Czechoslovak Mathematical Journal
Jaroslav Kurzweil; Garyfalos Papaschinopoulos
Structural stability of linear discrete systems via the exponential dichotomy
Czechoslovak Mathematical Journal, Vol. 38 (1988), No. 2, 280–284
Persistent URL: http://dml.cz/dmlcz/102223
Terms of use:
© Institute of Mathematics AS CR, 1988
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents
strictly for personal use. Each copy of any part of this document must contain these Terms of use.
This document has been digitized, optimized for electronic delivery and
stamped with digital signature within the project DML-CZ: The Czech Digital
Mathematics Library http://dml.cz
Czechoslovak Mathematical Journal, 38 (113) 1988, Praha
STRUCTURAL STABILITY OF LINEAR DISCRETE SYSTEMS VIA
THE EXPONENTIAL DICHOTOMY
JAROSLAV KuRzwEiL, Praha, and GARYFALOS PAPASCHiNOPOULOS, Xanthi
(Received April 1, 1986)
INTRODUCTION
Consider the difference equation
(1)
x(n + 1) = A(n) x{n),
n є N = {0, 1, ...}
where A(n) is an invertible k x fc-matrix for n є N such that
\А(п)\йМ,
\А'\п)\йМ9
M>0,
neN.
We denote by Wthe space of the systems of the form (1) and by | • | the Euclidean
norm.
Equation (1) is said to possess an exponential dichotomy if there exist a projec­
tion P (P 2 = P) and constants K > 0, 0 < p < 1, such that
\X(n) РХ~\т)\
й Крп~т ,
\X(n) (I - P) X'\m)\
n ^ m
m n
й Kp ~ ,
m ^ n
where X(n) is the fundamental matrix solution of (1) such that X(0) = L
Consider a system in W
{2)
y(n + 1) = B(n) y{n).
According to [5, p. 17], (l) and (2) are said to be topologically equivalent ifthere
exists a function h: N x Rk ^> Rk with the following properties:
(i) if |x| -^ oo, then |ft(n, x)| ^ oo uniformly with respect to n,
(ii) the map /?„(•) = h(n, •) from Rk to Rk is a homeomorphism for each и,
(iii) the map #„(•) = fyT*(') from Я к to jRfc also has property (i),
(iv) if x(n) is a solution of (1) then h(n, x(n)) is a solution of (2).
Equation (1) is called structurally stable if there exists ö > 0 such that if (2)
belongs to H^and \B(n) — A(n)\ < ô then (2) is topologically equivalent to (l). .
The results of this paper are:
(i) Ifequation (1) has an exponential dichotomy then it is topologically equivalent
•280
to the system
(3)
xL{n + 1) = eL xL{n),
i = 1, 2 , . . . , k
where eL = l/e or eL = e.
(ii) System (1) is structurally stable if and only if it has an exponential dichotomy.
The above results are the discrete analogues of those of Palmer [4] and [5]. We
denote that the first result is not derived directly form the continuous case.
We also note that some results on exponential dichotomy and structural stability
ofdiscrete systems are included in the papers [6], [7], [8], [9], [10], [11].
MAIN RESULTS
Proposition 1. If equation (1) has an exponential
logically equivalent to (3).
dichotomy then it is topolo-
Proof. Suppose that (1) has an exponential dichotomy. Let the rank of the cor­
responding projection P be /. Using the same method as in [2, pp. 39 — 41] we find
an invertible bounded matrix S(n) with bounded inverse such that the transformation
x = S(n) y transforms (1) into the system
(4)
y(n + 1) = diag (A,(n), A2(n)) y{n)
where A^n) is an / x 1 matrix and A2(n) is a (k — /) x (k — /) matrix.
Moreover, the system
(5)
y(n + l) = A,(n)y(n)
has an exponential dichotomy with a projection of rank equal to / and the system
(6)
y(n + i) =
A2(n)y(n)
has an exponential dichotomy with a projection of rank equal to 0.
