In this chapter, we explain the behavior of solutions of an autonomous system
dg
dt = f (y). We look at solution curves in the y-space rather than the (t, y-)-space.
Such curves are called orbits of the given system. In general, an orbit does not tend to a limit point as t -+ +oo. However, a bounded orbit accumulates to a set as t -4 +oo. Such a set is called a limit-invariant set. In §IX-1, we explain the basic properties of limit-invariant sets. In §IX-2, using the Liapounoff functions, we explain how to locate limit-invariant sets. The main tool is a theorem due to
J. LaSalle and S. Lefschetz [LaL, Chapter 2, §13, pp. 56-71] (cf. Theorem IX-2-1).
The topic of §IX-4 is the Poincare-Bendixson theorem which characterizes limit- sets in the plane (cf. Theorem IX-4-1). In §IX-3, orbital stability and orbitally asymptotic stability are explained. In §IX-5, we explain how to use the indices of the Jordan curves to the study of autonomous systems in the plane. Most of topics discussed in this chapter are also in [CL, Chapters 13 and 16], [Har2, Chapter 7], and [SC, Chapter 4, pp. 159-171].
IX-1. Limit-invariant sets
In this chapter, we explain behavior of solutions of a system of differential equations of the form
(IX.1.1)
dg
fW
where y" E Ill;" and the entries of the RI-valued function f (y) are continuous in the entire y-space II8". We also assume that every initial-value problem
dy
= Ay),
y(0) = if
dt
(IX.1.2)
has a unique solution y = p(t, y7). System (IX.1.1) is called an autonomous sys- tem since the right-hand side f (y) does not depend on the independent vari- able t.
Observe that p(t + r, r) is also a solution of (IX.1.1) for every real number T.
Furthermore, At + r, rl) = p(r, il) at t = 0. Hence, uniqueness of the solution of initial-value problem (IX.1.2) implies that p(t +r, rt) = p(t, p(-r, rt)) whenever both sides are defined. For each il, let T(rl) be the maximal t-interval on which the solution p(t, r) is defined. Set C(rt) = {y" = p(t, rl) : t E Z(rl)}. The curve C(rl) is called the orbit passing through the point il. Two orbits C(iji) and C(rj2) do not intersect unless they are identical as a curve. In fact,
(IX.1.3) C(ijl) = C(i@ if and only if 7)2 E C(il, ).
279
It must be shown that y" E L+(771.
(3) is closed: To prove this, suppose that
(2) The boundedness of C+(n7) follows from the boundedness of p(t,i7) immedi- ately.
lim yk = y for 9k E L+(1-7).k-+oo
Since g k E
it follows that ilk =
280 IX. AUTONOMOUS SYSTEMS
If f (n) = 0, the point ij is called a stationary point. If , is a stationary point, the orbit consists of a point, i.e., C(17) = Generalizing property (IX.1.3) of orbits, we introduce the concept of invariant sets which play a central role in the study of autonomous system (IX.1.1).
Definition IX-1-1. A set M C R is said to be invariant if i E M implies
C(n) C M.
For example, every orbit is an invariant set.
Remark IX-1-2. If M1 and M2 are invariant sets, then M1 UM2 and M1 f1M2 are also invariant. For a given set f2 C 1R", let Ma (A E A, an index set) be all
invariant subsets of S2, then U M>, is the largest invariant subset of Q.
AEA
Hereafter, assume that every solution p(t, nJ) is defined for t > 0, i.e., (t : t >
0) C Z(n). In general, lim Pit, nom) may not exist. However, if p(t, n) is bounded for
t > 0, the orbit C(171 accumulates to a set as t -+ +oo. This set is very important in the study of behavior of C(n) as t -+ +oo.
Definition IX-1-3. Let C+(i7-7)denote the set of all y" E II8" such that lim p(tk, n ) = y f o r some increasing sequence {tk : k = 1, 2, ... } of real numbers
k-++oo
tk = +oo. The set C+(n) is called the limit-invariant set for the
such that
lim
k-+oo
initial point r7.
