Nodes, saddle points, and spirals
Let’s look at three constant-coefficient homogeneous linear systems: w
~ ′ = P w,
~
where P is given below. Match each of the given systems with its phase portrait.
Hint: focus on the eigenvalues.
3 −2
−3 −2
3 −2
(a)
,
(b)
,
(c)
−2 −4
−2 −4
−2 4
y
2
2
1
1
0
-2
-1
0
0
1
2
-2
-1
0
x
-1
-1
-2
-2
2
1
0
-2
-1
0
-1
-2
1
1
2
1
2
The characteristic equation of a system is
r2 − tr(P )r + det(P ) = 0,
(1)
and so the eigenvalues are
1
2
√
tr ± 21 discr, where discr = tr2 −4 det .
(2)
Let’s make a chart of this data for the systems above.
(a)
(b)
(c)
tr det discr
−1 −16
65
−7
8
17
7
8
17
eigenvalues
signs
√
− 21 ± 21 √65 +, −
− 27 ± 21√ 17 −, −
1
7
+, +
2 ± 2 17
Hence the phase portrait for (a) must exhibit one eigensolution flowing towards
the origin, and the other away from it; the phase portrait for (b) must exhibit both
eigensolutions flowing towards the origin; and the phase portrait for (c) must
exhibit both eigensolutions flowing away from the origin.
Note that in all cases, each component of the solution is a linear combination of er1 t and er2 t , where r1 and r2 denote the eigenvalues. If (say) r1 is the
larger of these then as t → +∞ then the term er1 t will dominate the expression.
Hence almost all trajectories follow this corresponding eigenline asymptotically
as t increases. However, as t → −∞ then the other term er2 t will dominate.
Hence almost all trajectories follow this corresponding eigenline asymptotically as
t decreases.
The phase portrait for (a) is a saddle point. Almost all trajectories initially
flow towards the origin asymptotic to the eigenline corresponding to the negative
eigenvalue, then turn and flow out to infinity asymptotic to the eigenline corresponding to the positive eigenvalue.
The phase portrait for (b) is a stable node. All trajectories flow into the origin,
initially asymptotic to the eigenline corresponding to the smaller eigenvalue, then
turn and flow asymptotic to the eigenline corresponding to the larger eigenvalue.
The phase portrait for (c) is an unstable node. All trajectories flow to infinity,
initially asymptotic to the eigenline corresponding to the smaller eigenvalue, then
turn and flow asymptotic to the eigenline corresponding to the larger eigenvalue.
Before we leave these examples let’s recall some elementary algebra. If r1 and
r2 are the eigenvalues, then we may use these values to factor the characteristic
equation:
(r − r1 )(r − r2 ) = r2 − tr(P )r + det(P ).
(3)
Hence
r1 + r2 = tr(P ), r1 r2 = det(P ).
(4)
If discr(P ) > 0 then the eigenvalues are real. This always happens if det(P ) < 0.
Furthermore, in this case, the eigenvalues must be of opposite sign, since their
product is negative. If they are real and of the same sign then det(P ) > 0. In
either of these cases the eigenvalues have the same sign as tr(P ).
2
But what if discr(P ) < 0? In this case there are no eigenlines in our picture.
Consider the following two examples.
3 −2
−3 −2
(d)
,
(e)
2 4
2 −4
2
2
y
1
1
0
-2
-1
0
0
1
2
-2
-1
0
1
2
x
-1
-1
-2
-2
tr
(d) 7
(e) −7
det discr eigenvalues
√
1
7
15 i
16 −15
2 ± 2 √
16 −15 − 72 ± 21 15 i
sign of real part
+
−
Do these eigenvalues have any meaning, apart from alerting us that the solutions spiral around with no real eigenline? Yes! To see the physical interpretation,
we need Euler’s formula for complex exponentials.
All of the algebra — including that for the eigenvectors — remains valid even
with complex arithmetic. Of course this leads to solutions whose vector components have terms that look like ex+yi . What could this mean? Let’s use our power
series representation of the exponential to find out:
ex+yi = ex eyi = ex (1 + yi +
x
2
3
4
1
1
1
2! (yi) + 3! (yi) ) + 4! (yi)
1 2
1 3
1 4
1 5
2! y − 3! y i + 4! y + 5! y i
= e (1 + yi −
h
1 2
y +
= ex (1 − 2!
1 4
4! y
+ · · · ) + (y −
= ex [cos(y) + sin(y) i].
1 3
3! y
+
5
1
5! (yi)
+ ···)
+
1 5
5! y
+ ···)
+ ···)i
i
If we apply this to the eigensolutions for systems (d) we find that each component is a linear combination
i
h
√
√
7
e 2 t c1 cos( 12 15 t) + c2 sin( 12 15 t) ,
and similarly for (e) (with −7/2 in place of 7/2). Thus the real part indicates the
rate at which the trajectory flows to infinity or to the origin (respectively), and
the imaginary part indicates how fast it spirals as it does so.
Further reading
All of the types of phase portraits are summarized in section 9.1. Study the table
on page 492, then do problems 1–12, pages 492–493.
3
Reading quiz
1. Describe the phase portrait of a stable node.
2. What property of the eigenvalues characterizes a stable node?
3. Describe the phase portrait of an unstable node.
4. What property of the eigenvalues characterizes an unstable node?
5. Describe the phase portrait of a saddle point.
6. What property of the eigenvalues characterizes a saddle point?
7. Describe the phase portrait of a stable spiral.
8. What property of the eigenvalues characterizes a stable spiral?
9. Describe the phase portrait of an unstable spiral.
10. What property of the eigenvalues characterizes an unstable spiral?
Exercises
For each of the P given below, (a) sketch the phase portrait; (b) compute the
eigenvalues; and (c) describe the asymptotic behavior.
3 2
1. P =
.
1 1
3 −2
2. P =
.
−1 1
3 2
3. P =
.
−1 1
−3 2
4. P =
.
1 −1
−3 2
5. P =
.
1 1
4