Consider equation (5). Let Yj^n) be a fundamental matrix solution of (5). By
Gram-Schmidt orthogonalization of the columns of Yj(w), [2, p. 87], starting with
the first column, we obtain a unitary matrix U^n) and an upper triangular matrix
V{n) in which the diagonal elements are real and positive functions for all n e N
such that Ux{n) = Y^n) V(n). The change of variables y = U^n) z transforms (5)
into the system
(7)
z(n + 1) = Щ\п
_1
+ 1) A^n) Ui(n) z(n) = B^n) z(n) .
The matrix tŽ (w) is a matrix solution of(7) since
= TJ~l{n + 1) TJ(n). Therefore B^n) is an upper
diagonal are real and positive functions on N.
Consider the differential equation
rlog J5i(0) ,
logBi(l),
(8)
z = B(t) z , B(t) =
log B^n) ,
U^n) = Y^n) U(n). So Bt{n) =
triangular matrix in which the
0 й t < 1
1 й
t<2
n ^ t < n + 1
281
From [1, p. 39] we have
log B,(n) = — Jy (z/ - ^ ( n ) ) " 1 log z dz
Z7T1
where y is any simple closed curve which contains in its interior every characteristic
root of Bx(n) but not the origin.
We claim that log B^{n) is a real valued bounded matrix for n є N. It is obvious
that it is a real matrix since Bx{n) is an upper triangular matrix with real positive
diagonal elements. Since the matrices B^n),B^(n)
are bounded for neN there
exist constants X, fi > 0 such that X ^ XL(n) ^ fi, n eN, i = 1, ..., 1 and XL(n) are
the eigenvalues of Bj(n). We choose r > max {^(e + ^ — Я), ^(e + ^)}, Я > e > 0.
So —r + 8 + ^ < r < r — e + X. Consider z 0 e R+: r < z 0 < r — e + X. It is
obviousthat |z 0 — X\ < r — e and |z 0 — ^| < r — e. Let y be the sphere |z — z0\ =
= r. Then wehave \z — At(n)| = \z — z 0 + z 0 — At(w)| = | z ~~ z o| ~ | z o ~ ^(rc)| ^
^ r - (r - e) = г, i = 1,..., /. So if |J5i(n)1 ^ L, L > 0, neiV then [2, p. 47}
implies
1
\(zi - B,(n))-4 s
Ф'-Д'ООГ
<; Фо + г + ьу-*
|z — Aj(n)| ... \z — Я1(п)|
г1
с a constant.
Therefore log Б^и) is a bounded matrix for n є АГ. Hence our claim is proved.
Let Z^t) be the fundamental matrix of (8) such that Z x (0) = I. Then
f*logBi(n)ds
Z,(t) = e
Zi(í) =
Jn
(ř
e
-"~
Zi(w) =
1)logBl(n+1)
(Ґ
е
-" ) І 0 8 В і ( и ) Z i ( n ) ,
Zi(w + 1) ,
n й t < n + 1,
n + 1 ^ t < n + 2.
The above relations yield
Z,{t) = e ( í """ 1 ) , 0 i J , l ( " + 1 ) e logfîl(/J) Z i ( n ) ,
и + 1 â t < n + 2.
Take t = n + 1. Then we obtain
Zi(n + 1) = ^i(n) Z±(n) , и є iV .
Therefore Zi(w) is a fundamental matrix solution of(7).
We prove that equation (8) has an exponential dichotomy. If z(f) is a solution of
(8) then for t є R+ we have
z(t) = Zt(t) Z r * ( M ) Z M ) > M b e i n § t n e integral part of ř.
If |B(i)| й L then for teR+
we have {Z,(t)Z^([t])l
й e I ( ř ~ [ ř ] ) g e 1 . So for
ö
t *> 5 ^ 0 and provided e~ , a > 0, К > 0 are the constants of the exponential
dichotomy of (7) we have
|z(i)| й e £ |z(M)| й K e r e - e ( [ ř ] " w > | z([s])| ^
á K e V ^ - ^ " * > | z ( [ s ] ) | á Í?e 2I e 2a e~ fl(ř ~ s) |z(s)| ,
í ^ 5^ 0.
