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5.8
Discrete Dynamical Systems
The mathematical formalizations of dynamical systems, in spite of their elementary
character, reveal many subtle points related to the puzzling nature of time. Here
we give a brief outline of some basic concepts (see [168, 181, 165, 182] for more
advanced concepts).
Given a function
δ
:
S
→
S
and
i
∈
N
,the
i
th-iterates of
δ
are inductively defined
by the following equations, for every
s
∈
S
:
0
δ
(
s
)=
s
,
i
+
1
i
δ
(
s
)=
δ
(
δ
(
s
))
.
Definition 5.4.
An
autonomous dynamical system
is a pair
(
S
,
δ
)
where the set of
states
S
is also called the phase space and
δ
:
S
→
S
.
In an autonomous dynamical system, the dynamics
is a
next-state-function
from
the phase space
S
to itself. According to the definition above, an autonomous dy-
namical system with an initial state
s
0
“represents”
δ
n
m
N
if
δ
(
s
0
)
=
δ
(
s
0
)
for every
n
m
. In this perspective, the infinite enumeration process is based
on the possibility of starting from
s
0
and generating, with the
next-state-function
,a
state that is new at every application of this function. If the set of states generated by
the
next-state-function
is finite, the counting process cannot be developed beyond a
certain bound, because surely the dynamics has a cycle (reaching an already gener-
ated state). In this case, it is possible to go beyond the bound by restarting the cycle
after its end, and by counting, in some way, the number of times the cycle is iterated
(this is the principle of calendars and chronologies, based on astronomical cycles).
In autonomous dynamical systems, time is a consequence of their internal dy-
namics. Instead, in the following definition of timed dynamical system, an external
notion of time is assumed.
,
m
∈
N
, with
n
=
Definition 5.5.
A
timed dynamical system
D
is a triple
D
=(
S
,
T
,
δ
)
givenbyaset
S
of states, called
phase space
,
T
S
is a
dynamics
, which assigns to any time instant the state which the system takes at that
instant.
⊆
R
is a set of
time instants
,and
δ
:
T
→
Any function
f
:
determines a dynamical system, in a trivial way. Motions,
developments, growth processes, and morphogenesis are representable as dynamical
systems. In planetary motions phase space is
R
→
R
6
because the position and velocity
of the planet with respect to three cartesian axes identify the state of the system, and
the dynamics is the motion law giving the planet positions and velocities at every
instant (a system of two planets needs a phase state included in
R
12
). A dynamical
R
system
D
=(
S
,
T
,
δ
)
is
discrete
if
T
is a subset of
Z
.
Proposition 5.6.
An
autonomous
dynamical system
(
S
,
δ
)
determines a family of
timed dynamical systems D
s
=
{
(
S
,
N
,
δ
s
)
|
s
∈
S
},
where dynamics
δ
s
is called the
i
timing
of
δ
based on the
initial state
s
∈
S, and it is defined by
δ
s
(
i
)=
δ
(
s
)
.
