# bounded growth

## bounded growth

[¦bau̇n·dəd ′grōth]
(mathematics)
The property of a function ƒ defined on the positive real numbers which requires that there exist numbers M and a such that the absolute value of ƒ(t) is less than Mat for all positive values of t.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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In [8], Palmer showed that a nonautonomous linear differential system with bounded growth and decay, is exponentially dichotomic if and only if it is topologically equivalent to a standard autonomous system with evolution operator U(t, s) = [e.sup.-(t-s)] P + [e.sup.t-s] Q.
Using the constructions in [18] or in the book [16], in [6] it is proved that all the exponentially dichotomic evolution families, even without bounded growth and decay, are structurally stable.
The reader will easily observe that if we add the conditions of bounded growth and decay along the trajectories through R(t)E, i.e.
We say that the evolution family U = [{U(t, s)}.sub.t,s[member of]R] has bounded growth and decay iff there are constants K, M > 0 such that:
Notice that if U has bounded growth and decay, then the above norms are equivalent to the initial one, on P(t)E and Q(t)E, respectively (see [5]).
Suppose that the (reversible) evolution family U = [{U(t, s)}.sub.t,s[member of]R] is trichotomic and has bounded growth and decay (1).
If the evolution family U = [{U(t, s)}.sub.t,s[member of]R] is Sacker-Sell trichotomic and has bounded growth and decay, then it is topologically equivalent (using (iii) in Definition 3) to the evolution family generated by the standard autonomous differential equation:
In [5] it is proved that any exponentially dichotomic equation with structural projections P Q and with bounded growth and decay is topologically equivalent to the standard equation with the evolution operator
In this section we restrict our study to evolutions families U = [{U(t, s)}.sub.t,s[member of]R] with bounded growth and decay, that satisfy [beta]exponential trichotomy.
(ii) comes from the bounded growth and decay hypothesis for the family U, and
Suppose that the evolution family U is exponentially trichotomic, has bounded growth and decay, and verify in addition relation (13).
Notice that the finiteness of the Bohl exponents is equivalent to bounded growth, respectively bounded decay.

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