-1-
Y. Zarmi
11. The method of multiple time scales [18-30]
11.1 Introduction
The method of multiple time scales works in many cases. From the computational point of view,
it is more cumbersome than the method of averaging. However, in this method, it is easier to
formulate the perturbation problem and to keep track of the terms that correspond to different
degrees of approximation. In addition, its application does not necessitate the passage to first
order equations as is required for the method of averaging. For example, one can apply it
directly to the second order equation of a perturbed harmonic oscillator, without having to go
first to two first order coupled equations for the unknown, x, and its velocity, dx/dt, or to
equations for the polar coordinates r and . The method is applicable also to problems with
aperiodic transients, where use of the method of averaging is especially inconvenient, or often,
impossible. As we shall later on show, the two methods can be shown to yield the same results
at least through second order in the case of problems with periodic functions [30].
In many situations, the solution of a dynamical problem, which is affected by a small parameter
, depends on time and on  through combinations of the type Tntn (n=0,1,2....). Each Tn
represents a different time scale. For example, when T0t is of O(1), then T1t is O(), close to
zero. The higher time scales are then even closer to zero. If these time scales are well separated
from each other, one assumes that they are independent variables. This is similar to what one
does in the method of averaging. Instead of freezing the slow unknown variables when
integrating on time, here one assumes that while a processes occurs on a fast time scale, the
slower time scales are frozen. (Just like the slow motion of the minute and hour hands of a clock
when the second hand has moved just a few seconds). One also assumes that the dependence of
the solution on all time scales is analytic near Tn=0.
Consider the system
dx
   n f  n  t, t, 2 t,;x 
dt n1
(11. 1)
Now define the independent time variables
Tn   n t
(11. 2)
Now introduce the relation
d





dt T0
T1
 D0   D1 

 
Dn 

Tn 

(11. 3)
Define
f  n t, t, 2 t ,;x  f  n T0,T1,;x 
(11. 4)
T0 ,T1,T2 ,  x 1 T0 ,T1,T2 ,   2 x  2 T0 ,T1,T2 , 
(11. 5)
and expand
x T0 ,T1,T2 ,  x
 0
Nonlinear Dynamics
-2-
The independence of the new time scales introduces spurious freedom, which is exploited to
eliminate secular terms in the expansion, order by order. This ensures that x(n) is bounded
relative to x(n1) for all n up to a desired order. For an approximation with an error of O(n+1)
over a time scale of O(1/m), one has to include in the series terms as follows:
 0
 1
n  n
n1
x  x T0 ,,Tn m    x T0 ,,Tn m1     x T0 ,,Tm   O 
(11. 6)
t  O1/  m 
For example,
t  O 0 
x  0  T0   O 

 0

x T0 ,T1   O 
x    0 
x T0 ,T1 ,T2   O 
  0 
1 
2

x T0 ,T1 ,T2   x T0 ,T1   O 
t  O1 /  
(11. 7)
t  O1 /  2 
t  O1 /  
Now we insert the expansions (11.3&5) in Eq. (11.1)
D
0
  D1   2 D2 
x     x    
0
1
2
x  2 
 f
 1
  2 f  2  
(11. 8)
and compare terms order by order, to obtain through O(2):
D0x  0  0
(11. 9a)
D0x  1  D1x  0  f  1
(11. 9b)




D0x  2  D1x  1  D2x  0  f  2 T0,T1, ;x  0  xf  1 T0,T1, ;x  0 x  1
(11. 9c)
The formal solution for x(0) and x(1) is readily obtained as
x  0  x  0 T1,T2 ,
x
1 
  D1 x
 0
(11.10a)
T0
T1,T2 , T0   f  1 ,T1,d  x˜ 1 T1,T2,
(11.10b)
0
x(0) is independent of T0, and xfi(1) is a "constant of integration", also independent of T0. The
solution for x(1) leads to the evolution of a secular term in T0, unless f(1) has an average f(1)0, and
D1x 0T1, T2 ,
  f 10
The solution for the second order equation, Eq. (11.9c), is
(11.11)
-3-
x
2
 x˜ 
2
Y. Zarmi
T1,T2, 
T0


