0387986995Basic TheoryC

Theorem IV-4-1 (G. Floquet [Fl]). Under assumptions (1) and (2), the funda- mental matrix solution 4i(t) of (IV.4.1) defined by the initial-value problem (IV.4.3) has the form

where P(t) and B are n x n matrices such that

(a) P(t) is invertible, continuous, and periodic of period w in t,

(Q) B is a constant matrix such that 4'(w) = exp[wB].

Observation IV-4-2. As was explained in Observation IV-3-8, letting 4'(w) _ S + N = S(I + M) be the S-N decomposition of 4(w), we define log($(w)) by log(4?(w)) = log(S) + log(1 + M), where

In the case when A(t) is a real matrix, the unique solution 4 (t) of problem (IV.4.3) is also a real matrix. Therefore, the entries of 4'(w) are real. Since S and N are real matrices, the matrix M = S-'N is real. Therefore, log(I + M) is also real. Let us look at log(S) more closely. If \j is a complex eigenvalue of 4'(w), its complex conjugate A3 is also an eigenvalue of 4'(w). In this case, set A,+1 = A3. It is easy to see that the projection Pj+1(4?(w)) is also the complex conjugate of P,(4'(w)).

However, if some eigenvalues of 4'(w) are negative, log[S] is not real. To see this more clearly, rewrite log[s] in the form

log[S] _ E log[A,]Pi(4?(w)) + E log1A3]P'(4'(w))-

A, <0

uation, let us look at S2. By virtue of the relations given in Lemma IV-1-9, we

log(S2] _ E log[(Ai)2]Pj(4'(w)) + 2 E log[a3]P,(4(w))

is a real matrix. Therefore, we can find a real matrix

log[4'(w)2] = log[S2J + 2log(In + MI.

Now, observe that 4'(,)2 =' (2w) and 4'(t + 2w) = 4'(t). (2w). Thus, setting

we obtain the following theorem.

Theorem IV-4-3. In the case when the matrix A(t) is real, the matrix 4s(t) has

the form

P(t) exp[tB],

where P(t) and B are n x n real matrices such that

(a) P(t) is invertible, continuous, and periodic of period 2w in t,

(p) B is a constant matrix such that 4?(w)2 = exp[2wB].

Remark IV-4-4. In the case when 4?(t) = P(t) exp[tB], we have

dP(t) dt

A(t)P(t) - P(t)B. Therefore, (IV.4.1) is changed to

-dz- = Bz

(IV.4.8)

dt

by the transformation y" = P(t)!. As noted above, P(t) is periodic in t of period w (or, respectively, 2w when A(t) is real), the behavior of solutions of (IV.4.1) is derived from the behavior of solutions of (IV.4.8).

Definition IV-4-5. The eigenvalues p1.... , p of the matrix B are called the characteristic exponents of equation (IV.4.1). The eigenvalues pi = exp[wpjl, ... , p = of the matrix 4?(w) (or, respectively, when A(t) is real, the eigenvalues pi = exp(2wpi],... , P = exp[2wµn] of 4?(w)2) are called the multipliers of equation (IV.4.1).

The following basic result is a corollary of Theorem IV-3-1.

Corollary IV-4-6. All solutions of equation (IV.4.1) tend to 0' as t -. +oo if and only if Ipj I < 1 for all j (or R(pj) < 0 for all j).

Observation IV-4-7. Let be an eigenvector of 45(w) associated with a multiplier p. Then, the solution d(t) = t(t)p" of (IV.4.1) satisfies the condition ¢(t + w) = pi(t). The terminology multiplier came from this fact. (In the case when A(t) is real, choosing an eigenvector p" of 4;(w)2 associated with a multiplier p, we obtain

(t + 2w) = pi(t) if d(t) = 4i(t)p.)

Example IV-4-8. Consider the initial-value problem

(IV.4.9) dY = A(t)Y, Y(0) = 12,

-it-

where

90 IV. GENERAL THEORY OF LINEAR SYSTEMS

Since p.(\) = (a - 1) 2, Pt(4i) = 12, we obtain

exponents are 0, 0, and the multipliers of (IV.4.9) are 1, 1. Note that in this case, as both of two multipliers are positive, it is not necessary to take 4i(2a)2 and period

41r to find B.