Then by Palmer's Theorem in [4, p. 9] and arguing as in [5, p. 20] we prove that (8)
•282
and x[ = —XjL are topologically equivalent (c.f. [5, p. 17]). Let ht: R+ xR1 ^> Rl
be the corresponding homeomorphism. If z(t, 0, x) is a solution of (8) such that
z(0) = x we get
hx(t9 z(t, 0, x)) = e " % ( 0 , x), t є R+ , x є R1 .
Take t = n, n eN. So (7) and xx(n + 1) = c~1x1(n) are topologically equivalent.
Now consider system (6). Proceeding as in (5) we obtain a unitary matrix U2(n)
such that the change of variables y = U2(n) z transforms (6) into a system
<9)
z(n + 1) = B2(n) z{n)
where B2(n) is a (k — /) x (fe — /) upper triangular matrix in which the diagonal
elements are real and positive functions on N. Arguing as in (7) we prove that (9)
and x2(n + 1) = ex 2 (x) are topologically equivalent. Let h2:N x iť"" 1 ~> jRfc_1
be the corresponding homeomorphism. Therefore the system
z(n + 1) = diag (Bi(n), B2(n)) z{n)
is topologically equivalent to (3). Hence (4) is topologically equivalent to (3).
The corresponding homeomorphism is h(n, x) = (h^n, [U~~1(n) x} x ),
h2(n,{U~1(n)x}2))
where {U~i(n)x}l
and {U~1(n)x}2
are the components of
1
k
U-i(n)xin
R and R ~\ respectively, U(n) = d i a g ( ^ ( n ) , U2(n)). So (l) and (3)
are topologically equivalent. The corresponding homeomorphism is h(n, x) =
= h(n, S'1^) x) and the proof ofthe proposition is complete.
Proposition 2. Equation (l) is structurally
tial dichotomy.
stable ifand only if it has an exponen­
Proof. The proofofnecessity is given in [10, Proposition 2]. Using Proposition 1,
the roughness of the exponential dichotomy [3, p. 232] and the same argument as
in [5, p. 20] we can easily prove sufficiency.
References
[1] W. A. Coppel: Stability and Asymptotic Behaviour of Differential Equations, Heath.
Boston, 1965.
,[2] W. A. Coppel: Dichotomies in Stability Theory, Lecture Notes in Mathematics, No. 629,
Springer Verlag, Berlin, 1978.
[3] D. Henry: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathe­
matics, No. 840, Springer-Verlag, Berlin, 1981.
,[4] K. J. Palmer: A characterization of exponential dichotomy in terms of topological equi­
valence. J. Math. Anal. Appl. 69 (1979), 8 - 1 6 .
[5] K. J. Palmer: The structurally stable linear systems on thehalf-line are those with exponential
dichotomies. J. Differential Equations, 33 (1979), 16—25.
,[6] G. Papaschinopoulos and /. Schinas: Criteria for an exponential dichotomy of difTerence
equations, Czechoslovak Math. J. 35 (110) 1985, 295-299.
,[7] G. Papaschinopoulos and /. Schinas: A criterion for the exponential dichotomy of difTerence
equations, Rend. Sem. Fac. Sci. Univ. Cagliari, Vol. 54, fasc. 1 (1984), 61 — 71.
,[8] G. Papaschinopoulos and /. Schinas: Multiplicative separation, diagonalizability and struc-
283
tural stability of linear difference equations, Differential Equations: Qualitative theory
(Szeged 1984), Colloq. Math. Soc. Janos Bolyai, 47, North-Holland, Amsterdam—New
York.
[9] G. Papaschinopoulos: Exponential separation, exponential dichotomy and almost periodicity
oflinear difference equations, J. Math. Anal. Appl. 120 (1986), 276—287.
[10] G. Papaschinopoulos and / . Schinas: Structural stability via the density of a class of linear
discrete systems, J. Math. Anal. Appl. (to appear).
[11] /. Schinas and G. Papaschinopoulos: Topological equivalence for linear discrete systems via
dichotomies and Lyapunov functions, Boll. Un. Math. Ital. 6, 4 (1985), 61 — 70.
Authors" addresses: J. K u r z w e i l , 115 67 Praha 1, Zitna25, Czechoslovakia (Matematický
ústav ČSAV), G. P a p a s c h i n o p o u l o s , Democritus University of Thrace, 671 00 Xanthi,
Greece.
284