f  2  ,T1 ,T2 ,; x  0  T1 ,T2 , 


 x f  1 ,T1 ,T2 ,; x  0  T1 ,T2 ,  x 1  ,T1 ,;x  0  T1 ,T2 ,

D x  1  ,T ,;x  0  T ,T ,
1
1 2
 1







 d





0
T0  D2 x  0  T1 ,T2 ,
Employing Eqs. (11.10a&b) and (11.11), this becomes
x  2   x˜  2  T1 ,T2 , 
T0
 f   ,T ,;x     f   ,T ,;x    ˜x  T ,d 
2
0
1
1
0
x
0
D x˜
D1f
 1 0
T1 ,T0
2

1
 1
T1 ,  D2 x  0  T1 , T0 
 T
  D1 f  1  ,T1 ,;x 0  d 
1
1
2
1
1
T0
0


0
(11.13)
T0



 1
,T1 ,;x  0   f  1 s,T1 ,;x  0 ds  f  1 0   d
 x f

0





0
To avoid secular behavior, the sum of all the terms that may be proportional to T02 must vanish.
They occur in the last two lines of Eq. (11.13). Hence, we must require:


 D1 f  1 0 T1 ,; x 0  
T0

 1
,T1 ,;x 0  d 
 T0    D1f


0


2 T
lim 2  0
 0

T0   T


0 

 1
,T1 ,; x  0   f  1 s,T1 ,;x  0  ds  f  1 0   d 
 x f

0
 

0






(11.14)

If f(1) is a KBM field (as we have already assumed), and the limits in Eq. (11.14) exist, then this
constraint is an identity in most cases (certainly, in the case of periodic functions, where the
average is computed over one period). The only real constraint is the requirement of the
vanishing of all the terms that are linear in T0. These include the terms in the second and third
lines of Eq. (11.13) as well as nonleading contributions to the last two lines. These are not
known in general. Thus, there is no general form for the required constraints.
In the lowest order, the result parallels that of the method of averaging of Chap. 10:
Nonlinear Dynamics
-4-
x  0  T1 ,T2 ,  t 
D1 x  0   f 1  0
(11.15a)
x
1 
d
1
   M     
dt
T0
T ,T ,; x   f   ,T ,;x    f   d  ˜x   T ,T ,
 0
0
1
0
1
1 0
1
1
1
2
0
(11.15b)
 1

t,

T0
 f   ,  M   d
1
1

q 1 
0
The zero order depends on time only through the slow time scales, and the dependence is
determined by f(1)0, the average of f(1). The dependence on the fast time scale, T0, appears only
in the next order. In principle, one may go on to higher orders. However, as we have just seen,
there may not be a simple formulation for the higher orders in the general case.
In the case of periodic functions the formulation is simpler and can be carried out systematically
to all orders. Moreover, Morrison [30] has shown that the method of multiple time scales and
the method of averaging are equivalent in first and second orders in the case of periodic
functions.
11.2 Periodic problems: Equivalence of Multiple time-scales and Averaging
Consider a perturbed harmonic oscillator
Ý x   f x, xÝ;   0
xÝ
(11.16)
In polar coordinates this equation becomes

 


sint    f  1 r cost   , r sint     2 f  2  
rÝ 
   cost   
Ý 
 1
2
 2

 f r cost   , r sint     f


r
 


(11.17)
 f 1   2 f 2  
We first derive the relevant expressions in the method of averaging.
dependence appears only through t+. Defining the slow variable
˜r 
   
˜ 

we find M(1) of the method of averaging
The explicit time
(11.18)
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M 1 
Y. Zarmi

  1 2

 1
P1 r˜     sin  f r˜ cos ,r˜ sin  d 

 2 0

 
  
