0387986995Basic TheoryC

for Jpj > 2. From (VIII.4.6), it is concluded that the diagonal entries A1, ... , A,, of

Bo are eigenvalues of A and that this allows us to set A = B0 and P0 = In, where I,, is the n x n identity matrix. Then, (VIII.4.7) becomes

(PhAh)PP + 9p - Bo,6,

successively, where PP,, and gp,, are the j-th entries of the vector P. and

respectively, and FP,, are known quantities. If >phAh - Aj 36 0, set 9,j = 0 and

h=1

solve (VIII.4.9). If >phAh - A? = 0, then set 9P,, = Fr,, and choose PPJ in any

h=1

way. This completes the proof of Theorem VIII.4.2.

Observation VIII-4-3. Assume that R(AE) < 0 for j = 1, ... , r and R(A3) > 0

vector Bou + g"(u) is zero for t ¢ I, ... , r. In fact, look at g"(u"). Then, the f-

n

th entry of the coefficient g"p is zero if Al 96 >phAh. Note that, if t > r and

h=1

n

At = > Ph Ah, then (Pr+ 1, . Pn) 0 (0, ... , 0). Hence, in such a case, uP = 0 if

h=1

(ur+1, ... , u,) = (0, ... , 0). Therefore, the 1-th entry of the vector Boti + 9"(uis

zero forI>rif(ur+1,...,u,,)=(0,....0).

Observation VIII-4-4. Under the same assumption on the Aj as in Observation

VIII-4-3, set (ur+1, ... , un) = (0,... , 0). Then, the system of differential equations

members of (VIII.4.10) are polynomials in (ul,... ,ur).

Observation VIII-4-5. Assume that R[Ah+1J <_ R[Ah]. Then, (V111.4.10) can be written in the form

Hence, system (VIII.4.10) can be solved with an elementary method. To see the structure of solutions of (VIII.4.10) more clearly, change (ul, ... , ur) by u2 _ ea'ivi (j = 1,... , r). This transformation changes (VIII.4.10) to

dv)

This, in turn, shows that the general solution of (VIII.4.10) has the form u) _ ea'ii) (t, cl, ... , cr) (j = 1.... , r), where (cl,... , c,.) are arbitrary constants and

0,(t,c1,....c,-) are polynomials in (t.cl,... ,G)

An analytic justification of the formal series 15(ii) is given by the following the- orem.

Theorem VIII-4-6. In the case when the entries of the Z"-valued function f'(y-) on the right-hand side of (VI11.4.1) are analytic in a neighborhood of 0, under the same assumption on the A) as in Observation VIII-4-3, the power series P(d) is convergent if (ur+i, ... , u") = (0,... , 0).

The proof of this theorem is straight forward but lengthy (cf. [Si2J). The key fact is the inequality

A) - >PhAh

r

for some positive number a if E ph is large. In this case, a iajorant series for P

h=1

can be constructed.

The construction of such a majorant series is illustrated for a simple case of a system

dy

dt = Ay + f (y),

where A is a nonzero complex number and AM is a convergent power series in y given by (VIII.4.2). According to Theorem VIII-4-2, in this case there exists a

formal transfomation ff = d + Q(u) such that dddt = Ad, where

IpI?2

IeI>2

250 VIII. STABILITY

ui"A,. Then,

lal>2

It can be shown easily that Y(ul) is a convergent majorant series of Q(ur). This proves the convergence of Q(ii).

Putting P(u") and the general solution of (VIII.4.10) together, we obtain a particular solution y" P(#(t, cl) of (VIII.4.1), where

=

(t,cj = (e'\It7Pi(t,cl,... ,cr),... ,e*\1 'Wr(t,cl,... ,cr),0,... ,0).

This particular solution is depending on r arbitrary constants c = (cl,... , c,.).

Furthermore, this solution represents the stable manifold of the trivial solution of

(VIII.4.1)if W(Aj)>0for j#1,...,r.

Remark VIII-4.7. In the case when y, A, and AM are real, but A has some eigenvalues which are not real, then P(yl must be constructed carefully so that the

the eigenvalues of A are a ± ib. If IV = ['] is changed by ul = 1/i + iy2i and

=- iy, system (VIII.4.1) becomes

where gP,,- is the complex conjugate of g,,,p,. If a 54 0, using Theorem VIII-4-2, simplify (VIII.4.11) to

ddt

by the transformation

Now, system (VIII.4.12), in turn, is changed back to - = At by wI = V 2 V2

where ql,p,,p2 and g2,p,,p2 are real numbers. Similar arguments can be used in general cases to construct real-valued solutions. (For complexification, see, for example, [HirS, pp. 64-65].)

