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Y. Zarmi
6. Lie Transformations and perturbation theory
6.1 Where the problem lies
The structure and efficiency of a perturbation expansion depend on the choice of the zero-order
approximation. As an example, consider the one-dimensional dipole. The potential is given by:
V x  
1
1

x xd
(6. 1)
For a point P, far from the dipole (|x|»d), we define the small parameter (d/x). The structure of
the expansion in powers of  depends on the choice of the origin. If we place the origin at the
positive charge (the expansion is performed around x), V(x) is expanded in powers of  as:
V x  

1   2 

x

(6. 2)
If we change the location of the origin to a point at a distance d (0≤≤1) from the positive
charge, then the coordinate of the point P is x()x+d. As the definition of the distance of P
from the origin has changed, we also modify the definition of the small parameter to
  
d
x
(6. 3)
(the new  differs from the old one by an O(2) term). The expansion around x() becomes
V x  

1  2  1  3 2  3  1 2 
x 


(6. 4)
Fig 6.1 Choice of origin in one-dimensional dipole
=1 (origin at positive charge), corresponds to the expansion in Eq. (6.2). For =(1/2) (origin at
mid-point between the two charges), the second-order term in Eq. (6.4), vanishes and the
correction to the lowest order term is only O(3). A very special choice of the origin is:
x   x  x x  d      0  x  x  d
for which the expansion is terminated beyond first order:
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Y. Zarmi
 0 
(6. 5)
x
Thus, the structure of the expansion depends on the choice of the origin. In any expansion, the
lowest order term will have the same functional form, it's actual magnitude varying only by
second order corrections. The form of higher-order terms depends on the choice of the origin.
V
The issue (so trivial here) of:
1) The choice of the zero-order approximation
2) The interrelationship between this choice and the structure of the expansion
plays an important role in the perturbative analysis of nonlinear systems. In the latter, the
manner in which the initial condition is satisfied is also related to these two questions. However,
the issue is often masked by the complexity of the problem, leading to confusion when
rediscovered.
6.2 Lie transformations
The most general approach to this issue goes back to the work of Poincaré [13]. There is a
wealth of literature on this topic, and we mention here only four sources [14-17], where further
references can be found. Given a problem which depends on a small parameter
d
x  Xx;    k Xk
dt
k0
x, Xk R n
 «1
(6. 6)
Xk are assumed to have no explicit time dependence. We assume for x(t;) a near-identity
transformation
x t;   yt;    T1y  O 2 
T1 R n
(6. 7)
and look for a zero-order approximation, y(t;), such that the expansion of the solution x(t;)
around y(t;) has properties that suite our needs. This expansion will depend on the selected zero
order. y(t;) will satisfy a new equation
d
y    k Yk
dt
k 0
y, Yk  R n
(6. 8)
with y chosen so that Yk(y) satisfy some requirements. For instance, naive perturbation theory
corresponds to the choice in which y(t;) is the solution of the unperturbed problem
d
y  Xy;  0   X0 y
dt
(6. 9)
In that case y does not depend on  and Yk are given by
Y0 y  X0y
Yk1y   0
(6.10)
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Y. Zarmi
Other choices might be, for instance, that no secular terms develop up to some prescribed order
in . We shall soon see what this means.
We now insert Eqs. (6.7-8) in Eq. (6.6), expand all terms in powers of  and functions of y and
carry the analysis through first order, so as to find the constraints that Y0(y) and Y1(y) must
satisfy. Comparing coefficients of powers of , we find through O(1):
Y0 y  X0y
Y1y  X1y   X0,T1
where the square brackets is the Lie bracket, defined for two vector-functions as:
i
 k F i
i
k G 
F,G

G

F
   

 y k
y k 
F, G  R n
(6.11)
(6.12)
The following is a possible extension of these results to higher orders. Consider, first, a scalar
function f(x) that has a Taylor expansion:

am dm
f y  a  
m f y
m  0 m! dx
(The displacement a is constant). This can be formally written as a translation operation:
 d 
f y  a  exp a f y 
 dy 
Now generalize to n-dimensional vectors x and y and a displacement, a=xy, which is a function
of y, to be expanded in powers of  (a small parameter). A possible generalization of the above
is:
x  exp DT y
(6.13)
where DT is the generator of the transformation, given by
n
DT   Ti y; 
i 1

y i
Ti y;     Ti,k y
k
(6.14)
k 1
where i denotes the component of the vector, and k  the order in the expansion. This is a near
identity transformation. For 0 it goes continuously to the identity (y goes continuously to x).
For conservative systems the transformation is canonical [17].
Using Eqs. (6.6), the relations resulting through O(4)) are:
Y2  X2 
Y3  X3 
1
2
1
2
X1  Y1,T1  X0 ,T2 
X2  Y2,T1 12 X1  Y1,T2  121 X1  Y1 ,T1 ,T1  X0 ,T3
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Y4  X 4 
1
2
Y. Zarmi
(6.15)
X3  Y3 ,T1  12 X2  Y2 ,T2   12 X1  Y1,T3 
1
12
X
 X
 Y2 , T1 ,T1  121
2