2

1
  1 cos  f  1 r˜ cos , r˜ sin d

 Q1 r˜  2 ˜r 

r˜
 
0
r˜ 
 P1 r˜  
d
 
 1
Ý

     M    1

dt  
Q r˜ 
r˜ 1  
˜ 

 
t
1
(11.19)
(11.20)
 sin  f  1  r˜ cos ,  r˜ sin   P1 r˜  
 1
 r  1 
 (11.21)
  1  f t,  M  1   cos   1
1



t  
f r˜ cos , r˜ sin   Q1 r˜ 
 r˜

r˜
Observing the identity



t 
we can integrate Eq. (11.21) over , since its r.h.s. is known. This yields

 1
r 1   sin  f 1  ˜r cos , r˜ sin  P r˜ 

 1 
  
d  hr˜  
   
cos  f  1 r˜ cos , r˜ sin    1 Q r˜ 
  1  
1

r˜
r˜
 
0
r 1  , r˜  
 
 1  ˜ 

 , r 
  1 r˜ 
 
  1 ˜ 

 r 

(11.22)

  1 t,
q  1 
Here h(rfi) is an "integration constant", and the notation is that of the method of averaging. Also,
we choose r¶(1) and ¶(1) to be periodic in =T0+(0) with zero averages over one period (2):
2
 1
 r , ˜r d 
2
 
  , r˜ d  0
0
1
(11.23)
0
The second order calculation in the method of averaging yields
M
with
 2
f
 2 0
1

2
2
  f

0
 1

  1     1 f  1 0 d
(11.24)
Nonlinear Dynamics
-6-
f  2 0
 1 2 

 2
d 
  sin  f
2  0

 
2

1
 2

 ˜  cos  f d

2  r 0

(11.25)
Note that here the slow variable, , is the vector (rfi,fi). For the evaluation of M(2) of Eq.
(11.24) we note the following.
 f  1

f  1
sin


˜r

 
f  1
 cos   1

f

cos


2
 r˜
r˜
cos  f

sin
f
r˜
 1
1 
f 1  

 

cos  f 1  


r˜
 
 sin
 r  1   1
r  1 
 ˜ 

r
r˜
 

    1  

 1
  1
 1 




 r˜
r˜
 
2
2
1
1
1
f  
 1
sin  f d 
cos 
d  P1 r˜ 
2 0
2  0

2
1
cos  f
2 0
 1
2
2
 cos  f
1 
2
1
f  1
dP
sin 
d  1

2 0
˜r
dr˜
2
1 
 d  0
0
(11.27)
1
f 1 
d  
sin
d  Q1 ˜r 
2 0

1
f  1
dQ
cos 
d  1

2 0
˜r
dr˜
2
(11.26)
 sin f
1 
1 
 d 
0
(11.28)
2
r 1  1
   d
0
Eqs (11.22,23,25-28) yield for Eq. (11.24) the result
M  2


dP1  1
d  1 
P



P
 2

1
d˜r
d˜r

 
Q2
d Q1   1
d  1 
   P1
 

 r˜
dr˜  r˜ 
d˜r 
2

1
P2 
sin  f

2 0

2

1
Q2 
cos  f

2 0

 2

f  1  1 f 1  1  
r

 d
˜r


 2

f  1  1 f 1  1  
r

 d
˜r


(11.29)
-7-
Y. Zarmi
Now we turn to the method of multiple time scales. In Eq. (11.17), expand
r  r  0    r 1    2 r  2  
(11.30)
    0      1   2   2  
Employing Eq. (11.3) for the time derivative, we obtain the following equations order by order.
D0 r  0   0

r  0   r  0  T1 ,T2 ,
D0   0   0

  0     0  T1 ,T2 ,
(11.31)
r   cos , r   sin
cos 
D     D       f   r   cos , r   sin 
r
D0 r  1  D1 r  0   sin  f
1
 1
0
0
0
1
1
0
0
(11.32)
0
0
Thus, Eq. (11.31) implies that r(0) and (0) are independent of the fast time variable T0 (t).
Therefore, to avoid secular terms (i.e., terms proportional to T0) in r(1) and (1), we must have
 
 r
r  0   P1 r  
D1   0   
 0
  
Q1 r
0


 0  

(11.33)
which is the equivalent of Eq.(11.20) in the method of averaging. The solution for r(1) and (1) is
thus of the from
   