IV-5. Linear Harniltonian systems with periodic coefficients

This section is divided into two parts. In Part 1, we explain the structure of solutions of a linear Hamiltonian system. The basic facts are found in the theory of real symplectic groups and their Lie algebra. If we denote by J the real (2n) x (2n)

matrix I 0 O 1, where In is the n x n identity matrix, the Lie algebra of the

real symplectic group Sp(2n, R) of order 2n is the set of all real 2n x 2n matrices of the form JH, where H is a real (2n) x (2n) symmetric matrix. A linear system

T = A(t)f on y' E R2" is a linear Hamiltonian system, if there exists a real

(2n) x (2n) symmetric matrix H(t) such that A(t) = JH(t). Consequently, the fundamental matrix 0(t) of this system belongs to Sp(2n,R) if 0(0) = I. We explain all these elementary facts of Hamiltonian systems in Part 1. We also prove that if G E Sp(2n, R) and G = S+N = S(I2"+M) is the S-N decomposition of G, then S and 12" + M belong to Sp(2n, R). In Part 2, we refine the Floquet theorem

(cf. Theorem IV-4-1) for the linear periodic Hamiltonian systems in such a way that the periodic transformation y = P(t)i given in Remark IV-4-4 is canonical. This means that P(t) belongs to Sp(2n, R). The important consequence is that the

linear system dx = BE of Remark IV-4-4 is a Hamiltonian system.

Part 1

Since [G(t)-']T = [G(t)T]-1, we further obtain

[G(t)-']T [dG(t)]T(- J)= JdG(t )G(t)_1.

Observing that the left-hand side of this relation is the transpose of the right-hand side, we conclude that H(t) is symmetric. 0

Observation 1. If GT JG = J, then (G-' )T JG-' = J and GJGT = J. In fact,

GT JG = J implies that (G-' )T JG-' = J. Then, taking the inverse of both sides of the first relation, we obtain the second relation (cf. (IV.5.3)).

Definition IV-5-3. A (2n) x (2n) real invertible matrix G such that GT JG = J is called a real symplectic matrix of order 2n. The set of all real symplectic matrices of order 2n is denoted by Sp(2n, R).

It is easy to show that Sp(2n, R) is a subgroup of GL(2n, R) (cf. Observation 1).

Hence, Sp(2n, R) is called the real symplectic group of order 2n.

Observation 2. If we change Hamiltonian system (IV.5.1) by a linear transfor- mation

(rV.5.1) becomes

dt - {P(t)-1JH(t)P(t) - P(t)-1 dt)J i.

Furthermore, if P(t) E Sp(2n, R) for t E Z, this system becomes

Since P-(t)P(t) = 12 and P(t)T E Sp(2n,R), we obtain

1 P(t)-'J = JP(t)T.

Observe that -J dP(t) P(t) is symmetric since P(t) E Sp(2n, R) (cf. Lemma

IV-5-2). This implies that (IV.5.5) is a Hamiltonian system.

Definition IV-5-4. Transformation (IV.5.4) is called a canonical transformation if P(t) E Sp(2n, R).

Observation 3. For a given constant matrix G in Sp(2n, R), let us construct the

S-N decomposition G = S + N, where S is diagonalizable, N is nilpotent, and

SN = NS. Since G is invertible, S is invertible (cf. Remark IV-1-13). If we set

M = S-' N, then M is also nilpotent, and we derive the unique decomposition

94 W. GENERAL THEORY OF LINEAR SYSTEMS

by the transformation

(IV.5.9) y = P(t)f.

The main concern of Part 2 is to show that a matrix P(t) can be chosen so that the transformation (IV.5.9) is canonical. Note that Y = exp[tB] is the unique

solution of the initial-value problem = BY, Y(O) = I2,,. Therefore, if JB is

symmetric, the matrix exp[tB) is symplectic. Hence, if JB is symmetric, the matrix

P(t) = 4i(t) exp[-tB] is symplectic, since 4i(t) is symplectic. This implies that in order to show that the transformation (IV.5.9) is canonical, it suffices to show that we can choose P(t) so that JB is symmetric.

Hereafter we use notations and results given in §IV-4. For examples,

Pj(4'(w))2 = P

Taking the transpose of (IV.5.10), we obtain

\P(.t(w))T )2 = Pj(.0(W))T.

Note that pAr(A) = PA(A) for any square matrix A. Also,

log[4?(w)2] = log[S2] + 2Iog[I2r, + MI,

log[S2] _ log[(,\,)2]Pi(I(w)) + 2 E log[Aj]Pj('(w)),

ai <0 other ,

1 m+l

1og[I2n + M] = E (-) Mm,

m=1

and

B = log[4i(w)2].

In order to prove that JB is symmetric, it suffices to prove that J log[S2] and

J log[I2i + M] are symmetric. To do this, let us first prove the following lemma.

Lemma IV-5-10. For the symplectic matrix 4i(w), we have

where Aj (j = 1, 2,... , k) are distinct eigenvalues of 4i(w).