 Y1 ,T1 ,T2  X0 ,T3 
1
All the functions in Eq. (6.15) are n-dimensional. The indices denote the order in .
6.3 Naive perturbation theory
For a naive expansion the generators Tik(y) are chosen so that y obeys the unperturbed equation
(the limit =0), Eqs. (6.9-10). As an example, consider the constraint Y1(y)0:
n
 X
T 
(6.16)
X1 y   Tk ,1 k0  Xk , 0 1k   0
y
y 
k 1 
Here the index k corresponds to the component of the vector, and 0,1 correspond to the order in
. Next, assume that X0, Y1 and T1 are given by expansions in monomials in powers of yk:
X1 y  X 0 ,T1  0
X k,0 
a
k , j1
j1 ,
jn

y1 j1
b
Xk ,1 
yn j n
, jn
k , j1
j1 ,
Tk,1 
 ck , j1
j1 ,
jn
y1 j1
yn
jn
y1 j1
y n jn
, jn
(6.17)
jn
, jn
Inserting Eq. (6.17) in Eq. (6.16) we obtain an algebraic equation for the coefficients
bk, j1
jn

 q  c
l
m  q   j
1  n,  l
ml  q l  1 jl
l, m1
ml
mn
ak, q 1
ql
qn
 ml  al, q1
ql
qn
ck , m1
ml
mn
 0
(6.18)
This is a set of linear equations for the coefficients, c, of the expansion of T1. When the set is
singular (the determinant of the coefficient matrix of the set vanishes), the unknown c cannot be
determined. Of particular interest is the case of a linear lowest order term, X0:
X0  Ax
(6.19)
and a nonlinear perturbation in the higher orders only. Without loss of generality, assume that A
is diagonal. If not, a transformation can be found that transforms A into a diagonal, or a
triangular matrix (the proof can be extended to the latter):
1

A  
 0

0 



 n 
(6.20)
Eq. (6.18) now becomes
bk, j1
jn
 ck , j1
jn
  j, 0
k
n j, 
j,    jl l
l1
(6.21)
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Y. Zarmi
Thus, whenever one of the eigenvalues of the linear part of the problem resonates with a
combination of the others (k is equal to a combination of all l with any array of n non-negative
integers), one cannot solve for T1 in terms of a power series in the unknown functions yk. This is
when secular terms appear. In fact, solving Eq. (6.9) with A of Eq. (6.20), we obtain:
y1  exp 1 t 1 

  
y     

  

yn  exp  n t  n 
(6.22)
with 1,.....,n being the initial values. Eq. (6.16) now becomes

j1
bk , j1
jn 1
jn
n
j1
 n n exp j, t    k Tk ,1 y    l yl
j
l1
Bk , j1
yl
0
(6.23)
jn
 l yl 
Noting that
Eq. (6.23) can be rewritten as
B
k , j1
j1
Tk ,1
jn
dyl
dt
exp j, t    k Tk ,1 y 
jn
dTk ,1
dt
0
(6.24)
Eq. (6.24) is easy to solve. If the equality k=(j,) does not hold for any of the arrays of integers,
j1,...,jn , then
Tk, 1 t   Tk, 1 0exp  k t  

j1
Bk , j1
exp k t   exp j, t 
jn
jn
 k  j, 
(6.25)
When k resonates with any (j,) combination, the corresponding term in Eq. (6.25) becomes
Bk , j1
jn
t exp k t 
(6.25a)
This grows linearly with time relative to the pure exponential. The higher order equations yield
similar results, possibly with higher powers of time. Inserting this result for T1 in the expansion
Eq. (6.7), for the solution of the original problem, Eq. (6.6), we find that, when resonance occurs,
the first order term may grow faster than the zero-order term by an unbounded linear time
dependence proportional to (t). For t=O(1/), the latter is O(0), and the first order term is not
small (not O()) relative to the zero-order one. The expansion is valid for t≤O(0) only.
If Re k<0, then the linear divergence of the term of Eq. (6.25a) (or, any other power of t, for that
matter) is not a serious problem. The exponential damping kills this term at long times, and
ways of overcoming the problem may be found. Still, the uniform validity of the expansion in
time is not guaranteed. The problem is a serious one if k has a positive real part, or is pure
imaginary. The latter case is of special interest, since it corresponds to an oscillatory behavior.
Many of the nonlinear problems of interest are harmonic motions perturbed by nonlinearities.
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Y. Zarmi
It is not clear that expansion methods which are uniformly valid in time (i.e., that do not develop
secular terms for all times, or, at least, for long times, e.g. t≤O(1/)) always exist, but one tries to
develop them so as to attain better approximations than is obtained in a naive expansion.
6.3.1 Example-perturbed harmonic oscillatorautonomous case
The ideas discussed above are best exemplified in the case of a harmonic oscillator subjected to a
small nonlinear perturbation that does not depend explicitly on time (i.e., an autonomous
system):
Ý
xÝ 0 2 x   f x, xÝ; 
(6.26a)
which can be transformed into a first order, two-dimensional system:
xÝ  0 y
yÝ  0 x 