, r   r 
r  1  r 1  ,r  0     1 r  0 
 1    1
 0
 1
(11.34)
 0
which is identical in form to the solution in the method of averaging, Eq. (11.22), with r(1) and
(1) being periodic functions of =T0+(0).
The second order equations read
f  1   1
D0 r
 D1 r  D2 r
 sin  f
 cos   f
 sin   0  r 
r
f  1  1
sin 


cos 
sin 
cos 
D0   2   D1   1  D2   0    0  f  2    0    1 f  1    0  2 r  1 f  1 
r
r
r
 2
 1
 0
 2
 1
 1
 

cos  f
f
1 

 1  
 0 
 0 r
r

r

 1
  T
0
   0

 1
(11.35)
Nonlinear Dynamics
-8-
In Eq. (11.35) f(i) (i=1,2) are calculated at the point (r(0)cos,-r(0)sin). Now we can take
advantage if the freedom gained by the introduction of the many time scales. To avoid secular
terms in T0 in the solution of Eq. (11.35) for r(2) and (2), the sum of all terms in the equation
which may lead to a linear T0 dependence must vanish:
2
D2 r 0  

  2  f  1  1 f 1   1 
1 
  0 r 
  cos  f  1   1  D1r 1  d (11.36)
sin  f
2  
r




0
2
 0
D2 


1 cos  f



2 r  0  

 2
 
 
 d

 r 0  D1  1 
f  1  1  f  1 1 
1
  0 r 
   0 f
r

r
sin  f
 1
  1
 1
r
1 
(11.37)
0
We now use the identity
2
1
1
D1r 1 d 

2 0
2
2
 D r
 1
1

 D1 1 d 
0
2
1 r 1 
r 1 
d 1 
 0
 0
 0  
D
r

D


D
r

d 
2 0 r 0  1
 1
dr 0  1

 
(11.38)
2
1 
 1
Q1 r  0  1 2  r  1
1 r  1
 0  d
 0  d
P1 r
d

d

P
r

P
r
0
0
1
0
1
0
2  0 r  
r   2 0 
dr 
dr 
Two terms in Eq. (11.38) vanish: the first one - owing to periodicity (see Eq. (11.23)) and the
second one - trivially. In addition, we note that
 
 0
2
 
2

 

 
1
1
cos f  1  1  d 
cos f  1  1  1  d  Q1 r  0   1 


2 0
2 0
(11.39)
where Eqs. (11.28) have been used. Employing Eqs. (11.38&39), Eq. (11.36) becomes
D2 r
 0
 P2  Q1 
 1
d  1 
 P1  0 
dr
(11.40)
Similarly,
2
1
d  1
 1
D  d  P1  0 
2  0 1
dr
(11.41)
and
2
1
 cos  f
2 r   
0
2
0
1 
r
 1
1
d 
 0
2 r
2
 sin f
0
 1
  1 d 
Q1
r 
 0
2
  1 
P1
r
 0
  1 (11.42)
-9-
Y. Zarmi
With Eqs. (11.41&42), Eq. (11.37) becomes
D2   0  
Q2
d 1  P1  1  Q1  1 

P
  0   0 2
1
r  0
dr  0 
r
r
(11.43)
The methods of averaging and of multiple time scales will be equivalent through second order if
M(2) of Eq. (11.29) and Eqs. (11.40&43) are equivalent:
 0
1
P2 r˜ 
dr
P1 ˜r

