Case III. Complex conjugate roots are of minor practical importance, and we discuss the derivation of real solutions from complex ones just in terms of a typical example
Case 2. Damped Forced Oscillations
4.4 Criteria for Critical Points. Stability
Furthermore, the product representation of the equation gives
.
Hence pis the sum and qthe product of the eigenvalues. Also from (6).
Together,
(7) , , .
This gives the criteria in Table 4.1 for classifying critical points. A derivation will be indicated later in this section.
¢ ⫽(l1⫺l2)2 q⫽l1l2
p⫽l1⫹l2
l1⫺l2⫽ 1¢ l2⫺pl⫹q⫽(l⫺l1)(l⫺l2)⫽l2⫺(l1⫹l2)l⫹l1l2
SEC. 4.4 Criteria for Critical Points. Stability 149
Table 4.1 Eigenvalue Criteria for Critical Points (Derivation after Table 4.2)
Name Comments on
(a)Node Real, same sign
(b)Saddle point Real, opposite signs
(c)Center Pure imaginary
(d)Spiral point Complex, not pure
imaginary
¢⬍0 p⫽0
q⬎0 p⫽0
q⬍0
¢⭌0 q⬎0
l1, l2
¢⫽(l1⫺l2)2 q⫽l1l2
p⫽l1⫹l2
Stability
Critical points may also be classified in terms of their stability. Stability concepts are basic in engineering and other applications. They are suggested by physics, where stability means, roughly speaking, that a small change (a small disturbance) of a physical system at some instant changes the behavior of the system only slightly at all future times t. For critical points, the following concepts are appropriate.
D E F I N I T I O N S Stable, Unstable, Stable and Attractive
A critical point of (1) is called stable2if, roughly, all trajectories of (1) that at some instant are close to remain close to at all future times; precisely: if for every disk of radius with center there is a disk of radius with center such that every trajectory of (1) that has a point (corresponding to say) in has all its points corresponding to in . See Fig. 90.
is called unstable if is not stable.
is called stable and attractive (or asymptotically stable) if is stable and every trajectory that has a point in approaches as . See Fig. 91.
Classification criteria for critical points in terms of stability are given in Table 4.2. Both tables are summarized in the stability chart in Fig. 92. In this chart region of instability is dark blue.
t:⬁ P0
Dd
P0 P0
P0 P0
DP
t⭌t1 Dd
t⫽t1, P1
P0
d⬎0 Dd
P0 P⬎0
DP
P0 P0
P0
2In the sense of the Russian mathematician ALEXANDER MICHAILOVICH LJAPUNOV (1857–1918), whose work was fundamental in stability theory for ODEs. This is perhaps the most appropriate definition of stability (and the only we shall use), but there are others, too.
We indicate how the criteria in Tables 4.1 and 4.2 are obtained. If , both of the eigenvalues are positive or both are negative or complex conjugates. If also , both are negative or have a negative real part. Hence is stable and attractive. The reasoning for the other two lines in Table 4.2 is similar.
If , the eigenvalues are complex conjugates, say, and
If also , this gives a spiral point that is stable and attractive. If , this gives an unstable spiral point.
If , then and . If also , then , so
that , and thus , must be pure imaginary. This gives periodic solutions, their trajectories being closed curves around , which is a center.
E X A M P L E 1 Application of the Criteria in Tables 4.1 and 4.2
In Example 1, Sec 4.3, we have a node by Table 4.1(a), which is
stable and attractive by Table 4.2(a). 䊏
yr⫽c⫺3 1
1 ⫺3d y, p⫽ ⫺6, q⫽8, ¢⫽4, P0
l2 l1
l12⫽ ⫺q⬍0 q⬎0
q⫽l1l2⫽ ⫺l12 l2⫽ ⫺l1
p⫽0 p⫽2a⬎0
p⫽l1⫹l2⫽2a⬍0
l2⫽a⫺ib.
l1⫽a⫹ib
¢⬍0
P0
p⫽l1⫹l2⬍0
q⫽l1l2⬎0
P1
P0
∈ δ
Fig. 90. Stable critical point P0of (1) (The trajectory initiating at P1stays
in the disk of radius ⑀.)
P0
∈ δ
Fig. 91. Stable and attractive critical point P0of (1)
Table 4.2 Stability Criteria for Critical Points Type of Stability
(a)Stable and attractive (b)Stable
(c)Unstable p⬎0 OR q⬍0
q⬎0 p⬉0
q⬎0 p⬍0
q⫽l1l2 p⫽l1⫹l2
q
p Δ= 0
Δ > 0 Δ < 0 Δ < 0 Δ > 0
Δ= 0 Spiral
point
Spiral point
Node Node
Saddle point
Fig. 92. Stability chart of the system (1) with p, q, ⌬defined in (5).