f x, xÝ; 
0
(6.26b)
To apply a Lie transformation to these equations, one needs two generators, Tx and Ty (see Eq.
(6.14). However, it is convenient to go over to complex notation
z x
i
y
0
(6.27)
which converts Eq. (6.26) into
Ý
z   i  0 z  Zz,z * ; 
Zz, z * ;  i

f x, y,; 
0
(6.28)
Now z and z* are the two independent functions. (The equation for z* is the complex conjugate
of Eq. (6.28).) Assume for Z(z,z*;) an expansion
Zz, z * ;     n Zn z,z * 
(6.29)
n1
and search for a transformation emanating from w, the zero-order approximation,
T    n Tn w,w * 
(6.30)
n1
such that
z  exp DT w
DT  T


T *
*
w
w
(6.31)
Note that the two generators are complex conjugate of each other:
Tw  T , Tw *  T *
F,T  T
F
F
T
T
T*
 F*
*  F
*
w
w
w
w
(6.31a)
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Y. Zarmi
Now demand that w obeys the unperturbed equation (i.e., naive perturbation, see Eq. (6.9)):
wÝ  i 0 w
(6.32)
which is solved by
w  exp  i  
  const.
  0 t  0
(6.33)
In addition, we must satisfy Eq. (6.10) for the generators. As an example, consider the first order
term. Eq. (6.11) with W0=-i0w yields
Y1  Z1 w,w *   i  0 T1 w,w *  i  0 w
T1
T
 i  0 w 1*  0
w
w
(6.34)
T
(6.35)
Now write Z1 and T1 as power series in w and w*
Z1 
Z
k ,m  0
km
w k w*
m
T1 
km
w k w*
m
k , m 0
where k,m stand for powers of w and w*, respectively. For simplicity, the subscript 1 has been
omitted in the expansion coefficients in Eq. (6.35).
For Eq. (6.34) to be obeyed, the coefficient of every monomial wkw*m must vanish:
Zkm  i 0 k  m  1Tkm  0
(6.36)
which is the form Eq. (6.21) assumes here.
All monomials wkw*m in Z1 can be eliminated by corresponding monomials in T1 with
coefficient
Tkm   
Zkm
i  0 k  m  1
(6.36a)
except when the eigenvalue i0 resonates with the combination (m i 0+k(-i 0)) (m=k1).
Thus, monomials of the form wkw*k1 cannot be eliminated. We therefore focus our attention on
the case in which Z1 contains only resonant terms:
Z1 w, w *   Zk w k  1 w *  w Fww * 
k
(6.37)
with F a complex function of =|w|. Using Eq. (6.33), Eq. (6.34) becomes
F 2 exp  i   i  0 T1   0
Eq. (6.38) is solved by
T1
0

(6.38)
-8T1  Gexp  i  
Y. Zarmi
F 2  exp  i 
0
(6.39)
G() is arbitrary. The freedom to add arbitrary solutions of the homogeneous part of Eq. (6.38)
is related to the choice of the zero-order term. It affects the manner in which the initial
conditions are satisfied. This will often show up in later chapters.
Thus, through first order, the solution to the original problem, Eq. (6.28), is approximated by:


