
0
t
0
As a result,
t
1 
1
1
1
1
 fT ,d   f ,d  d  f s  ,   f s  ,s    ds 
T
 0
0
0
t
T
t
0
I
T
(9.22)

1   1
1
ds  f ,  d   f , d
T   t

0
s t
s
0
II
Denoting the bound for f1 (for xD and 0≤t≤L/y M, and that f1 obeys the Lipschitz condition
with some constant, 1, Eq. (9.36) yields (I and II denote corresponding parts in the equation):
t
T
1 
 f ,d   f ,d  d  ds s      MT )
T  0
0
0
II
t
t
1
T
1
0
I
We employ Eq. (9.16c) for  to re-write the part denoted by I, to obtain
t
T
1 
1
1
 fT ,d   f ,d  d  ds 
T  0
0
0
t
t
s
 f s    f s,sds  MT
0
1
1
T

II
0
I
1
With N denoting an appropriate bound for fT , we finally obtain
1
1
1
 fT ,d   f ,d  2 1 M  1 N   M T  Const.T
t
t
0
0
(9.23)
Employing Eqs. (9.16b&c), we obtain
t
t
wt   t     f w   f d   1   f , w   fT ,d  O1
0
0
0
1
0
1
2

Nonlinear Dynamics
-6-
(The O(1()]2) error in Eq. (9.16b) becomes O(1()]2) after integration, since t may be as
large as (1/).) With the aid of Eq. (9.37), we now get
t
wt   t     f w   f d 
0
0
0
t
t
 1   f ,w   f , d   1   f ,  fT , d  O1 2 
1
1
1
0
0
t
t
  f
1
0
w   f 0  d   1  f 1,w  f 1, d 
0
0
 1  Const.T  O1 2 
(9.24)
Employing the Lipschitz condition for f0 and f1 (with Lipschitz constants 0 & 1) we obtain
t
wt   t    0  1 1  w    d   1  Const.T  O1
2

(9.25)
0
The Gronwall lemma, applied to Eq. (9.39), yields


wt   t    1 Const.T  O1 2  exp  0  1  1 t 


Const.  1 Const.T  O1 2 
t  L  
(9.26)
We complete the proof of Eq. (9.17), by finding a bound for ||z(t)z(t)||. Eqs. (9.16c&d) lead to
t
t
t   zt     f    f z  d  1   fT ,   f
0
0
1
0
10
z d
(9.27)
0
Next, expand, and use Eq. (9.14), to obtain
t
t
1
 f , d  T  d
0
0
 T
 f s,sds 
1
T
t
t
1

 T
1
 f , d  T  d  f s,s  f , ds
10
1
0
t
t
 f ,d   f , d
0


0
1
T
10
10
0

2  
  L
t 2 2
T
 T
t  L  
(9.28)
Now use the Lipschitz condition for f0 in Eq. (9.41) and obtain
t   zt 
t

 0    z  d 
0
The Gronwall lemma yields,
1  2  L
T
(9.29)
-7-
t   zt 

Y. Zarmi
1  2  L
  2  
exp 0 t  K 1
T
T
t  L  
K  L exp  L
(9.30)
0
This completes the verification of Eq. (9.17), thereby completing the proof of the theorem.
9.3 Specific Consequences
9.3.1 Improvement on first order
The additional information about f1 (that it is a KBM field, with average f10) enables us to
improve the first order averaging result. (This is similar to what has been found in the error
estimate theorem of Chapter 4, namely, that adding information about the relevant functions may
improve our results.) In fact, compare the three equations
(9.31a)
xÝ  f t,x 
x0  x0
0
(9.31b)
yÝ  f y 
y0   x 0
0
10
(9.31c)
zÝ  f z    1 f z 
z0  x 0
First order averaging states:

x  y  O 1  
12

t  L  
We now show how the information about can be exploited to improve this error bound. Based
on te result of the second order averaging theorem, we have [see Eq. (9.15)]
x  z  1u t,zt   O1  1 2
1
2
12
  O 
1
t  L  
(9.32)
Moreover, Eqs. (9.31b&c) lead to
t
zt   yt     f y   f
0
t
0
z d   1  f 10z  d
0