Stable and attractive: The second quadrant without the q-axis.
Stability also on the positive q-axis (which corresponds to centers).
Unstable: Dark blue region
E X A M P L E 2 Free Motions of a Mass on a Spring
What kind of critical point does in Sec. 2.4 have?
Solution. Division by mgives . To get a system, set (see Sec. 4.1).
Then . Hence
, .
We see that . From this and Tables 4.1 and 4.2 we obtain the following results. Note that in the last three cases the discriminant plays an essential role.
No damping. , a center.
Underdamping. , a stable and attractive spiral point.
Critical damping. , a stable and attractive node.
Overdamping.c2⬎4mk, p⬍0, q⬎0, ¢⬎0, a stable and attractive node. 䊏 c2⫽4mk, p⬍0, q⬎0, ¢⫽0
c2⬍4mk, p⬍0, q⬎0, ¢⬍0 c⫽0, p⫽0, q⬎0
¢ p⫽ ⫺c>m, q⫽k>m, ¢⫽(c>m)2⫺4k>m
det (A⫺lI)⫽ 2 ⫺l 1
⫺k>m ⫺c>m⫺l2⫽l2⫹ c ml⫹k
m⫽0 yr⫽c⫺k0>m ⫺c1>md y
yr2⫽ys⫽ ⫺(k>m)y1⫺(c>m)y2
y1⫽y, y2⫽yr ys⫽ ⫺(k>m)y⫺(c>m)yr
mys⫹cyr⫹ky⫽0
SEC. 4.4 Criteria for Critical Points. Stability 151
1–10 TYPE AND STABILITY OF CRITICAL POINT
Determine the type and stability of the critical point. Then find a real general solution and sketch or graph some of the trajectories in the phase plane. Show the details of your work.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11–18 TRAJECTORIES OF SYSTEMS AND SECOND-ORDER ODEs. CRITICAL POINTS
11. Damped oscillations. Solve . What
kind of curves are the trajectories?
12. Harmonic oscillations. Solve Find the trajectories. Sketch or graph some of them.
13. Types of critical points. Discuss the critical points in (10)–(13) of Sec. 4.3 by using Tables 4.1 and 4.2.
14. Transformation of parameter. What happens to the critical point in Example 1 if you introduce as a new independent variable?
t⫽ ⫺t ys⫹19y⫽0.
ys⫹2yr⫹2y⫽0
y2r⫽ ⫺5y1⫺2y2
y2r⫽4y1⫹4y2
y1r⫽y2
y1r⫽4y1⫹y2
y2r⫽3y1⫺2y2
y2r⫽2y1⫹y2
y1r⫽ ⫺y1⫹4y2
y1r⫽y1⫹2y2
y2r⫽ ⫺9y1⫺6y2
y2r⫽ ⫺2y1⫺2y2
y1r⫽ ⫺6y1⫺y2
y1r⫽ ⫺2y1⫹2y2
y2r⫽5y1⫺2y2
y2r⫽ ⫺9y1
y1r⫽2y1⫹y2
y1r⫽y2
y2r⫽ ⫺3y2
y2r⫽2y2
y1r⫽ ⫺4y1
y1r⫽y1
15. Perturbation of center. What happens in Example 4 of Sec. 4.3 if you change A to , where I is the unit matrix?
16. Perturbation of center. If a system has a center as its critical point, what happens if you replace the matrixAby with any real number (representing measurement errors in the diagonal entries)?
17. Perturbation. The system in Example 4 in Sec. 4.3 has a center as its critical point. Replace each in Example 4, Sec. 4.3, by . Find values of bsuch that you get (a)a saddle point, (b)a stable and attractive node, (c)a stable and attractive spiral, (d)an unstable spiral, (e)an unstable node.
18. CAS EXPERIMENT. Phase Portraits. Graph phase portraits for the systems in Prob. 17 with the values of b suggested in the answer. Try to illustrate how the phase portrait changes “continuously” under a continuous change of b.
19. WRITING PROBLEM. Stability.Stability concepts are basic in physics and engineering. Write a two-part report of 3 pages each (A) on general applications in which stability plays a role (be as precise as you can), and (B) on material related to stability in this section. Use your own formulations and examples; do not copy.
20. Stability chart. Locate the critical points of the systems (10)–(14) in Sec. 4.3 and of Probs. 1, 3, 5 in this problem set on the stability chart.
ajk⫹b
ajk k⫽0 A~
⫽A⫹kI
A⫹0.1I