1
z  exp  i     G 
F  2  exp  i   O 2 
0


(6.40)
As  is essentially linear in time, the expansion of z in a power series in  has a secular term
proportional to t. It remains small only for t≤O(0). For longer times, t=O(1/), the perturbation
expansion is not valid. Thus, if we insist on killing all monomials (resonant ones as well) in the
dynamical equation, the resulting generators of the transformation include terms that increase
indefinitely with time. Similar results are obtained in the higher orders. The n'th order term may
give rise to secular terms of all types: nt, nt2,.....ntn. Technically speaking, resonant terms
cannot be eliminated by a T1 which is expanded in monomials of w and w* because , which
appears as a multiplicative factor in Eq. (6.39), cannot be expanded in a power series in w and
w*:
   12 i logw * w
6.4 Normal forms = "skeleton equations"
The previous analysis implies that often it is not wise to eliminate all the terms in a nonlinear
differential equation, as this may lead to secular terms in the expansion of an approximation for
the solution. In going over from the original problem, Eq. (6.6), to a transformed one, Eq. (6.8),
it is advisable to retain terms that resonate with the zero-order part, X0, leading to
dy
 X0 y   U1 y   2 U 2 y 
dt
where Uk are pure resonant terms. This is called a normal form.
(6.41)
With a particular transformation T(y;), the given dynamical system, Eq. (6.6), is converted to
the normal form, Eq. (6.41). However, starting with a given normal form, different choices of
the transformation generate a wealth of "original" nonlinear equations. That is to say, the
collection of all nonlinear differential equations, with right hand sides that can be expanded in
powers of the unknown functions (and this corresponds to a great part of the problems of
interest), is equivalent to the much smaller subset of normal form equations. As we shall later on
see, normal forms capture all the intrinsic interesting dynamical features characteristic of
nonlinear problems.
6.4.1 Example-perturbed harmonic oscillatorautonomous case
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Y. Zarmi
The procedure of Section 6.3.1 eliminates all terms in Z1 that do not resonate with Z0=i0z.
The procedure of eliminating all non resonant terms in higher orders in  similarly yields:
z  exp DT w
dw
  i  0 w    n Fn w w * w   i  0 w   Fww * w
dt
n1
Ý  R 2; 

w  exp  i  
R 2;   Re F 2; 
Ý    I  2; 

0
(6.42)
I 2;   Im F 2 ; 
Eliminating all non-resonant terms in the original equation (Eq. (6.28)) we end up with a
nonlinear equation which may be more complicated than the original one: higher order terms that
are not present in the original equation may be generated by the transformation. However, all
terms in the normal form have the same time dependent phase as the linear term. They all
resonate with it. That is why we do not eliminate them. Their elimination leads to the
generation of secular terms.
Eq. (6.42) has one important consequence. Although it is nonlinear, the nonlinearity is only
through its dependence on the amplitude, =|w|. The dependence on the phase is linear (all terms
in the equation have w as a common factor). As a result, if the solution is periodic, only one
frequency is excited. This happens in the conservative case, where R(2;)0. The amplitude, ,
is constant then and the equation for  yields a frequency update only. Another way to see this is
by defining w=q+i p, and writing the equations for q and p:

 


2
2
qÝ  R w ; q   0   I w ; p




(6.42a)

pÝ   0   I w ; q   R w ;  p
2
2
For a conservative system (R(2;)0), this yields
Ý  q  0
qÝ
2

   0   I w ;
2

(6.42b)
In dissipative problems, where R≠0, the amplitude, , is not constant and the (single) frequency
is time dependent. If a limit cycle exists with a special value of  for which R0, then the
equation is reduced again to a mere frequency update. In either case, all higher harmonics are
generated in the original unknown, z=x+iy, only through the inverse of the Lie transformation
that generated the equation for w. Now, suppose someone gave us the normal form to start with .
We can now invent an infinity of Lie transformations that will take us back to an infinity of
"original" problems. Thus, the family of nonlinear dynamical problems is equivalent to the
smaller subset of normal forms.
6.4.2 Some parts of the transformation are indeterminate
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Y. Zarmi
From Eq. (6.18) or (6.21) in the general case and Eq. (6.36) in the perturbed oscillator case, we
see that the coefficient of any monomial in the transformation T1 (and similarly in higher orders)
with a combination of powers that yields resonance (k=(j,)) cannot be determined. This is
seen most easily in the case of the perturbed harmonic oscillator (Eq. (6.36)). Whenever m=k-1,
the coefficient, Tk,k-1 can be chosen freely. The normal form is determined up to an additive term
of the form G(|w|2)w. This freedom in the choice of the generator (i.e., of the perturbation
series) is related to the freedom in the choice of the zero-order term and to the manner in which
the initial condition is satisfied.
6.5 The Duffing equation
The Duffing equation is the simplest example of interest for a nonlinearly perturbed harmonic
motion. Through it we demonstrate the general statements made above. The equation is
x0  a
Ý x   x 3
xÝ
xÝ0  b
*
Ý
z   i z  18 i z  z 

z0  a  i b
3
z  x  i xÝ
x
(6.43)
z  z 
*
1
2
The perturbation in Eq. (6.43) is O():