0
(9.33)
t
zt   yt    0  z  y  d   1   Nt
t  L  
0
where the Lipschitz condition was applied to f0, and f10 was majorized by its bound, N. The
Gronwall lemma now yields
zt   yt  
1   N
N
exp 0 t  1

exp  0 L  11   
0
0
O1 
(9.34)
t  L  
Eqs. (9.32&34) imply
x  z  x  y  y  z  O1 
(9.35)
Nonlinear Dynamics
-8-
instead of the O(11/2) result of the first order averaging theorem (Chapter 8).
9.3.2 Periodic case
When the function f(t,x) on the r.h.s. of our original problem, Eq. (9.16a), is periodic in t for
fixed x, we have z, [f1T and f10 coincide in Eq. (9.16c&d). Also, as shown in Chapter 8, in
that case 1()=. Therefore, in second order averaging, the error is O(2) [see Eqs. (9. 15&20):
xt   z t   1 u 1t, zt   O1 2  1  2 1 2   O 2 
9.3.3 Adding higher order terms to the equation (without proof)
Until now have we assumed that f(t,x) on the r.h.s. of our differential equation did not depend on
, so that higher order effects were all due to the expansion of the approximate solution. We now
introduce an explicit dependence on e in the equation, by adding higher order terms. Given
xÝ  f t,x    2 gt,x   3 Rt,x; 
x0  x0
(9.36a)
zÝ  f 0z    1 f 10z   2 g 0z
z0   x 0
(9.36b)
with f, g, R, bounded for xD and t≤(L/) by -independent bounds and where f, f1 and g are
KBM fields, then
x t   zt   1  u 1 t,zt   O1 2  1  2 1 2  3 1 2 
t
3  sup 
vD
t  L  
2
 g,v  g
0
vd
t  L  
(9.37)
0
The addition of an explicit 2 term (in addition to f10, which is generated by the expansion
method) yields an error of a similar type. Note that the additional bounded 3 term does not spoil
the degree of approximation over the given time scale [t=O(1/)]. Its contribution is bounded by
3Mt (M being a bound for R(t,x;)), which is O(2) for t≤(L/), while the errors in Eq. (9.37) are
at best O(2) (usually they will be larger, see examples in Chapter 8). Again, if f(t,x) and g(t,x)
are periodic in t with period T for fixed xD, then all errors in Eq. (9.37) are O(2).
9.4 First order approximation for t=O(1/2) - the periodic case
Until now we have dealt with the issue of improving the accuracy of the approximation from first
to second order, over the time interval t=O(1/). One may ask whether it is possible to extend
the approximation to longer time intervals (e.g., t=O(1/2)). With the tools we have developed,
this is possible, but there is a price to pay. Extending the validity to longer time scales comes at
the expense of the accuracy, unless, of course, one retains more terms in the expansion. We
discuss here the case of periodic functions, since it is of wider interest. We will show that when
-9-
Y. Zarmi
one takes into account all effects through O(2), then the same approximation that yields an
O(2) error for t=O(1/), also provides an O() error for t=O(1/2).
Consider, again, Eqs. (9.36a&b). In addition to the properties of f and g, postulated in Section
9.3, we require that the two are periodic with period T for fixed vD:
f t  T,v   f t,v
(9.38)
gt  T,v   gt,v
For t≤(L/), in addition to the result of Chapter 8:
1   
xt   yt   O 
we now also have
xt   z t   1 u 1t,z t   O 2 
From Eqs. (9.1&2), and the fact that 1()=, one may now write u1 as
t
t
u t,v   f ,v   f vd   f ,vd  t f v
1
0
0
0
(9.39)
0
Note that, for fixed v, u1(t,v) is also periodic in t with period T, because the average part of f(,v)
has been subtracted in Eq. (9.39):
u t  T ,v 
t T
 f , vd  t  T f
1
0
v 
0
t T
T
 f ,vd   f , vd  t  T f
0
0
v 
T


