4.3. Classifying Simple Singular Points
Consider again the trajectory equation for a 2-D phase space
dy Q  x, y 

dx P  x, y 
(4.10)
A singular point, say r0   x0 , y0  , is defined by the condition
Q  x0 , y0   P  x0 , y0   0
For a point r   x, y  in the neighborhood of r0, we have
x  x0  u
y  y0  v
where u and v are small.
(4.11)
Hence
dy Q  x0  u, y0  v 

dx P  x0  u, y0  v 
dv

du
(4.12)
1
1
Q,u u  Q,v v  Q,uu u 2  Q,vv v 2  Q,uv uv 
2
2
1
1
2
P,u u  P,v v  P,uuu  P,vv v 2  P,uvuv 
2
2
(4.13)
where we have made use of the "comma" notations for partial derivatives:
F,z 
F
z
F,zw   F, z ,w 
  F   2 F


w  z  wz
A simple singularity is one that can be described by the linearized version of (4.13):
dv Q,uu  Q,v v

(4.14)
du P,uu  P,v v
Obviously, (4.14) breaks down if there exists u, v  0 such that both the denominator
and numerator vanish.
P,u
P,v
Q,u Q,v
0
The condition for this is simply
(4.15)
Note that the determinant is just the Jacobian J 
Thus, a singularity is simple if and only if J  0 .
motion near the singularity reduce to
  P, Q 
evaluated at r0.
  u, v 
In which case, the equations of
u  P,u u  P,v v
v  Q,uu  Q,v v
(4.18)
where the partials are evaluated at r0.
From the 1st equation, we get
1
v   u  P,uu 
P,v
v
1
 u  P,uu 
P,v
Putting these into the 2nd equation in (4.18), we get
1
1
u  P,uu   Q,uu  Q,v  u  P,uu 

P,v
P,v
or
u   P,u  Q,v  u   P,uQ,v  P,vQ,u  u  0
u  u  Ju  0
where
(4.19a)
 P,u P,v 
J

 Q,u Q,v 
  P,u  Q,v  TrJ
J
  P, Q 
 P,uQ,v  P,vQ,u  det J
  u, v 
(4.20a)
With the ansatz u  e  t , (4.19a) becomes
 2    J  0
with solutions
 

1
   2  4J
2
(4.21a)

(4.22a)
which gives

1
 C  exp   
2


 4J  t

1

u  C exp     2  4 J t 
2

v

2
(4.22b)
See Example 4-2 of the text for a matrix approach using the following notations:
a b 
  p,
J
J  q,

c d
The accompanying Mathematica file is 04-2.nb.
Analysis
For a simple singularity, J  0 so that   0 .
The nature of the solution is of
course determined by the discriminant     4J . Thus,
2
Case 1.
>0

both  are real.
Case 2.
0

 are complex conjugates.
Case 3.
0

 are real and equal (or degenerate).
If, furthermore, J > 0 and   0 , then for
Case 1.
 > 0.
Since
 2  4J   , both  have the same sign.
Hence, r0 is a nodal point.
Case 2.
  0.
r0 is a focal point.
Case 3.
  0.
r0 is a nodal point.
In all three cases, the singular point is stable if   0, unstable if   0. If   0,
both  are purely imaginary so that r0 could be a vortex point if the Poincare’s
theorem for vortex [§4.3.1] is satisfied.
If J  0, then
 2  4J   so that both  are real but of the opposite signs.
Thus, r0 is a saddle point, which is always unstable.
These situations are illustrated in Fig.4.8.
Finally, these results should be compared with the Lyapunov's theorem:
If the real parts of the characteristic roots obtained from the linearized equations
of motion in the neighborhood of a stationary point are not zero, then the stationary
point is stable only if all real parts are negative. It is unstable if at least one real part
is positive.
4.3.1 Poincare's Theorem for the Vortex (Center)
Saddle Points

0
Unstable Nodal Points
0
Unstable Focal Points
0
Stable Focal Points
0
Stable Nodal Points
Fig.4.8.
=0
J
=0
The -J plane for establishing types of simple singular points.
(  = 2  4J )