Z1   18 i z  3z z  3 zz
3
2
*
*2
z
*3

(6.44)
We therefore start with a first order analysis:
 
z  exp  DT1 w
(6.45)
To kill nonresonant terms in O(), Eq. (6.36) determines all the coefficients Tkm of the
transformation T1, except for T21 which is undetermined (k=2, m=1, km1=0). We find
2
T1  161 w 3   w 2 w *  163 w w * 
1
32
w*
3
(6.46)
T21 = is a free coefficient. The resonant terms in this order are left intact. Note that the free
term in T1 may have monomials of the form wkw*k-1 for any k, with arbitrary coefficients. We
avoid this excessive freedom, as it does not add to the clarity of presentation.)
The original equation, Eq. (6.43), has only an O() term. However, the transformed equation for
w will have new terms of higher orders . This is the price we pay for simplifying the equation in
the lowest order. Using Eqs. (6.11,15,25,31), the equation for w through O(2) is:


wÝ  i w  83 i  w 2 w *  12  2  12 i w  w *   38 i w 2 w * ,T1  O 3 
3
6.47)
Most of the contributions to the 2 term are non-resonant, and can be killed by a transformation:
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Y. Zarmi
 
w  exp DT2 v
(6.48)
However, to find the normal form through second order, it is not necessary to find T2. The
resonant part of the 2 term in the normal form is unaffected by the T2 transformation. Thus, in
order to identify that part, all one has to do is write the term in square brackets in Eq. (6.47)
explicitly. It turns is of fifth degree monomials. Of these, only the w3w*2 term is resonant (k=3,
m=2, 3-2-1=0). Thus, identifying the coefficient of this contribution in Eq. (6.47) immediately
yields the normal form through second order, without having to compute T2:

vÝ  i v  38 i  v 2 v *  i  2

 83    *  v 3 v *  O 3 
51
256
2
(6.49)
We have to find T2 only if calculation of the 3 term in the normal form is required. That
depends on the level of accuracy desired and the duration in time over which this accuracy is to
prevail in the computation of the solution. For example, to find an approximation for z, with an
O(2) error over t≤O(1/), we need not know the 3 term at all. Hence, there is no need to
compute T2. In fact, let M3 be a bound for the 3 term over the time period considered
(boundedness of the terms involved is assumed). The contribution of this term to the solution of
Eq. (6.49) is bounded by
 3M t
For t≤O(1/), this constitutes an O(2) error. Thus, if we want an approximation to the solution
of Eq. (6.47) with an O(2) error over t≤O(1/), we do not need to know both the 3 term and T2.
In polar coordinates,
v = exp(  i )
(6.50)
we find
Ý O 3 

Ý 1  38  2  


51
256

3
8
   
*
4
  O
2
3
(6.51)

The latter is solved by
  0  O 2 

  0  1  38 0 2  

51
256

3
8
   
*
4
0
2



t  O 2 


t  O1  

t  O
t  O1 
(6.52)
If lower accuracy is sufficient, we may also write

  0  1  38 0  
2

51
256

3
8
   
*
4
0
2
2

(6.52a)
In the present problem of a conservative system,  is constant in all orders of the expansion. The
normal form induced by the perturbation generates a frequency correction only. Although the
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Y. Zarmi
normal form is nonlinear, only the fundamental frequency is excited, and no higher harmonics
are.
Now that v is solved for, we can write for z, the original unknown,

t  O1  2 
v  O
z  
*
2

t  O1  
v   T1 v, v   O 
(6.53)
Again, the level and the duration in time of the accuracy are connected.
We now examine the effect of the choice of the zero-order term and the transformation on the
manner in which the initial conditions are satisfied. Eqs. (6.50 & 53), yield
z  e  i     3 161 e 3 i   e  i  
3
16
ei
1
32
e 3 i    O 2
t  O1  
(6.54)
For the sake of simplicity, assume as initial conditions
x 0  A
xÝ0  0

(6.55)
z0  A  i  0
That is, z(0) is purely real. It is the maximum amplitude of the oscillation (turning point).
Implementation of the initial condition depends on the choice of . If we choose =5/32 and
0=0, the -term in Eq. (6.54) vanishes at t=0 and we obtain:
0   A
0   0
(6.56)
A 4  2  O 3 
(6.57)
The basic frequency is now
   1  38 A  2 
21
256
If, on the other hand, we choose for  any other value, then the initial condition will not be
satisfied by the zero-order term alone. Take, for instance, =0, 0=0. The result is
x t  0  A   
5
32
   O
3
2


  A  325 A 3   O 2 
(6.58)
and
   1  38  2 
51
256
 4  2  O 3 
(6.59)
In Eqs. (6.57) and (6.59) the first order corrections to the frequency have the same functional
form, but different values (the difference is O(2)). The second order corrections have different
forms (recall the example of the dipole of Section (6.1)). But, through O(2) the values
calculated in both cases are the same since  and 0 are related (Eq. (6.58)). In the first case, the
zero-order approximation, v, satisfies
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Y. Zarmi
v 0  A  i 0
while in the second one it is
v 0    i 0
In this example, the two zero-order approximations have the same initial velocity, 0, but different
initial positions. As a result, the functional form of the second order terms in the corresponding
solutions are different.
A choice of special interest is =17/64, which eliminates the O(2) correction to the ,
yielding
A 
7
64
   O
3
2

  A  647  A  O

3
2

    1  38  2  O 3 
6.5.1 Why quadratic nonlinearities are not interesting in lowest order
Consider a quadratically perturbed harmonic oscillator
Ý x   x 2
xÝ
*
zÝ  i z  14 i  z  z 
2