The basic properties of
are given in the following theorem.
Theorem IX-1-4. If p(t, n) is bounded fort > 0, then C+(n) is nonempty, bounded, closed, connected, and invariant.
Proof.
(1) C+ (1-71 is nonempty: In fact, let {sk : k = 1,2,... } be an increasing sequence
of real numbers such that lim sk = +oo. Since p(t, n) is bounded for t > 0,
k-.+oo
there exists a subsequence {tk : k = 1, 2.... } such that lim tk = +oo and that
lim p(tk, exists. This limit belongs to f_+ (17). k-.+oo
lira p(tk,t, r 7 ) for some {tk,t : I = 1, 2, ... } such that
lim
tk,l = +00. Choose tk
t-.+oo
t-+oo
so that
lim tk,t,, = +oo and 19k - P(tk,t,,,'Th !5 ! Then, since ly - P(tk,t4, n)I -
k-.+oo
+ ly - ykl, we obtain
k
k lim oP(tk,l,,, n) = y E C+(1_7)
(4) C+(777) is connected: Otherwise, there must be two nonempty, bounded, and closed sets Sl and S2 in R" such that
2. LIAPOUNOFF'S DIRECT METHOD
281
(1) S1 n S2 = 0,
(2) S1 U S2
L+(rf)
yl E S1, y2 E S2}. Note that d >
Set d = distance(S{{l, S2) = min{lyl - y21
Set also So = g: distance(y, SI) =
:
0.
1.
Then, So is not empty, bounded,
and closed.
Furthermore, So n C+(i) = 0.
Choose two points y"I E SI and
2
k = 1,2,...) of
y ' 2 E S 2 and t w o sequences {tk
: k = 1, 2, ... } and {sk :
real numbers so that tk < Sk (k = 1, 2, ... ),
lim tk = +oo,
lim sk = +oo,
k-+oo
k-+oo
limp P(tk,
y"1, and k lim G p1sk, r7) = 92. Assume that
r), SO <
k
2 and distance(p(sk, r), SI) > 2.
Then, there exists a Tk for each k such that
tk < Tk < sk and p-(Tk, rl) E So.
Choose a subsequence (at
: e = 1,2.... ) of
{rk
: k = 1, 2, ... } so that lim at = +oo and lim p(at, r) = # exist. Then,
Y E So n C+ (Y-)) = 0. This is a contradiction.
(5) C+ (1-7) is invariant: It must be shown that if ff E C+ (r), then p1t, y) E L+(1-7) f o r all t E 11(y). In fact, there exists a sequence {tk : k = 1, 2, ... } of real numbers
such that Iii o tk = +oo and k lim o P-14, r) = 17. From (IX.1.1) and the continuity
k
of P(t,y) of y", it follows that for each fixed t.
k li +co
tk, n) = k
UM P(t, Pptk, r1)) = Pit, y) E L+(7).
El
The materials of this section are also found in [CL, Chapter 16, §1, pp. 389-3911 and [Har2, Chapter VII, §1, pp. 144-1461.
IX-2. Liapounoff's direct method
In order to find the behavior of pit, r)) as t
+oo, it is important to locate
L+(rl). As a matter of fact, if p(t, r)) is bounded for t _> 0 and if a set M contains
,C+ (1-7), then p(t. r)) tends to M as t -+ +oo, i.e.,
lim inf(1p(t, r)) - yj : y E M) = 0.
t
+a:
Otherwise, there must be a positive number co and a sequence {tk . k = 1, 2.... } of
real numbers such that
k
lim tk = +00, limo P140 7) exists, and ipjtk, >)) -y"1 eo
for all y E Jul. Hence,
k
lim
-1) V M. This is a contradiction. Keeping this
k-.too
fact in mind, let us prove the following theorem (cf. [LaL, Chapter 2, §13, pp.
56-71]).
