Libro de Resúmenes / Book of Abstracts (Español/English)

Resumenes 146
• Si X es compacto entonces : f periódicamente
densa⇒
f periódicamente
densa
• f es mezclante ⇔ f es mezclante
• f tieneentropía
topológica positiva ⇔ f tieneentropía
topológica positiva
También, diversos aspectos teóricos y aplicados en abierto serán discutidos.
Individual and collective dynamics: chaotic relations 2
It is well known that a numerous class of real problems are modelled by a
discrete dynamical system
xn + 1 = f ( xn
), n = 0,
1,
2,...
(1)
where ( X , d)
is a metric space and f : X → X is a continuous function.
The basic goal of the theory of discrete dynamical systems (DDS) is to
2
n
understand the nature of all orbits x,
f ( x),
f ( x),
...., f ( x)
as n becomes large
and, generally, this is an impossible task. In certain sense, the study of the
orbits in a DDS says us as are moved the points in the base space X and, in
many cases, these orbits presents a chaotic structure.
On the other hand, sometimes is not sufficient to know as are moved the
points in the base space X but it is necessary to know as are moved the
subsets of X what carries us to the problem of analyzing the dynamics of
set-valued dynamical discrete systems (SDDS).
For example, it is usual today to put special transmitters on different types
of biological species to know their movements within their habitats. Thus, if
we denote by X the habitat of a given specie and we choose an individual
representative x ∈ X , then we can obtain information about the movement
of x in the ecosystem X and, of this way, we can construct an individual
displacement function f : X → X . Eventually, this function could be chaotic
or not, but the question now it is the following: What is the connection
between the dynamic of individual movement and the dynamic of collective
movement?
In this direction, given a dynamical discrete system (1) we consider the setvalued
dynamical discrete system (SDDS) associated to f :
An + 1 = f ( An
), n = 0,
1,
2,...
(2)
where f : K(
X ) → K(
X ) , f ( A)
= f ( A)
= { f ( a)
/ a ∈ A}
, is the natural extension
of f to K (X ) .
In this context, our main goal in this work is studying the following
fundamental question: individual chaos implies collective chaos? and
conversely?
As a partial response to this question above, if we consider the space
( K ( X ), H ) of all nonempty compacts subsets of X endowed with the
Hausdorff metric induced by d , then the following chaotic relationships
between f and f are showed:
• f transitive ⇒ f transitive
• f sensitively
dependent ⇒ f sensitively
dependent
• If X is compact then : f periodically
dense ⇒ f periodically
dense
2 Trabajo financiado por Conicyt-Chile vía Proyecto 1040303 y Dipog-UTA vía Proyecto 4731-04.