(6.60)
No term in the perturbation (z2, z.z*, z*2) resonates with the zero-order term. Hence, the whole
quadratic term can be eliminated by a transformation, z=exp(DT1).w. Eq. (6.36) yields
wÝ  i w  O 2 
(6.61)
T1   w  ww 
1
4
2
1
2
*
1
12
w
*2
(The O(2) correction contains only cubic monomials in w and w*.) Thus,
w  e  i   O
  const.


  t   0 
t  O1  
(6.62)
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Y. Zarmi
Fig. 6.2 Numerical solutions of Eq. (6.60)
The zero-order approximation is not affected by the perturbation for a long time. There is no
change in the frequency in O(). A change occurs first in second order, where the transformation
T1 generates a w2w* term which resonates with w. The first order correction to z, T1, yields
z  e  i     2  14 e2 i  
1
2
 121 e 2 i    O 2 
(6.63)
The frequency is changed only in O(2). Further analysis yields the normal form through 2 as:
wÝ  i w 
w  e
i 

5
12
i  2 w 2 w *  O 3 
Ý O

3

Ý 1  125  2  2  O 3

(6.61a)
Eq. (6.63) generates a real shift in z by (2)/2) and higher harmonics. This is shown in Fig. 6.2,
where numerical solutions of Eq. (6.60) are displayed for =0.2, xÝ(0)=0, and x(0) = 0.5, 1.0, 1.5.
As the amplitude grows, the distortion of the circle owing to presence of the higher harmonics
becomes greater, and the shift of the origin grows.
However, the change in the frequency is small. Fig. 6.3 shows the time dependence of x(t) for
the x(0)=1.5, xÝ(0)=0. The period remains very close to 2. Forx(0)=1.5, evaluating Eq. (6.63)
at t=0, one finds =1.35, and that, in O(2), the angular frequency changes from 1 to 0.969625.
Thus, a nonlinear perturbation has to have odd terms, at least cubic, in z and z* in order to
contribute in its own order in . In terms of the real variables, this happens when one or more of
3
2
2
3
the terms x , x xÝ, x xÝ , xÝ appears in the perturbation.
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Y. Zarmi
Fig. 6.2
6.6 Conservative vs. dissipative (still autonomous) systems
Consider, again, the perturbed harmonic oscillator. In the most general form, its equation is
Ý x   Fx, xÝ;   0
xÝ
(6.64)
with F - a real function. Multiplying Eq. (6.64) by dx/dt and integrating we obtain
t
1
2
xÝ2  12 x 2    F x, xÝ; xÝdt  E
(6.65)
0
If F does not depend on the velocity, then Eq. (6.65) represents motion in a potential, V(x), with
1
2
xÝ2  V x   E
(6.66)
x
V x   12 x 2    F x ; dx 
0
The significance is best seen in the complex notation:
Ý
z   i z  i F
 z  z , z  z ;
1
2
*
1
2
*
z  x  i xÝ
(6.67)
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Y. Zarmi
To be more specific, let us study, side by side, two similar equations:
Ý x   x 3  0
xÝ
Ý x   xÝ3  0
xÝ
(6.68)
The left hand equation (Duffing) is clearly integrable, corresponding to motion in the potential
V(x)=(1/2) x2+(/4) x4. The right hand equation corresponds to motion with friction. Now
change to complex notation, and the two equations become:
*
Ý
z   i z  18 i  z  z 
3
*
zÝ  i z  18  z  z 
3
(6.69)
Next, we have to find the Lie transformations that generate the normal forms. To obtain the
normal forms through O() only, we do not have to find T1. All we have to do is identify the
resonant part in the perturbation, and erase all other terms. This yields:
wÝ  i w  38 i  w 2 w *  O 2 
wÝ  i w  38  w 2 w *  O 2 
(6.70)
In Eq. (6.69), the original perturbation is purely imaginary in the left hand equation and purely
real in the right hand equation. Studying the structure of higher-order terms generated by
successive transformations, one can see that the normal forms will have the following generic
structure
   i  I w w
 w
wÝ  i w  i  I1 w
wÝ  i w   R2 w
2
2
2
2
(6.71)
2
where Ri and Ii are real functions. The corresponding equations for w* are:
 w
*
*
wÝ   i w  i  I1 w
2
   i  I w w
*
*
wÝ   i w   R2 w
*
2
2
2
2
*
(6.72)
Now multiply Eqs. (6.71) by w* and Eqs. (6.72) by w, and add up the results, to obtain