   Q    r   
dM1  r  0
  P r 
P2 r
 0
 0
1
1
M
0
(11.44)
dP1 r˜  1

A ˜r
d˜r
1 
dA
d˜r
   P r d r  P r  r  Q r  r  
r
dr
r
r 
Q2 r  0 
 0
 1
M
 0
 0
 0
1
 0
1
1 
M
 0
 0
 0
1
1 
M
2
 0
 0
(11.45)
Q2 r˜ 

r˜
P1 r˜ 
d A r˜ 
d˜r
1
dQ1 r˜   A r˜ 
d˜r
r˜
1 


Q1 r˜  A r˜ 
2
r˜ 
 1
In Eqs. (11.44&45) the subscript "M" stands for multiple time scales, and "A" Stands for
averaging. These equations can be satisfied by the choice
r˜  r  0 
1 1 
A 
 M    A    M   0
1
A
1
1
In particular problems for which f(1) satisfies
Q1
dQ1
dP
 P1 1
dr˜
dr˜
one can also have
r˜  r  0 
A1  M1  
 A1   M1  
dQ1
  P1 
dr˜
dP1
  Q1 
dr˜
11.3 Example - The Duffing equation
Consider again the Duffing equation
Ý x   x 3  0
xÝ
Expand the solution x
(11.46)
Nonlinear Dynamics
-10-
x  x
n
 n
T0 ,T1,
(11.47)
n0
and the time derivative in terms of the multiple time scales
d2
2
2
2
2  D0   2 D0 D1   D1  2 D0 D2  
dt



n
Tn   t
Dn 
Tn


(11.48)
The resulting equations for x(n) through second order are:
D0 2 x  0  x  0  0
(11.49a)
 
D0 2 x  1  x  1  2 D0 D1 x  0  x  0
3
(11.49b)
 
D0 2 x  2  x  2  2 D0 D1x  1  D1 2  2 D0 D2 x  0  3 x  0 
2
x  1
(11.49c)
Eq. (11.49a) implies that x(0) is independent of the slow time scale, T0:
x  0  AT1 ,T2 ,e i T0  c.c
(11.50)
where "c.c" stands for complex conjugate of the expression to the left. Inserting Eq. (11.50) in
Eq. (11.49b) we obtain
D0 2 x  1  x  1  2 i D1 A e i
 A 3 e 3 i T0  3A 2 A e i T0  c.c.
T0
(11.51)
where a bar over a symbol denotes its complex conjugate. The solution for x(1) will develop
secular terms unless all resonant contributions on the r.h.s. of Eq. (11.51) (that is, all
contributions with an exp(iT0) dependence on T0) cancel. Hence,
2 i D1 A  3 A 2 A  0
2 i D1 A  3 A 2 A  0
(11.52)
Multiplying the first of Eqs. (11.52) by A¶ and the second by A, and adding the resulting
equations, we find
D1 A  0
2
(11.53)
implying that A depends on T1 only through its phase:
A  A T2 ,e
i  T1 ,
From Eq. (11.52) we find that  satisfies the equation
(11.54)
-11-
D1 
3
2
A
2

Y. Zarmi

3
2
A T1   0 T2 ,
2
(11.55)
Thus,  depends linearly on the slow time scale, T1= t. If we write
x  x0 cosT0   
x0 T2, 
A
(11.56)
  38 x0 2 T1   0T2,  38 x 0 2  t   0 T2 ,
(11.57)
1
2
then
generating a frequency correction identical in form to that in previous chapters (e.g., Eq.
(10.47)).
Once the terms that could have led to a secular behavior have been eliminated from Eq. (11.51),
the solution for x(1) can be written as a sum of a solution of the homogeneous part of the equation
and a particular solution of the inhomogeneous equation as
B  BT ,T ,
x 1  Be i T0  18 A 3 e 3 i T0  c.c.
1
2
(11.58)
Inserting Eq. (11.50) for x(0) and Eq. (11.58) for x(1) in Eq. (11.49c), we obtain
D0 2 x  2  x  2 
2 D0 D1 18 A 3 e3 i T0  Be i T0   D1 2  2 D0 D2 A e i
T0
18 A 5 e 5 i T0  14 A 4 A  A 2 Be 3 i T0 