Theorem IX-2-1. Let V(y) be a real-valued, continuous, and continuously differentiable function for Jyl < ro, where 0 < ro < +oo. Set
fDt = J#: V (y-) < e}, St = {y" E DI : V-(y) J(y) = 0}.
Mt = the largest invariant set in St,
F' a`,
yl
fl(
where Vi(y) ' f(y) _ ayJ fJ(y), y
, and f( y) _
. Suppose
J-1
yn
fn (y)
1Q1+00
Proof.
It suffices to consider the case when p(to,rl > ro for some to > 0. There are two possibilities:
lim V (y) = +oo, then all solutions p(t, r)7) are bounded for t > 0.
advance. To do this, the following theorem is useful.
Theorem IX-2-3. If V (y) is a real-valued and continuously differentiable function for all y" E lR^ such that VV(y) f(y) < 0 for Iyi > ro, where ro is a positive number,
must be shown in
for all y E C+(t)). Since C+(71 is invariant, V(p(t, y)) = uo for all y E L+(171 and t > 0. Therefore, VV(p(t, y)) &I t, y)) = 0 for all y E C+ (q) and t > 0. Setting t = 0, we obtain Vc(y) f (y) = 0 if y E C+(rl, i.e., C St if i E Vt. Hence,
C+() C Mt for if E Vt.
The following theorem is useful in many situations and it can be proved in a way similar to the proof of Theorem IX-2-1.
Theorem IX-2-2. Let V (y) be a real-valued and continuously differentiable func- tion for ally E lR' such that VV(y) f (y) < 0 for all y" E R". Assume also that the orbit p(t, rte) is bounded fort > 0. Set S = 1g: Vi(y) f(y) = 0) and let M be the
lim u(t) = uo exists and no > po. This implies that V(y) = uo
and that
largest invariant set in S. Then, L+ (7-1) C M.
The proof of this theorem is left to the reader as an exercise. In order to use Theorem IX-2-2, the boundedness of p(t,
it E 1)t. Hence,
closed, connected, and invariant (cf. Theorem IX-1-4). Furthermore,
Let po be the minimum of V(rl') for Iy7 < r. Then, po < u(t) and
282
IX. AUTONOMOUS SYSTEMS
that there exist a mat number t and a positive number r such that 0 < r < ro,
Dt C {y : Iy1 < r}, and Vg(y-) f (y) < 0 on De. Then, L+ (1-7) C Me for all n E Vt.
Proof.
Set u(t) = V(p(t,rl")). Then,
dot)
= V9(r(t,r7))'
dtp(t,i) =
Therefore,
du(t) < 0 as long as p(t, t) E Vt. This implies that u(t) < V(1-7) < e
dt
Hence, p(t, rl E Dt for t > 0 if ij E Vt.
for r) E Vt as long as p(t, n-) E Vt.
Consequently, Ip(t, rlI < r for t > 0. It is known that C+ (171 is nonempty, bounded,
C Vt.
d t) < 0 if
(1)IP-(t,r1)I > ro fort > to,
(2)IP(t,n')I > ro for to < t < t, and Ip1ti,17)1 = ro for some tl > to.
Case (1). In this case, dt V (p( t, r)1) = Vj(p(t, rl) f (p(t, 4-7)) < 0 fort > 0. Therefore,
V (r(t,
V (r(to, t))) for t > to. Hence, p(t, tt") is bounded for t > 0.
in a way similar to Case (1). The details are left to the reader as an exercise.
0
IX-3. Orbital stability
In this section, we introduce a concept of stability (respectively asymptotic stability) in a sense different from the stability (respectively the asymptotic stability) of Chapter VIII (cf. Definitions VIII-1-1 and VIII-1-3). We denote again by C(rf the orbit passing through fl. Also, assume that every solution p(t, if) is defined for
t>0.
Definition IX-3-1. An orbit C(sjo) is said to be orbitally stable as t -+ +00 if, for any given positive number e, there exists another positive number b(e) such that
distanee(p"(t, rl"), C(rlo)) < c for t > 0 whenever 14 - rlo1 < b(e).