d 2
w 0
dt

d 2
2
w  2  R2 w ; w
dt
2
(6.73)
The conservative problem yields a zero-order term with a constant amplitude (the perturbation
corresponds to a frequency update only). In the dissipative problem, the zero-order term has a
time dependent amplitude that grows or decreases depending on the sign of R2. In the general
case, (Eq. (6.67)), a conservative system corresponds to F=F(z+z*;), i.e., independent of zz*.
6.7 Example of equivalent equations
As mentioned in Chapter 1, the equation developed by Lord Rayleigh for the description of the
generation of sound waves in organ pipes
Ý
Ý 1 Ý2
xÝ
1  x1   x1  3 x1 
and the equation
(6.74)
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Y. Zarmi
2
Ý
Ý
xÝ
2  x 2   x2 1  x 2 
(6.75)
developed by Van der Pol for the description of electrical current oscillations in triodes, are
equivalent. Indeed, taking the time derivative of Eq. (6.74) and defining x 2= xÝ1 , one derives Eq.
(6.75). The equivalence of the two equations also manifests itself in the fact that they have the
same normal form. In complex notation (z=x+i xÝ1 ), we obtain the corresponding equations:
z  z *
Ý
z   i z   
 2 

z  z  
3
*
1
3


8
*
z  z *  z  z 

Ý
z  i z  
1
2 
4

2




(6.76a)
(6.76b)
The resonant terms in both equations are identical, having the form
1
2

z  18 z 2 z *  12 z 1  14 z
2

Thus, through O(), the normal forms corresponding to Eqs. (6.85) and (6.86) are the same:

wÝ  i w  12  w 1  14 w
2
 O 
2
(6.77 )
where z and w are related by a Lie transformation, z=exp(DT)w. Note that through O(), we
need not find the generators of the transformations. It is enough to retain the resonant terms in
the original equations. The zero-order approximation to the solutions of both is the same, found
by solving Eq. (6.77 exactly through O(e). In polar coordinates we have
 
2




2
Ý 12 1  14  2   O 2  
1  e   t 4 / 0   1

 
Ý 1  O 2 


    0  t
w  e  i 



 O



(6.78)
The solution of both equation exhibits a limit cycle. The O() error in equation (6.78) is
estimated as follows. The error source is the 2-term, neglected in Eq. (6.76a). Let 2M be an
upper bound for the latter. Its contribution to the integrated equation is bounded by 2Mt. For
t≤O(1/), this induces an O() error. If such an error is permitted, the zero-order approximation
is sufficient and the problem is solved. Improvement of the accuracy of the approximation to,
say, O(2), requires that the O() contribution to the generator T1 be calculated. One finds its
effect on the O(2) terms in the equation, and solves the equation through O(2), to obtain
z  w   T1w,w *  O 2 
t  O1  
(6.79)
The difference between the solutions of the Rayleigh and Van der Pol equations resides in the
generator T1(w,w*) Using Eq. (6.36) we find:
-18 w   w 2 w *  i

T1  
2
*

 w   w w  i
w
w
1
4
*

1
4
*

w 3  161 ww * 
1
96
w*
w 
1
32
*3
1
48
1
16
Y. Zarmi
2
3
1
16
ww
*2

w


3
Rayleigh
(6.80)
Van der Pol
, , ,  are coefficients of arbitrary terms. However, it is meaningless to use T1 in Eq. (6.79)
unless the normal form is developed and solved through O(2).
6.9 Non-autonomous systems:
periodic solutions for equations with periodic coefficients
The dynamical equations of autonomous systems do not depend explicitly on time. In nonautonomous systems they do. This difference affects the properties of the solutions. The case of
periodic solutions is especially interesting. We analyze the latter using the method of Lie
transformations, which has been developed above for autonomous systems, and can be extended
to non-autonomous ones as well. Consider
dx
  Fx,t; 
dt
x, F R n
(6.81)
This system is converted into an autonomous n+1 dimensional one by defining a new
"unknown", t, yielding
dx
  Fx,; 
dt
d
1
dt
(6.82)
One can now apply Lie transformations in the n+1 dimensional space of (x,). The resulting
transformed equation depends on  (i.e., on time).
Systems with equations that have an explicit periodic time dependence are of special interest. In
Section 6.4.1 we have seen that in the autonomous case of the perturbed harmonic oscillator, the
normal form has two important properties: (i) the amplitude and the phase are uncoupled; (ii)
once the equation for the amplitude is solved, the frequency update and the phase are computed
by a straightforward quadrature (Eq. (6.42)). This is not the case in non-autonomous case.
Consider
Ý x   Fx, xÝ,t;
xÝ
x0  x0
xÝ0  xÝ0
(6.83)
2


Fx, xÝ,t 
;   Fx, xÝ,t; 