t
T f 0 v    f   T,v d  t  T f 0 v 
(9.40)
0
t
 f ,vd  t f
0
v  u 1 t v
0
As a result, u1(t,v) is bounded for all vD. This property of u1 is also a result of the fact that u1
is an integral over time of the deviation of f(t,v) from its average over a period. Suppose that, by
the time t, n whole periods T have elapsed, t=nT+t (t<T), Eq. (9.54) leads to
nT
u t  n T   t,v   f ,v d 
1
0
which is bounded.
nT   t
 f ,vd  n T   t f
nT
t
0
v   f ,v d   t f 0 v 
0
Nonlinear Dynamics
-10-
Employing Eq. (9.3) in Eq. (9.36a) for dx/dt, yields (here 1()=):
1
2
1
3
xÝ  f t,wt    u t,wt   gt,wt    u t,wt  O  
 ft, wt    2 wf t,wt u 1 t,wt   gt,wt  O 3 
(9.41)
In addition, Eq. (9.3) implies
xÝ wÝ  wu 1 t,wt w
Ý 
I   
w
u 1 t,wt 

t
u 1 t,wt wÝ  f t, wt   f 0 wt 
(Note that in this equation there is no approximation: Eq. (9.3) is assumed as an exact relation,
with w(t) unknown). Under inversion of the matrix I   wu 1t,wt  (I is the unit matrix),
this yields through O(2):


Ý I   w u 1 t, wt  xÝ  f t, wt   f 0 wt   O 3
w
(9.42)
The term in square brackets in Eq. (9.42) starts with O(). Therefore, there is no need to keep the
2 term in the inversion of the matrix. Substituting Eq. (9.41) for dx/dt, we find
Ý=  f 0 wt    2 f 1t,wt   gt, wt  O 3 
w
(9.43)
with f1 defined by Eq. (9.10).
We are now ready to show what additional assumptions are required to make z(t) [Eq. (9.36b)] a
good approximation for x(t) [Eq. (9.36a)] over an extended time scale, t=O(1/2). We begin with
x  z  x w  w z
(9.44)
||xw|| is easy to handle. From Eq. (.3), we have
x  w   u 1t, wt 
(9.45a)
As u1(t,w(t)) is O(1), we conclude that, for all times for which w and z exist:
xt   wt  = O
(9.45b)
For the second term, we employ the differential equations (9.36b) and (9.43), to obtain
t
t
w  z    f 0 w  f 0 z d   2  f 1 ,w   f 10 w  d 
0
0
t

2
t
 f w   f z  d    g, w  g w d 
10
10
0
2
0
0
t

2
 g w   g z   d  O t
0
0
0
3
(9.46)
-11-
Y. Zarmi
The second and fourth integrals in Eq. (9.46) are always O(1), since they are integrals over the
difference between a periodic function and its average. Therefore, these terms constitute O(2)
contributions for all times. For the remaining integrals, in the absence of additional information,
the best we can do is use the Gronwall lemma in the following manner. We use the Lipschitz
condition for f0, f10, and g0 with Lipschitz constants 0, 10, and g, respectively, to obtain:

 w  z d  O  O t
wt   zt    0   10   g 
2
t
2
3

0


O 3 
O 3 
2
2


wt   zt   
 O exp  0   10  g  t 
2
 0   2 10  g 
 0   10  g 

(9.47)
At best, this result implies


wt   zt   O 2 
t  O1  
(9.48)
One cannot beyond t=O(1/), as the exponential in Eq. (9.47) becomes unbounded. The obstacle
is the 0 term in the exponential, which arises from the O() contribution of the first term in Eq.
(9.46). To improve upon Eq. (.48), we must get rid of the 0 term. This is achieved by
assuming that f0(v), the period average of f(t,v) for fixed v, vanishes. Then the first integral in
Eq. (9.46) vanishes identically. So does the third integral, since f10, the average of f1(t,x)
vanishes. The fifth integral contributes O(2). Eq. (9.47) is now replaced by
O 3 

O 3 
2
2
2
wt   zt    2
 O exp  g t  2
 O 
t  O1/   (9.49)





g

g
provided w and z exist for this time scale. We now add the last required assumption:


The solution z(t) of Eq. (9.36b) exists for t≤L/2, with L a constant.
This implies that, for t≤L/2, w(t) has a unique solution, and is different from z(t) by O().
Consequently, a unique solution exists for x(t) over t≤L/2. Eqs. (9.45&49) imply
xt   z t   O 
t  O1/  2 
(9.50)
The analysis presented above is summarized in the following.
Theorem
Given
xÝ  f t,x    2 gt,x   3 Rt,x; 
zÝ  f 0z    1 f 10z   2 g 0z
x0  x0
z0   x 0
1) f, g and R are bounded for xD and t≤(L/2) by -independent bounds;
2) f and g are periodic in t for fixed x;
3) f0(x) (the average of f(t,x)) vanishes;
(9.51a)
(9.51b)
Nonlinear Dynamics
-12-
4) The averaged equation, Eq. (9.51b), has a solution, z(t) for t=O(1/2).
Then Eq. (9.51a) has a solution x(t) for t=O(1/2), and
x t   zt   O 
t  O1/  2 
(9.52)
9.5 Example for second order averaging - Van der Pol equation
Consider, again, the Van der Pol equation
Ý x   1  x 2xÝ
xÝ
0  «1
(9.53)
The transformation to polar coordinates yields