 3 
3
2
2
i T0
1


A
A

2
A
A
B

A
B
e
 
 8
(11.59)
 c.c.
Elimination of the resonant terms (to avoid secular terms in the solution for x(2)) requires:
2 i D1 B  D1  2 i D2 A  6 A A B  3 A B  38 A A  0
2
2
3
2
(11.60)
and a corresponding complex conjugate equation. Utilizing Eq. (11.52) we find
D1 2 A   94 A 3 A 2
(11.61)
Eq. (11.60) and its complex conjugate now become
2 i D1 B  2 i D2 A  6 A A B  3 A 2 B  158 A 3 A 2  0
2 i D1 B  2 i D2 A  6 A A B  3 A 2 B  158 A 3 A 2  0
(11.62)
Multiplying the first of Eqs. (11.62) by A¶ and the second by A, adding the resulting equations
and using Eq. (11.52) repeatedly, yields
D1A B  A B  D2 A  0
2
(11.63)
Recalling that |A| does not depend on T1, a secular term in T1=t will appear unless each term in
Nonlinear Dynamics
-12-
Eq. (11.64) vanishes separately:
D2 A 2  0
D1 A B  A B   0
A  A T3 ,

(11.64)
A B  A B  F T2 ,T3 ,

with F  a real function.
From Eqs. (11.53, 54, 55, & 64) we deduce that the amplitude B can depend on T1 only through
its phase with the same dependence as in A:
B  B T2,T3,e
i 

3
2
A T1  0 T2 ,
2
(11.65)
Employing Eqs. (11.54, 55&65), Eq. (11.62) yields
D20  3 A B cos0   0   15
0
16 A
4
(11.66)
The solution now depends on the choice of |B| and 0. In other words, it depends on the choice
of the homogeneous term in x(1). This freedom will affect the choice of the zero-order term and
the form of the second-order contribution to the frequency. Such freedom is encountered in all
expansion methods. For instance, choosing B=0, Eq. (11.66) yields
15
˜
˜
 0   15
16 A T2  0 T3 ,   256 x0  t  0 T3 ,
4
4
2
(11.67)
This gives for x(t) the approximation
x  x  0  T0 ,T1 ,T2 ,   x 1  T0 ,T1,T2 ,  O 2  


15
˜0 
x0 cos t 1  38  x 0 2  256
 2 x0 4  
1
32

 x 0 3 cos 3 t 1  38  x 0 2 
15
256
˜0
 2 x0 4   
(11.68)
 O 
2
t  O1/ 
Let us now impose a simple initial condition: the turning point
x 0  a
xÝ0  0
(11.69)
Satisfaction of this condition implies
˜0 0

x0 
1
32
 x0 3  O 2  a

x0  a  321  a3  O 2 
(11.70)
Thus, the convenient choice B=0 is not the most appropriate one for initial conditions as simple
as the turning point of the trajectory. The reason is that satisfaction of the initial conditions
involves contributions from higher orders in the expansion. If, on the other hand, we require
B cos  0   0    18 A
which is obtained, for instance, by the choice
3
-13-
B  A A   A e
1
8
2
3
1
8
Y. Zarmi
3
i A T1  i  0
2

 0  0  
then Eq. (11.66) yields
21
21
˜ 1T3,   256
˜ 1T3 ,
 0   16
A T2  
x 0  t  
4
4
2
(11.71)
With this choice we find
x0  a
x 1 t  0   xÝ 1  t  0  0
(11.72)
˜ 1 T3 ,  0

Thus, a different choice of the homogeneous term in x(1) enables us to satisfy the initial
conditions by the zero-order term alone.
11.4 Example - The Van der Pol equation
Consider now the Van der Pol equation
Ý x   1  x 2xÝ
xÝ
(11.73)
With the expansions (11.47&48), Eq. (11.73) yields the following equations through second
order.
D0 2x  0  x  0  0
2
D0 x
 1
x

 1

 1x
D0 2 x  2  x  2  1  x  0 
2
 0 2
D x
0
D x
0
 1
 0
(11.74a)
 2 D0 D1x
 0
(11.74b)

