teorema liou ville
TRANSCRIPT
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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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