yÝ  x   1 x 2 xÝ


x  r cost   
y  r sint  
xÝ y


r   12 r 1  14 r 2  cos 2 t    14 r 2 cos 4 t    
d   

  
   f r, 


dt   1
1 2
1 2
 2 sin 2t     4 r sin2 t    8 r sin 4 t  

(9.54)
First Order . The averaged equation is obtained by integrating the r.h.s. of Eq. (9.54) with
respect to t, over one period (0≤t≤2). The averaged equation is, therefore,
2
d r˜  12 r˜ 1  14 r˜ 
    
   f
˜  
dt 
0

0
r˜, ˜ 
(9.55)
r˜ =0 is an unstable solution of the differential equation. Any small deviation of r˜ from zero will
cause r˜ to grow, because d r˜ /dt is positive near r˜ =0. The nontrivial limit cycle solution is:
2
r˜  
  
2
    1  4 r0   1 exp  t 
  
˜  

0
 


12





(9.56)
For long times, r˜ 2, independently of the initial value, r0. r˜ =2, is a stable fixed point: It
attracts the solution towards it from either side:
 0
dr˜
2
  12 r˜ 1  14 ˜r  
dt
 0
The result for x(t) is
r 2
r 2
(9.57)
-13-
x t  
Y. Zarmi
2
1  4 r  1exp t
12
2
cost   0   O 
t  O1  
(9.58)
0
Note that the limit cycle solution, 2cos(t+0), is not a solution of Eq. (9.53) even to O().
Second Order. We now need to calculate u1(t,x) (note that because this is a problem with a
periodic f(t,x), in the general formulation of the theory, 1()=).
t
1  u 1 t, x    f,x   f 0 xd 
0
t
12 r  cos 2 t     14 r 2 cos 4 t  



 
d 
1 sin 2 t    1 r 2 sin2 t    1 r 2 sin4 t   

 4

 8


2
0
 14 r sin2 t     sin2   321 r 3 sin4 t    sin 4 



 

 1

2
1 2
1
 4 cos 2 t     cos 21  2 r  32 r cos 4 t    cos 4 
(9.59)
Eq. (9.59) is based on the theory developed previously. In the case of periodic functions, one can
modify u1 so that it has zero average over a 2 cycle, by an appropriate choice of the of the zeroorder term (see next Chapter). This yields
 1 r sin2 t    1 r 3 sin 4 t   

4
32


1
 u t, x   

 1

2
1 2
1
 4 cos 2 t   1  2 r   32 r cos 4 t   
(9.60)
Using Eqs. (9.10,54,55,60), we find

f 10   1
 8 

2
4 
3
11
16 r  256 r 
0
(9.61)
The equation for r and  through second order is therefore,
1
1 2


2  1  4 r 
d r 
0
2 10

    f   f  
 2  1  3 r 2  11 r 4 
dt 



8
16
256

(9.62)
The solution for the radius, r , is the same as in Eq. (9.56) (no second order correction). An
explicit solution for  is cumbersome, and conveys no important information. However, the
behavior of the phase, , on the limit cycle, i.e., (t∞, r 2) is interesting. Eq. (9.62) yields
   161  2t   0
(9.63)
Nonlinear Dynamics
-14-
This is the updating of the phase for asymptotic times. As a result, we have on the limit cycle
r  r 
       u 1 r ,  O 2 
 



2
2
1
1
1
 1
2

  2 sin2 1  16  t   0  4 sin 4 1  16  t   0
 1 2
 
2
1
1
1
 16  t   0  
4 cos 2 t    8 cos 4 1  16  t   0






 O
2

(9.64)
Eq. (9.64) is not applicable for t0, if the initial radius, r0, differs from the limit cycle radius by
more than O(2).