This definition is independent of the choice of the point ijo on the orbit C(ilo). If
C(i-lo) = C(#,) and if the condition of this definition is satisfied in terms of ilo, then the same condition is satisfied also in terms of , with a different choice of b(e).
The following example illustrates these new concepts.
where it = [c?]. It is easy to show that if r(n) - r(q) = b > 0 and
9(ili) = 9(ij2), then distance(p(t, f2), C(#,)) = b. Hence, the orbit C(ij) is orbitally stable as t -+ +oo for every i ) . However, i h ) - p (b ff) I = r(il') + r(')2 )
since 1r(ffi)t+9(ili)1-1r(il2)t+9(il2)j = 6t (cf. Figure 1). Therefore, every nontrivial solution p"(t, i)) is not stable as t +oc in the sense of Definition VIII-1-1.
Definition IX-3-3. An orbit C(fo) is said to be orbitally asymptotically stable as t - +00 if
(i) C(ilo) is orbitally stable as t +oo,
C(i )) = 0
(ii) there exists a positive number 6o such that lim
whenever {il - iloj < 60.
This definition is independent of the choice of the point 4 on the orbit. However, the choice of 6o depends on rjo.
Consider a system of differential equations
(IX.3.1)
= f(y,
284
IX. AUTONOMOUS SYSTEMS
where the entries of the R"-valued function f is continuously differentiable on the entire y-space W1. Assume also that the solution At, rjo) is periodic in t of period
1 (i.e., #(t + 1, ito) = p1t, i"Jo) for -oo < t < +oo) and that f'(p1t, ijo)) 54 0. The system of linear differential equations
di7
Of
(IX.3.2)
=
is called the first variation of system (IX.3.1) with respect to the solution pit, rlo).
The coefficients matrix of (IX.3.2) is periodic in t of period 1.
Let pi, p2, ... , p be the multipliers of (IX.3.2) (cf. Definition IV-4-5). Since
2 p-(t,
(pit, o)) d p1t, ijo), linear system (IX.3.2) has a nontrivial periodic
solution fjt, im). This implies that one of the multipliers must be 1. Set pl = 1. The following theorem gives a basic sufficient condition for orbitally asymptotic stability.
Theorem IX-3-4. If n - 1 multipliers p2,... , p, satisfy the condition ipiI < 1
(j = 2,... , n), then the periodic orbit C(rjo) is orbitally asymptotically stable as t-4+oo.
Prof.
We prove this theorem in five steps. It suffices to find an (n - 1)-dimensional
manifold M in a neighborhood of the point
so that
(1)M is transversal to the orbit C(i) at ilo; i.e., the tangent of C(yo) is not in the tangent space of M at ilo,
(2)there exist two positive numbers K and r such that
Iu(t,n - p1t,i )I <- KIn -
(cf. Figure 2).
iole-°:
for i EM and t>0
FIGURE 1.
FIGURE 2.
Step 1. There exists an invertible n x n matrix P(t) whose entries are real-valued, continuously differentiable, and periodic in t of period 1 (or 2) such that the transformation
(1X.3.3)
6 = P(t)v"
3. ORBITAL STABILITY
285
changes (IX.3.2) to
(IX.3.4)
dt = AV,
A = [ 0 B,
where B is a constant (n - 1) x (n - 1) matrix (cf. Theorems IV-4-1 and IV-4-3).
Since the absolute values of n -1 multipliers are less than 1, there exist two positive numbers Co and a such that
(IX.3.5)
Iexp[tB] I < Coe-tot
for
t > 0.
A fundamental matrix solution of (IX.3.2) is given by
1/(t) = P(t) exp[tA],
exp[tA] =
[1
0
1
o exp[tBj J
This implies that the first column vector of P(t) is a periodic solution of (IX.3.2) and all the other columns of W(t) are not periodic. Hence, assume that
(IX.3.6)
P(t) _ [d AtA), Q(t),
where Q(t) is an n x (ii - 1) matrix. Note that det[P(t)] 56 0.
