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(
S
is an
initial
state
, conventionally associated to time 0. With a clock, it is intrinsically assumed
that the application of its next-state-function can be considered as a uniform pro-
cess, in such a way that the transition from a state to the next one happens according
to the same law, with no appreciable difference in the quantity of time (whatever
this could mean) necessary to perform that transition. In other words, clocks are
physical entities where we can observe
regular
events of transition from a state to
another state. If
k
is a bound such that no physical clock is available at a level inferior
to 10
−
k
seconds (the time measurement unit), what reason can be used to assume
the existence of time at smaller temporal scales?
One of the most important concepts in dynamical systems is that of
attractor
.
There are many ways to formalize this notion. Intuitively, an attractor is a quasi
state (a set of states) such that when a trajectory reaches it (a state in it), then this
trajectory remains inside it. Moreover, this quasi state is included in a bigger quasi
state, called its
basin
, such that any trajectory running through the basin, after a
while (or according to a limit process), tends to “fall” into the attractor. A basin
is a case of
dynamically invariant
set
B
, that is, in a dynamical system
S
,
δ
,
s
0
)
where
(
S
,
δ
)
is an autonomous dynamical system, and
s
0
∈
(
S
,
δ
)
,if
x
B
. An attractor of basin
B
is a sort of minimal or limit subset of
B
where dynamics could remains eternally confined. This means that attractors can be
viewed as “dynamical states”, or even, as
second level states
. For example, a living
organism in a stable situation, performing life functions, moves along a trajectory,
which macroscopically seems just a state, but corresponds essentially to an attractor.
In mammals there are around 200 cell types. The stable states of these cells can be
seen as different attractors where cell dynamics may fall, in order to satisfy some
specific conditions corresponding to their biological role.
∈
B
,then
δ
(
x
)
∈
Example 5.8.
The Collatz dynamical system [191] is the autonomous system on pos-
itive natural numbers where dynamics is given, for any
x
∈
N
,by3
x
+
1if
x
is odd,
and
x
/
2if
x
is even. . The Collatz conjecture claims that
{
1
,
2
,
4
}
is the attractor
of this system (starting from any number you fall into the 1
1 loop, and
then you remain there). It has been proved to be true for numbers until around 2
60
(a web computational project on this topic is actively testing the Collatz conjecture,
see
http://www.ericr.nl/wondrous/
).
−
4
−
2
−
A simple notion of attractor can be defined in the following way.
Definition 5.9.
Given a dynamical system of dynamics
δ
, an attractor
A
of basin
B
,
with
A
⊆
B
, is a set of states such that, for any
x
∈
A
,
δ
(
x
)
∈
A
,andforany
x
∈
B
n
there exist a natural number
n
such that
δ
(
x
)
∈
A
.
An attractor
A
of basin
B
is called
minimal
if
A
C
for any attractor
C
of basin
B
.
Usually, attractors are intended as minimal attractors. It can be easily proved that a
minimal attractor is consisting of only one state or of a cycle.
Time measurement is present as soon as the first human civilizations are devel-
oped. In any organized society, especially when it reaches a certain level of com-
plexity, humans need to dominate the flow of time, in order to plan, project and
organize events and activities. The common experience of the day/night alternation,
⊆
