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MATH 623 Ordinary Differential Equations Fall 2010Instructor: Georgi Medvedev

5 Lecture 5. Linear systems and Floquet theory.

5.1 Linear systems

Consider the homogeneous system of ODEs

x= A(t)x and (5.1)

and inhomogeneous systems of differential equations

x= A(t)x + h(t), x Rn, (5.2)

whereA(t) Rnn is a continuous matrix andh(t) Rn is a continuous vector.

Definition 5.1. Suppose{x(1), x(2), . . . , x(n)}is a set of solutions of (5.1). Let

X(t) = col (x(1), x(2), . . . , x(n)) Rnn

such thatdetX(t0)= 0for somet0 R. ThenX(t)is called fundamental matrix solution of (5.1).

Definition 5.2. Fundamental matrix solution of (5.1),X(t), is called a principal matrix solution of (5.1) attimet0 R ifX(t0) =I.

Lemma 5.3. The rank ofX(t)does not depend ont.

Proof. Let x(1), x(2), . . . , x(r) be solutions of (5.1). Then

x(t) =c1x(1)(t) + c2x

(2)(t) + + crx(r)(t) (5.3)

solves (5.1). Ifx(t0) = 0for somet0 R then

c1x(1)(t) + c2x

(2)(t) + + crx(r)(t) = 0t R, (5.4)

by uniqueness of solutions of the intial value problems for (5.1). Therefore,x(1)(t), x(2)(t), . . . , x(r)(t)are

linearly independent for everyt iff they are linearly independent for some t0.

We will need the following properties of solutions of (5.1) and (5.2).

Lemma 5.4.

1. IfX(t) is a fundamental matrix solution then X(t)X1(t0) is the principal matrix solution at timet0.

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2. The variation of constants formula:

x(t) =X(t)X1(t0

)x(t0

) + t

0X(t)X1(s)h(s)ds. (5.5)

3. The Liouvilles formula:

detX(t) = detX(t0)exp{

tt0

trA(s)}. (5.6)

Proof.

1. follows by verification.

2. By a change of variable in (5.2)

x(t) =X(t)y(t), (5.7)we have

Xy+ Xy= AX y+ h. (5.8)

PlugX=AXin (5.2) to obtainy= X1(t)h(t). (5.9)

The statement in 2) follows after integrating (5.9) and using (5.7).

3. Use product rule to verify the Leibniz formula for determinants:

(det X) =n

i=1

det

X1,1 X1,2 X1,n...

......

Xi,1 Xi,2 X

i,n

......

...

Xn,1 Xn,2 Xn,n

. (5.10)

Since X=AX,

Xi,k =n

j=1

ai,jXj,k, i, k {1, . . . , n},

and

( Xi,1, . . . , Xi,n) =n

j=1

ai,j(Xj,1, . . . , X j,n), i {1, . . . , n}. (5.11)

Thus,

(det X) =n

i=1

det

X1,1 X1,2 X1,n...

......n

j=1 a1,jXj,1n

j=1 a2,jXj,2 n

j=1 an,jXj,n...

......

Xn,1 Xn,2 Xn,n

. (5.12)

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Subtract from rowi on the rhs of (5.13) the linear combination:

n

j=1j=i

ai,j(Xj,1, . . . , X j,n)

to obtain (5.13)

(det X) =n

i=1

det

X1,1 X1,2 X1,n...

......

ai,iXi,1 ai,iXi,2 ai,iXi,n...

......

Xn,1 Xn,2 Xn,n

= trA(t)det X(t) (5.13)

The statement in 3) follows by integrating (5.13).

Example 5.5. Consider

x= A(t)x, A(t) =

1 4(cos 2t)2 2 + 2 sin 4t2 + 2 sin 4t 1 4(sin2t)2

. (5.14)

From the characteristic equation

2 + 2 + 1 = 0 (5.15)

one finds that the eigenvalues ofA(t)are negative1,2 = 1. Nonetheless, the equilibrium at the origin is

unstable, because

x(t) =et

sin2tcos2t

(5.16)

solves (5.14).

5.2 Linear systems with periodic coefficients

Consider

x= A(t)x, x Rn, (5.17)

whereA(t)is continuous periodic matrix with the least period >0,A(t + ) =A(t) tR

.

Theorem 5.6. A fundamental matrix solution of (5.17) can be written as

X(t) =P(t)etB, (5.18)

whereP(t)is periodic andB (in general complex) is a constant(n n)matrix.

Remark5.7. Equation (5.18) implies that the stability of the equilibrium at the origin is deteremined by the

spectrum ofB .

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Proof. Note thatX(t) =X(t + )is a fundamental matrix solution of (5.17):

X(t) =A(t + )X(t + ) =A(t)X(t).

Thus, for some constant nonsingular matrix C Rnn,

X(t + ) =X(t)C. (5.19)

Set

C=eB (5.20)

and define

P(t) =X(t)etB (5.21)

ThenP is periodic, because

P(t + ) =X(t + )eBetB =X(t)etB =P(t).

Remark5.8. When Chas negative eigenvalues, Bas defined in (5.20) is complex. However, if one redefines

C=X(t)1X(t + 2)thenB can be chosen real.

Definition 5.9. MatrixC in (5.19) is called the monodromy matrix. The eigenvalues

1, 2, . . . , n

ofCare called characteristic multipliers. Complexk is called a characteristic exponent if

k =ek.

Theorem 5.10.

12 . . . n = exp{

0

trA(t)dt}, (5.22)

ni=1

i = 1

0

trA(t)dtmod 21i (5.23)

Proof. Let X(t)be a principal matrix solution. Then by Liouvilles formula,

det X(t) = exp{ 0

trA(s)ds}. (5.24)

On the other hand

det X(t) = det P(t)etB.

Thus,

det P()eB = exp{

0

trA(s)ds}.

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But,

det P() = det P(0) = det X(0) = 1.

Finally,12 . . . n = det P()e

B = exp{

0

trA(s)ds}.

This shows (5.22). Equation (5.23) follows from (5.22).

For2Dflows Theorem it allows to compute the characteristic multipliers. To show this, consider

x= f(x), x R2, (5.25)

where as usualf(x) = (f1(x), f2(x))T is a smooth vector field. Supposex = u(t)is a periodic solution of(5.25) with period1. The variational equation is obtained by linearizing the flow near O = {x= u(t) : t

S1}, i.e., pluggingx = u(t) + (t)in (5.25) and keeping the linear term:

= A(t), A(t) =Df(u(t)) . (5.26)

We are interested in stability of the trivial solution = 0 of the variational equation (5.25). By differentiatingboth sides of

u= F(u(t)) ,

we find that= u(t)solves (5.26). Therefore, one can choose the fundamental matrix solution as

X(t) = (u(t)).

Thus,

X(t + ) = (u(t + )) = (u(t)) =

1 0

. (5.27)

From (5.27) conclude that one multiplier cahrachteristic multiplier is1 = 1. Theorem 5.2 then yields thesecond multiplier

2 = exp{

0

trDf(u(s))ds}= exp{

0

divf(u(s))ds}.

Therefore, the sign of0 divf(u(s))dsdetermines the stability of the periodic orbit of a 2Dflow.

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