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Unformatted text preview: 394 Chapter 6 Linear Systems of Differential Equations Apology for a Special Case As we come to the end of this section, you may think we have lavished a lot of
attention on a situation that is very special in that the system is linear, it has only
one equilibrium point, and that equilibrium point is at the origin. There are good
reasons for all of this. We have chosen in this chapter to deal with linear systems having constant
coefﬁcients because we can obtain explicit quantitative solutions to work with,
Such quantitative representations allow us to see properties and interrelationships
that are harder to Spot from purely numerical or even graphical results, helpful as
they are. Even more special Was our focus on 2—dimensioriai systems, but this is
where we see nonambignous pictures. Having only one equilibrium point helps us to focus on the variety of typical
configurations, one to a system. This result is a feature of linearity, as long as the
system is unforced. With this detailed anatomy of equilibrium points established, however, we will
ﬁnd that it can be transformed, with modest adjustments, to help analyze nonlinear
systems with multiple equilibrium points, many of them not at the origin. Look
well to your nodes and spit‘alsmthey will come back, often a little squeezed,
sttetched, or twisted, in broader contexts as we go on, 4" of? waffle»  7 if". fansegm é."  m» ,3 _  _ . gamer  We deﬁned stability, asymptotic stability, and instability for equilibrium solutions of autonomous systems
52’ = ﬂit), Then, for 2 x 2 linear homogeneous constant coeﬁicienr systems 32’ = ASE, we studied the geometric
configurations that characterize the equilibrium solutions, namely nodes (including degenerate nodes and star
nodes), spiral points, centers, and saddle points. Finally, we classiﬁed these gedmetries according to their
stability properties. 6.4 Problems Classiﬁcation Veriﬁcation For the systems in Problems b6, ﬂ, 0 1 _ _ _
verzﬁirhatthe equilibriumpoimattiieoriginhasthegeomeiric 6' X = [_} _1]x {5933; 130"“)
character claimed, and determine its stability behavior.
7, Undamped Spring Convert the equation of the urn L i, m mi}? (saddie point) damped mass—spring system,
n X + (1%.: = 0,
2 i! = 0 1 i (center) into a system Determine its equilibrium point or points,
""1 0 and classify the geometry and stability of each.
_. __ —2 0 _. 8, Damped Spring Convert the equation of the damped vi
3' x! _ l 0 ~12] x (Star Bede) brating spring,
mﬁc' «i~ bit «4» kx = 0,
a, “ ~2 l u I
4. x m x (degenerate node) . _ .
0 ‘2 into a system (mass m, damping constant I), and spring
constant k are all positive). Show that the origin is an equi»
_ .. __ 2 l a libiiurn solution, and classify its geometry and stability as
3' x] _ i3 4] x (“0%) functions of m, b, and k.  in] Section 64 Stabiiity and Linear Ciassification 395 9. One Zero Eigenvalue Suppose that A1 = 0 but A3 74—— O,
and show the following: (a) There is a line of equilibrium points
(b) Soiutions starting offthe equilibrium line tend toward the tine if hr; < O, away from it if A; > 0.
10. Zero Eigenvalue Exampie Consider the system
m 0, = —.I; l I?“ x I Ida—x (a) Show that the eigenvalues are in = 0 and it; m It
(in) Find the equilibrium points of the system. (c) Obtain the general solution of the system“ (ti) Show that the solution curves are straight lines" Both Eigenvalues Zero In Problems 11—14, you will investi
gate the nature of the solution of the 2 x 2 system it’ m A)?
when both eigenvalues ofA are zero Find the solution, plot any
ﬁxed points on the phase portrait, and indicate the pertinent
ity‘ormation in your sketch 11. a: [0 lie 0 o
12. are H is. tr: :gii
14. rt: [33} 15. Zero Again Consider the linear system 35—1—2)":
_ E m2 .v (:1) Find the eigenvalues and eigenvectors
('0) Draw typical solution curves in the phase planet I6. All Zero Describe the phase portrait of the system
ﬁr m 0 O i
w. 0 0 . 17. Stabiiity2 Classify geometry and stability properties of _,__k at
X—O_1X for the following values of parameter k. (a)k<——i (b)k=l (e) —1<k<0
(d)km0 {e)k>0 18. Bifurcation Point Bifurcation points are vaiues of a pa
rameter of a system at which the behavior of the solutions
change qualitatively Determine the bifurcation points of thesystetn
sh— “ i 5’:
M _1 k t 19. Interesting Relationships The system i’ m ASE, where a b
A W [C d]’
has eigenvalues It] and Ag. Show the following:
(a) TIA m l: + 2L2 (13) HI = hill Interpreting the TraceDeterminant Graph For 52’ = A52.
Fig 6.4 I 7 represents possible cornbinations of values of trace
TrA and determinant iAli In Problems 20:27, establish the facts about the equilibrium solution at it w 0. 20. If MI > 0 and ("tiA)2 w 41A] > o, the origin is a node
21. If [Al < O, the origin is a saddle point" 22. lf‘TrA ¢ 0 and (TiAF — 4§Al < 0, the origin is a spiral
point. 23“ If TrA = 0 and A > 0, the might is acenten 24. If (TIA)2 — 41A = O and TIA .75 0, the origin is a degen
erate or stat node 25. if‘TrA > O or iAl < O, the origin is unstable 26. if IA] > 0 and TrA = 0, the origin is neutraiiy stable,
This is the ease of purely imaginary eigenvalues” 27. if TrA < 0 and IA} > 0, the origin is asymptotically
stabte, 28. Suggested Journal Entry Whatcan you say about the re—
lationship between diagonalization of matrix A and the ge
ometry and stability of the equilibrium solution at the
origin of the system? Deveiop your response using speciﬁc
examples" 2Inspired by Steven H Strogatz, Nonlinear Dynamics and Chaos (Reading: AddisonnWesEey, I994), an excellent treatment of‘dynaznicai systems at a more advanced level. ...
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 Fall '07
 RickRugangYe

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