For classical works related with the materials in this section as well as more general problems, see, for example, [Du).

VIII-5. Two-dimensional linear systems with constant coefficients

Throughout the rest of this chapter, we shall study the behavior of solutions of nonlinear systems in 1R2. The R2-plane is called the phase plane and a solution curve projected to the phase plane is called an orbit of the system of equations. A diagram that shows the orbits in the phase plane is called a phase portrait of the orbits of the system of equations. As a preparation, in this section, we summarize the basic facts concerning linear systems with constant coefficients in R2.

Consider a linear system

dy =

(VIII.5.1) Ay, dt

where E R2 and A is a real, constant, and invertible 2 x 2 matrix. Set p = trace(A) and q = det(A), where q ¢ 0. Then, the characteristic equation of the matrix A is

1\2 - pA + q = 0. Hence, two eigenvalues of A are given by

It is known that

p = AI +A2, q = AIA2, and Al - A2 =2 P2 - q.

4

252 VIII. STABILITY

Also, let t and ij be two eigenvectors of A associated with the eigenvalues Al and A2, respectively, i.e., At = All;, l 0 0, and Au = A2ij, ij 0 0. Observe that, if y(t) is a solution of (VIII.5.1), then cy(t +r) is also a solution of (VIII.5.1) for any constants c and T. This fact is useful in order to find orbits of equation (VIII.5.1) in the phase plane.

Case 1. Assume that two eigenvalues Al and A2 are real and distinct. In this case, two eigenvectors { and ij are linearly independent and the general solution of differential equation (VIII.5.1) is given by

where cl and c2 are arbitrary constants and -oo < t < +oo.

2

la: In the case when Al > A2 > 0 (i.e., p > 0, q > 0 and 4 > q), the phase portrait of orbits of (VIII.5.1) is shown by Figure 4. The arrow indicates the direction in

lb: In the case when 0 > Al > A2 (i.e., p < 0, q > 0 and 4 > q), the phase

portrait of orbits of (VIII.5.1) is shown by Figure 5. The trivial solution 0 is stable as t --i +oo. The point (0, 0) is called a stable improper node.

4

1c: In the case when Al > 0 > A2 (i.e., q < 0), the phase portrait of orbits of

Note that as t -+ -oo (or +oo), only two orbits of (VIII.5.1) tend to 0. The point

(0, 0) is called a saddle point.

2

Case 2. Assume that two eigenvalues Al and A2 are equal. Then, q = 4 and

Al=A2=29& 0.

2a: Assume furhter that A is diagonalizable; i.e., A = 212i where 12 is the 2 x 2

identity matrix. Then, every nonzero vector c is an eigenvector of A, and the general solution of (VIII.5.1) is given by y"(t) = exp [t]e. In this case, the phase portrait

of orbits of (VIII.5.1) is shown by Figures 7-1 and 7-2. As t -+ +oo, the trivial

solution 6 is unstable (respectively stable) if p > 0 (respectively p < 0). Note that,

for every direction n, there exists an orbit which tends to 6 in the direction n" as t tends to -oo (respectively +oo). The point (0, 0) is called an unstable (respectively stable) proper node if p > 0 (respectively p < 0).

identity matrix and Nris a 2 x 2 nilpotent matrix. Note that N 0 and N2 = O.

Hence, exp[tA] = exp 12t] {I2 + tN}. Observe also that a nonzero vector c' is an

eigenvector of A if and only if NcE = 0. Since N(Nc) = 0, the vector NcE is either 6 or an eigenvector of A. Hence, NE = ct(-) , where is the eigenvector of A which was given at the beginning of this section and a(c) is a real-valued linear homogeneous function of F. Observe also that at(c) = 0 if and only if c' is a constant multiple of the eigenvector

by y(t) = exp [t] {c + ta(c7}, where c" is an arbitrary constant vector. In this

case, the phase portrait of orbits of (VIII.5.1) is shown by Figures 8-1 and 8-2. The trivial solution 6 is unstable (respectively stable) as t - +oo if p > 0 (respectively p < 0). The point (0, 0) is called an unstable (respectively stable) improper node if p > 0 (respectively p < 0).

Case 3. If two eigenvalues \I and .12 are not real, then q > and

4

Al =a+ib, A2=a-ib, a= 2,

4

Note that b > 0. Set A = a12 + B. Then, B2 = -b212, since two eigenvalues of B