 D1 x  0   2 x  0  x  1 D0 x  0 
2 D0 D1 x
 1
 D1  2 D0 D2 x
2
 0
(11.74c)
Eq. (11.75a) is solved by
x  0  AT1,e i T0  c.c.
(11.75)
Insertion of Eq. (11.76) in Eq. (11.75b) yields
D0 2x  1  x  1  i A  A 2 A  2 D1 Ae i T0  i A 3 e3 i T0  c.c.
(11.76)
Requiring that no secular terms evolve we obtain the "secularity" condition
2 D1 A  A 2 A  A  0
(11.77)
Nonlinear Dynamics
-14-
Writing A as
A Aei
(11.78)
Eq. (11.77) yields
  A
D1 A
A
2
4
 A 0
2
A0
A 1 expT  exp T 
1/ 2
0
1
D1  0

  T2 ,
A0  A0 T2 ,
(11.79)
1
The amplitude |A| tends to 1 when T1∞. Writing x(0) in terms of a trigonometric function, as in
Eq. (11.56), we find that in the zero-order, the amplitude x0 tends to 2. Thus, again, as we have
already seen in both the method of normal forms, and the method of averaging, the motion,
which does not start as a periodic one, goes asymptotically to a limit cycle the radius of which
equals 2 in the zero-order. Now, Eq. (11.76) can yield a solution for x(1) which is free of secular
terms:
x 1  Be i T0  18 i A 3 e 3i
T0
 c.c.
(11.80)
Employing Eqs. (11.67&80) in Eq. (11.75c) we find
D0 2 x  2  x  2  58 A 5 e5 i T0
 54 A 2 D1 A  83 A 3  3 i A 2 B  34 A 4 A e 3 i T0
2

2 i D1 B  D1 A  2 i D2 A  i B  D1 A 
 i T
  1 3 2
e 0
2
2

 8 A A  i A B  2 A A B  D1 A A  

(11.81)
 c.c.
Now require that the coefficient of exp(iT0) vanishes. Employing Eq. (11.77) we find
D1 2A  14 A  A 2 A  34 A 3A 2
(11.82)
With Eq. (11.82), the secularity condition becomes
2 i D2 A  14 A  A 2 A  78 A 3 A 2  2 i D1 B  i B  2 A A B  i A 2 B  0
(11.83)
Writing A as in Eq. (11.78), we obtain the following equation for |A|:
D2 A  D1A B  A B  A B  A B 1  A A   i A A A B  A B   0
2
(11.84)
A possible choice is B=0 (no homogeneous term in x(1)). The equations |A| and for  are now
D2 A  0

A  A T1 ,T3 ,
(11.85)
2
4
2
4
D2    18  12 A  167 A

   18  12 A  167 A T2   0 T3 ,


-15-
Y. Zarmi
For very large T1, when A tends to 1, the T2 dependence becomes
   161 T2   0   161  2 t   0
(11.86)
which is the same as the result we found in the method of averaging.
The Van der Pol oscillator is not an energy conserving one. Consequently, fulfillment of the
initial conditions is not of great interest, since eventually, the system converges onto the limit
cycle. This is why the choice B=0 is rather convenient here, despite the fact that it is not the best
one for imposing the initial conditions. In general, the homogeneous terms in each order can be
chosen arbitrarily (with one constraint  that they do not spoil the quality of the approximation).
The choice is a matter of convenience. As an example, if we choose B=A, then, with the help of
Eq. (11.79), Eq. (11.83) yields for the amplitude:
D2 A  0
A  A T1, T3,

2
(11.87)
and for the slowly varying phase:

D2   
 

1
8
1
8

3
2
A  167 A
2
 23 A 
2
7
16
A
4
4

T


A 1
2
15
16

  0 T3 ,
(11.88)
 t   0 T3 ,
2
If, on the other hand, we choose B=(i/8)A2A¶, then Eq. (11.83) yields
D2 A 
2
1
4
A 0
6

A 
2
A0
2
1
T1.T3,
1
2
4
(11.89)
A0 T2
Thus, what seems to be an appealing choice turns out to be valid only for T22t which is not too
large. For T2=O(1/|A0|4) (or t=O(1/(2|A0|4)) the choice leads to a divergent result.