 
Since F is periodic in time with period (2/), it can be expanded in a Fourier series. In
complex notation, Eq. (6.83) may be cast in the form:
Ý
z   i z  i  Gz,z *, v,v *; 
vÝ  i  v
z0  x0  i xÝ0
(6.84)
v 0 1
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Y. Zarmi
where vexp(-it) and F is expanded in a Fourier series
Gz, z * ,v,v * ;   Fx, xÝ,t;  
G
z k z * e i m  t  c.c. 
k
klm
k , l, m 0
G
klm
z k z * v m  c.c.
k
(6.85)
k , l, m 0
The original two-dimensional problem becomes an autonomous four-dimensional one, with two
components (v and dv/dt) easily solved for. (In this spirit, the general case, Eq. (6.82), is
equivalent to an infinite number of degrees of freedom: In general, its Fourier expansion
includes a continuum of frequencies).
We now employ Lie transformations to eliminate nonresonant terms in the perturbation in a
given order. Resonance may occur in more combinations than in the autonomous case. The
eigenvalues of the linear part of Eq. (6.84) are ±i and ±i. Resonance in the equation for z
occurs when integers k,l,m≥0 exist such that,
i  1  k 1  l 2  m3  i k  l  m
(6.86)
For simplicity, we choose =1. In this case, resonant monomials in the normal for w, the Lie
transform of z, are of the form
l
w k w* v mv *
n
k  l  m n 1
The general structure of the normal form is:
wÝ  i w  
W
klmn
k  l m  n1
 w k w * v m v *
l
n
(6.87)
In polar coordinates,
w  e  i 
v  e i
t
we also write the phase, , as a sum of its fast and slow components:
 t 
to obtain
Ý i  
Ý i 

leading to
W
klmn
k l  m  n1
  k  l e i k  l  1
Ý    k  l Rkl cosk  l 1  Ikl sink  l  1  R,; 

k ,l
Ý    k  l Rkl sink  l  1  Ikl cosk  l 1  I,;

k, l
(6.88)
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Y. Zarmi
where
Rkl 
 ReW
 Im W
Ikl 
klmn
m, n
k  l  m  n1
klmn
m ,n
k  l  m  n1
The amplitude, , and the slow phase, , of the zero-order term are coupled in non-autonomous
systems. As a result, the simple behavior observed in autonomous systems is not guaranteed
here.
6.9.1 Periodic solution in case of stable fixed point
Assume that a point (0,0) exists for which d/dt=d/dt=0. Eq. (6.88) implies
R0, 0 ;  I 0, 0;   0
(6.89)
The implicit function theorem guarantees the existence of a unique solution of Eq. (6.89) for the
amplitude and the (constant) slow phase if the Jacobian of the two equations is nonzero at the
vicinity of (0,0):

R

R

I

I
  0
0
(6.90)
  0
When this happens, we have a periodic solution with amplitude 0 and the frequency is not
modified relative to the unperturbed one (note the difference from periodic solutions in
autonomous problems).
To study the stability of the periodic solution, we expand Eq. (6.88) to first order around (0,0):
= 0 + 
 = 0 + 
d
R
R
 
 
 
dt
 0
 0
(6.91)
d
I
I
 
 
 
dt
 0
 0
where the derivatives are evaluated at (0,0). If both eigenvalues of  have negative real parts,
(0,0) is a stable fixed point in the  plane and the periodic solution is stable. A small
perturbation modifying  and , will decay in time so that
 , 
t •
 0 , t   0  .
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Y. Zarmi
As an example, consider Eq. (6.75) (Van der Pol). Its normal form, Eq. (6.77), generates a limit
cycle (a circle of radius 2 in lowest order). The slow phase is an arbitrary constant in O(), and
develops a linear time dependence in O(2). Now modify this equation to:
Ý x   xÝ1  1  a cos2t  x 2 
xÝ
v  e i
z  x  i xÝ

t
(6.92)
2
*
z  z *  
v 2  v * z  z 
1  1  a
Ý
z  i z  

2 
2 
4
 
2




(a=O(1); a>0 assumed without loss of generality). The resonant terms are those of Eq. (6.77) as
well as the two mixed terms

 161 a z v
3
*2
 zz
*2
v
2

yielding a normal form (through O()):


2
2
2 *
*
2
wÝ  i w  12  w 1  14 w  18 a w v  w v
2
 O 
2
6.93)
In polar coordinates this equation becomes
w  e  i 
t
Ý 12  1  14  2   O 2 

(6.94)
Ý  18  a 2 sin2  O 2 

Thus, in O(0), the radius is the same as in the Van der Pol equation. However, the slow phase
exhibits a peculiar behavior. For >0, 0=0, ±2,... are stable, while for <0 0=±/2, ±3/2,...
are stable. The slow phase is not arbitrary in this order. By the Poincaré - Lyapounuv theorem,
we expect this periodic orbit to be stable against sufficiently small perturbations even when
higher order effects are incorporated for sufficiently small ||.
Exercises
6.1 Show that the normal forms for Eqs. (6.76a&b) are the same in O(2) as well.